2.1 Manifolds

• An n-manifold without boundary is a ‘nice’ topological space with the property that each point in it has a neighborhood topologically equivalent to the usual n-dimensional Euclidean space . In other words an n-manifold resembles locally .
• Similarly, an n-manifold with boundary is ‘nice’ topological space with the property that each point in it has a neighborhood topologically equivalent either to (if the point lies in the interior) or (if the point lies on the boundary).

2.2 Homeomorphisms

In Section 2.1 by ‘topologically equivalent’ we mean the following: two n-manifolds X and Y are homeomorphic or topologically equivalent if there exists a homeomorphism between them, namely a function f: X → Y with the properties that:

• f is continuous
• There exists the inverse function f⁻¹: Y → X (equivalently f is 1-1 and onto)
• f⁻¹ is also continuous

Intuitively the homeomorphism f is an elastic deformation of the space X to the space Y, not involving any self-intersections or any ‘cutting’ and ‘regluing’ (see also Appendix A).

2.3 Properties of manifolds

An n-manifold, M, is said to be:

• connected if it consists of only one piece,
• compact if it can be enclosed in some k-dimensional ball,
• orientable if any oriented frame that moves along any closed path in M returns to a position that can be transformed to the initial one by a rotation.

The above notions are more rigorously defined in Appendix A.

2.4 n-spheres and n-balls

In each dimension the basic connected, oriented, compact n-manifold without boundary is the n-sphere, Sⁿ. Also, the basic connected, oriented n-manifold with boundary is the n-ball, Dⁿ. The boundary of a n-dimensional ball is a n − 1−dimensional sphere, ∂Dⁿ = Sⁿ⁻¹, n >= 1. In Fig 1, this relation is shown for n = 1, 2 and 3. As shown in Fig 1(1), the space S⁰ is the disjoint union of two points. By convention, we consider these two one-point spaces to be {+1} and {−1}: S⁰ = {+1} ∐ {−1}.

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Fig 1. (1) A segment D¹ is bounded by two points S⁰ (2) A disc D² is bounded by a circle S¹ (3) A 3-ball D³ is bounded by a sphere S².

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Besides the relation of Sⁿ with Dⁿ⁺¹ described above, the n-sphere Sⁿ is also intrinsically related to the Euclidian space via the notion of compactification.

2.5 The compactification of

Compactification is the process of making a topological space into a compact space. For each dimension n, the space with all points at infinity compactified to one single point is homeomorphic to Sⁿ. So, Sⁿ is also called the one-point compactification of . Conversely, a sphere Sⁿ can be decompactified to the space by the so-called stereographic projection. For example, for n = 1 we have that the circle S¹ is the one-point compactification of the real line , see Fig 2(1), while for n = 2 the sphere S² is the one-point compactification of the plane , see Fig 2(2). The compactification of is discussed and illustrated in Section 7.1.1 (see Appendix A for details on the one-point compactification of ).

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Fig 2. (1) S¹ onto (2) S² onto .

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2.6 Product spaces

The product space of two manifolds X and Y is the manifold made from their Cartesian product X × Y (see also Appendix A). If X, Y are manifolds with boundary, the boundary of product space X × Y is ∂(X × Y) = (∂X × Y) ∪ (X × ∂Y).

For example the next common connected, oriented, compact 2-manifold without boundary after S² is the torus, which can be perceived as the boundary of a doughnut, and it is the product space S¹ × S¹. Analogously, a solid torus, which can be perceived as a whole doughnut, is the product space S¹ × D². A solid torus is a 3-manifold with boundary a torus:

Other product spaces that we will be using here are: the cylinder S¹ × D¹ or D¹ × S¹ (see Fig 3), the solid cylinder D² × D¹ which is homeomorphic to the 3-ball and the spaces of the type S⁰ × Dⁿ, which are the disjoint unions of two n-balls Dⁿ ∐ Dⁿ.

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Fig 3. Two ways of viewing a cylinder.

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All the above examples of product spaces that are of the form S^p × D^q can be viewed as q-thickenings of the p-sphere. For example the 2-thickening of S⁰ comprises two discs, while the 3-thickening of S⁰ comprises two 3-balls. It is also worth noting that the product spaces S^p × D^q and D^p+1 × S^q−1 have the same boundary: ∂(S^p × D^q) = ∂(D^p+1 × S^q−1) = S^p × S^q−1(⋆).

2.7 Embeddings

• An embedding of an n-manifold Nⁿ in an m-manifold M^m is a map f: N ↪ M such that its restriction on the image f(N) is a homeomorphism between N and f(N). The notion of embedding allows to view spaces inside specific manifolds instead of abstractly. Embeddings even of simple manifolds can be very complex. For example, the embeddings of the circle S¹ in the 3-space are the well-known knots whose topological classification is still an open problem of low-dimensional topology.
• An embedding of a submanifold Nⁿ ↪ M^m is framed if it extends to an embedding Nⁿ × D^m−n ↪ M.
• A framed n-embedding in M is an embedding of the (m − n)-thickening of the n-sphere, h: Sⁿ × D^m−n ↪ M, with core n-embedding e = h_|: Sⁿ = Sⁿ × {0} ↪ M. For example, the framed 1-embeddings in comprise embedded solid tori in the 3-space with core 1-embeddings being knots.
• Let X, Y be two n-manifolds with homeomorphic boundaries ∂X and ∂Y (which are (n − 1)−manifolds). Let also h denote a homeomorphism h: ∂X → ∂Y. Then, from X ∪ Y one can create a new n-manifold without boundary by ‘gluing’ X and Y along their boundaries. The gluing is realized by identifying each point x ∈ ∂X to the point h(x) ∈ ∂Y. The map h is called gluing homeomorphsim, see Appendix A. One important example is the gluing of two n-discs along their common boundary which gives rise to the n-sphere, see Fig 4 for n = 1, 2. For n = 3, the gluing of two 3-balls yielding the 3-sphere S³ is illustrated and discussed in Section 7.1.2. Another interesting example is the gluing of solid tori which also yield the 3-sphere. This is illustrated and discussed in Section 7.1.3.

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Fig 4. (1) D¹ ∪_h D¹ = S¹ (2) D² ∪_h D² = S².

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As we will see in next section, the notions of embedding and gluing homeomorphism together with property (⋆) described in 2.6 are the key ingredients needed to define topological surgery. It is roughly the procedure of removing an embedding of S^p × D^q and gluing back D^p+1 × S^q−1 along their common boundary.

3.1 1-dimensional 0-surgery

We only have one kind of surgery on a 1-manifold M, the 1-dimensional 0-surgery where m = 1 and n = 0: The above definition means that two segments S⁰ × D¹ are removed from M and they are replaced by two different segments D¹ × S⁰ by reconnecting the four boundary points S⁰ × S⁰ in a different way. In Figs 5(a) and 6(a), S⁰ × S⁰ = {1, 2, 3, 4}. As one possibility, if we start with M = S¹ and use as h the standard (identity) embedding denoted with h_s, we obtain two circles S¹ × S⁰. Namely, denoting by 1 the identity homeomorphism, we have , see Fig 5(a). However, we can also obtain one circle S¹ if h is an embedding h_t that reverses the orientation of one of the two arcs of S⁰ × D¹. Then in the substitution, joining endpoints 1 to 3 and 2 to 4, the two new arcs undergo a half-twist, see Fig 6(a). More specifically, if we take D¹ = [−1, +1] and define the homeomorphism ω: D¹ → D¹;t → −t, the embedding used in Fig 6(a) is which rotates one D¹ by 180°. The difference between the embeddings h_s and h_t of S⁰ × D¹ can be clearly seen by comparing the four boundary points 1, 2, 3 and 4 in Figs 5(a) and 6(a).

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Fig 5. Formal (a) 1-dimensional 0-surgery (b₁) 2-dimensional 0-surgery and (b₂) 2-dimensional 1-surgery using the standard embedding h_s.

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Fig 6. Formal (a) 1-dimensional 0-surgery (b₁) 2-dimensional 0-surgery and (b₂) 2-dimensional 1-surgery using a twisting embedding h_t.

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Note that in dimension one, the dual case is also an 1-dimensional 0-surgery. For example, looking at the reverse process of Fig 5(a), we start with two circles M′ = S¹ ∐ S¹ and, if each segment of D¹ × S⁰ is embedded in a different circle, the result of the (dual) 1-dimensional 0-surgery is one circle: χ⁻¹(M′) = M = S¹.

3.2 2-dimensional 0-surgery

Starting with a 2-manifold M, there are two types of surgery. One type is the 2-dimensional 0-surgery, whereby two discs S⁰ × D² are removed from M and are replaced in the closure of the remaining manifold by a cylinder D¹ × S¹, which gets attached via a homeomorphism along the common boundary S⁰ × S¹ comprising two copies of S¹. The gluing homeomorphism of the common boundary may twist one or both copies of S¹. For M = S² the above operation changes its homeomorphism type from the 2-sphere to that of the torus. View Fig 5(b₁) for the standard embedding h_s and Fig 6(b₁) for a twisting embedding h_t. For example, the homeomorphism μ: D² → D²; (t₁, t₂) → (−t₁, −t₂) induces the 2-dimensional analogue h_t of the embedding defined in the previous example, namely: which rotates one D² by 180°. When, now, the cylinder D¹ × S¹ is glued along the common boundary S⁰ × S¹, the twisting of this boundary induces the twisting of the cylinder, see Fig 6(b₁).

3.3 2-dimensional 1-surgery

The other possibility of 2-dimensional surgery on M is the 2-dimensional 1-surgery: here a cylinder (or annulus) S¹ × D¹ is removed from M and is replaced in the closure of the remaining manifold by two discs D² × S⁰ attached along the common boundary S¹ × S⁰. For M = S² the result is two copies of S², see Fig 5(b₂) for the standard embedding h_s. Fig 6(b₂) illustrates a twisting embedding h_t, where a twisted cylinder is being removed. In that case, taking D¹ = {h: h ∈ [−1, 1]} and homeomorphism ζ: the embedding h_t is defined as: . This operation corresponds to fixing the circle S¹ bounding the top of the cylinder S¹ × D¹, rotating the circle S¹ bounding the bottom of the cylinder by 180° and letting the rotation propagate from bottom to top. This twisting of the cylinder can be seen by comparing the second instance of Fig 5(b₂) with the second instance of Fig 6(b₂), but also by comparing the third instance of Fig 5(b₁) with the third instance of Fig 6(b₁).

It follows from Definition 1 that a dual 2-dimensional 0-surgery is a 2-dimensional 1-surgery and vice versa. Hence, Fig 5(b₁) shows that a 2-dimensional 0-surgery on a sphere is the reverse process of a 2-dimensional 1-surgery on a torus. Similarly, as illustrated in Fig 5(b₂), a 2-dimensional 1-surgery on a sphere is the reverse process of a 2-dimensional 0-surgery on two spheres. In the figure the symbol ↔ depicts surgeries from left to right and their corresponding dual surgeries from right to left.

4.1 Introducing dynamics

The formal definition of 1-dimensional 0-surgery gives a static description of the initial and the final stage whereas natural phenomena exhibiting 1-dimensional 0-surgery follow a continuous process. In order to address such phenomena or to understand how 1-dimensional 0-surgery happens, we need a non-static description.

Furthermore, in nature, 1-dimensional 0-surgery often happens locally, on arcs or segments. That is, the initial manifold is often bigger and we remove from its interior two segments S⁰ × D¹. Therefore, we also need dynamics that act locally.

In Fig 10, we introduce dynamics which explain the intermediate steps of the formal definition and extend surgery to a continuous process caused by local forces. The process starts with the two points specified on the manifold (in red), on which attracting forces are applied (in blue). We assume that these forces are created by an attracting center (also in blue). Then, the two segments S⁰ × D¹, which are neighborhoods of the two points, get close to one another. When the specified points (or centers) of two segments reach the attracting center, they touch and recoupling takes place giving rise to the two final segments D¹ × S⁰, which split apart. As mentioned in previous section, we have two cases (a) and (b), depending on the homemorphism h.

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Fig 10. Introducing dynamics to 1-dimensional surgery.

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As mentioned in Section 3.1, the dual case is also a 1-dimensional 0-surgery as it removes segments D¹ × S⁰ and replace them by segments S⁰ × D¹. This is the reverse process which starts from the end and is illustrated in Fig 10 as a result of the orange forces and attracting center which are applied on the ‘complementary’ points.

Remark 1 It is worth mentioning that the intermediate steps of surgery presented in Fig 10 can also be viewed in the context of Morse theory [11]. By using the local form of a Morse function, we can visualize the process of surgery by varying parameter t of equation x² − y² = t. For t = −1 it is the hyperbola shown in the second instance of Fig 10 where the two segments get close to one another. For t = 0 it is the two straight lines where the reconnection takes place as shown in the third instance of Fig 10 while for t = 1 it represents the hyperbola of the two final segments shown in case (a) of the fourth instance of Fig 10. This sequence can be generalized for higher dimensional surgeries as well, however, in this paper we will not use this approach as we are focusing on the introduction of forces and of the attracting center.

These local dynamics produce different manifolds depending on where the initial neighborhoods are embedded. Taking the known case of the standard embedding h_s and M = S¹, we obtain S¹ × S⁰ (for both regular and dual surgery), see Fig 11(a). Furthermore, as shown in Fig 11(b), we also obtain S¹ × S⁰ even if the attracting center is outside S¹. Note that these outcomes are not different than the ones shown in formal surgery (recall Fig 5(a)) but we can now see the intermediate instances.

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Fig 11. 1-dimensional surgery on one and two circles.

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4.2 Explaining 1-dimensional phenomena via dynamics

Looking closer at the aforementioned phenomena, the described dynamics and attracting forces are present in all cases. Namely, magnetic reconnection (Fig 9) corresponds to a dual 1-dimensional 0-surgery (see Fig 10(b)) where g: D¹ × S⁰ ↪ M′ is a dual embedding of the twisting homeomorphism h_t defined in Section 3.1 of Section 3. The tubes are viewed as segments and correspond to an initial manifold M = S⁰ × D¹ (or M = S¹ if they are connected) on which the local dynamics act on two smaller segments S⁰ × D¹. Namely, the two magnetic flux tubes have a nonzero parallel net current through them, which leads to attraction of the tubes (cf. [12]). Between them, a localized diffusion region develops where magnetic field lines may decouple. Reconnection is accompanied with a sudden release of energy and the magnetic field lines break and rejoin in a lower energy state.

In the case of chromosomal crossover (Fig 7), we have the same dual 1-dimensional 0-surgery as magnetic reconnection (see Fig 10(b)). During this process, the homologous (maternal and paternal) chromosomes come together and pair, or synapse, during prophase. The pairing is remarkably precise and is caused by mutual attraction of the parts of the chromosomes that are similar or homologous. Further, each paired chromosomes divide into two chromatids. The point where two homologous non-sister chromatids touch and exchange genetic material is called chiasma. At each chiasma, two of the chromatids have become broken and then rejoined (cf. [13]). In this process, we consider the initial manifold to be one chromatid from each chromosome, hence the initial manifold is M = S⁰ × D¹ on which the local dynamics act on two smaller segments S⁰ × D¹.

For site-specific DNA recombination (see Fig 8), we have a 1-dimensional 0-surgery (see Fig 10(b)) with a twisted homeomorphism h_t as defined in Section 3.1 of Section 3. Here the initial manifold is a knot which is an embedding of M = S¹ in 3-space but this will be detailed in Section 8. As mentioned in [14], enzymes break and rejoin the DNA strands, hence in this case the seeming attraction of the two specified points is realized by the enzyme. Note that, while both are genetic recombinations, there is a difference between chromosomal crossover and site-specific DNA recombination. Namely, chromosomal crossover involves the homologous recombination between two similar or identical molecules of DNA and we view the process at the chromosome level regardless of the knotting of DNA molecules.

Finally, vortices reconnect following the steps of 1-dimensional 0-surgery with a standard embedding shown in Fig 10(a). The initial manifold is again M = S⁰ × D¹. As mentioned in [15], the interaction of the anti-parallel vortices goes from attraction before reconnection, to repulsion after reconnection.

4.3 Defining solid 1-dimensional surgery

There are phenomena which undergo the process of 1-dimensional 0-surgery but happen on surfaces, such as tension on membranes or soap films and the merging of oil slicks. In order to model topologically such phenomena we introduce the notion of solid 1-dimensional 0-surgery. Solid 1-dimensional 0-surgery on the 2-disc D² is the topological procedure whereby a ribbon D¹ × D¹ is being removed, such that the closure of the remaining manifold comprises two discs D² × S⁰. The reader is referred to Fig 5(a) where the interior is now supposed to be filled in. This process is equivalent to performing 1-dimensional 0-surgeries on the whole continuum of concentric circles included in D². More precisely, and introducing at the same time dynamics, we define:

Definition 2 We start with the 2-disc of radius 1 with polar layering: where r the radius of a circle and P the limit point of the circles, that is, the center of the disc. We specify colinear pairs of antipodal points, all on the same diameter, with neighborhoods of analogous lengths, on which the same colinear attracting forces act, see Fig 12(1) where these forces and the corresponding attracting center are shown in blue. Then, in (2), antipodal segments get closer to one another or, equivalently, closer to the attracting center. Note that here, the attracting center coincides with the limit point of all concentric circles, which is shown in green from instance (2) and on. Then, as shown from (3) to (9), we perform 1-dimensional 0-surgery on the whole continuum of concentric circles. The natural order of surgeries is as follows: first, the center of the segments that are closer to the center of attraction touch, see (4). After all other points have also reached the center, see (5), decoupling starts from the central or limit point. We define 1-dimensional 0-surgery on the limit point P to be the two limit points of the resulting surgeries. That is, the effect of solid 1-dimensional 0-surgery on a point is the creation of two new points, see (6). Next, the other segments reconnect, from the inner, see (7), to the outer ones, see (8), until we have two copies of D², see (9) and (10). Note that the proposed order of reconnection, from inner to outer, is the same as the one followed by skin healing, namely, the regeneration of the epidermis starts with the deepest part and then migrates upwards.

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Fig 12. Solid 1-dimensional surgery.

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The above process is the same as first removing the center P from D², doing the 1-dimensional 0-surgeries and then taking the closure of the resulting space. The resulting manifold is which comprises two copies of D².

We also have the reverse process of the above, namely, Solid 1-dimensional 0-surgery on two discs D² × S⁰ is the topological procedure whereby a ribbon D¹ × D¹ joining the discs is added, such that the closure of the remaining manifold comprise one disc D², as illustrated in Fig 12. This process is the result of the orange forces and attracting center which are applied on the ‘complementary’ points. This operation is equivalent to performing 1-dimensional 0-surgery on the whole continuum of concentric circles in D² ∐ D². We only need to define solid 1-dimensional 0-surgery on two limit points to be the limit point P of the resulting surgeries. That is, the effect of solid 1-dimensional 0-surgery on two points is their merging into one point. The above process is the same as first removing the centers from the D² × S⁰, doing the 1-dimensional 0-surgeries and then taking the closure of the resulting space. The resulting manifold is which comprises one copy of D².

5.1 Introducing dynamics

In order to model topologically phenomena exhibiting 2-dimensional surgery or to understand 2-dimensional surgery through continuity we need, also here, to introduce dynamics. In Fig 14(a), the 2-dimensional 0-surgery starts with two points, or poles, specified on the manifold (in red) on which attracting forces created by an attracting center are applied (in blue). Then, the two discs S⁰ × D², neighborhoods of the two poles, approach each other. When the centers of the two discs touch, recoupling takes place and the discs get transformed into the final cylinder D¹ × S¹.

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Fig 14. Introducing dynamics to 2-dimensional surgery.

(a) 2-dimensional surgeries with standard embeddings (b) 2-dimensional surgeries with twisted embeddings.

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As mentioned in Example 3.3, the dual case of 2-dimensional 0-surgery is the 2-dimensional 1-surgery and vice versa. This is also shown in Fig 14(a) where the reverse process is the 2-dimensional 1-surgery which starts with the cylinder and a specified cyclical region (in red) on which attracting forces created by an attracting center are applied (in orange). A ‘necking’ occurs in the middle which degenerates into a point and finally tears apart creating two discs S⁰ × D². As also seen in Fig 14(a), in the case of 2-dimensional 0-surgery, forces (in blue) are applied on two points, or S⁰, while in the case of the 2-dimensional 1-surgery, forces (in orange) are applied on a circle S¹.

In Fig 14(b), we have an example of twisted 2-dimensional 0-surgery where the two discs S⁰ × D² are embedded via a twisted homemorphism h_t while, in the dual case, the cylinder D¹ × S¹ is embedded via a twisted homemorphism g_t. Here h_t rotates the two discs while g_t rotates the top and bottom of the cylinder by 3π/4 and −3π/4 respectively. More specifically, if we define the homeomorphism ω₁, ω₂: D² → D² to be rotations by 3π/4 and −3π/4 respectively, then h_t is defined as the composition . The homeomorphism g_t: D¹ × S¹ → M is defined analogously.

These local dynamics produce different manifolds depending on the initial manifold where the neighborhoods are embedded. Taking M = S², the local dynamics of Fig 14(a) are shown in Fig 15(a) and 15(b) producing the same manifolds seen in formal 2-dimensional surgery (recall Fig 5(b₁) and 5(b₂)). Note that, as also seen in 1-dimensional surgery (Fig 11(b)), if the blue attracting center in Fig 15(a) was outside the sphere and the cylinder was attached on S² externally, the result would still be a torus.

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Fig 15. (a) 2-dimensional 0-surgery on M = S² and 2-dimensional 1-surgery on M′ = S⁰ × S² (b) 2-dimensional 1-surgery on M = S² and 2-dimensional 0-surgery on M′ = S⁰ × S².

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Looking back at the natural phenomema happening on surfaces, an example is soap bubble splitting during which a soap bubble splits into two smaller bubbles. This process is the 2-dimensional 1-surgery on M = S² shown in Fig 15(b). The orange attracting force in this case is the surface tension of each bubble that pulls molecules into the tightest possible groupings.

5.2 Defining solid 2-dimensional surgery

Most natural phenomena undergoing 2-dimensional surgery do not happen on surfaces but are three-dimensional. Therefore we introduce, also here, the notion of solid 2-dimensional surgery. There are two types of solid 2-dimensional surgery on the 3-ball, D³, analogous to the two types of 2-dimensional surgery.

The first one is the solid 2-dimensional 0-surgery which is the topological procedure of removing a solid cylinder homeomorphic to the product set D¹ × D², h(D¹ × D²) (such that the part S⁰ × D² of its boundary lies in the boundary of D³) and taking the closure of the remaining manifold D³\h(D¹ × D²), which is a regular (or twisted) solid torus. See Fig 5(b₁) where the interior is supposed to be filled in. The second type is the solid 2-dimensional 1-surgery which is the topological procedure of removing a solid cylinder homeomorphic to the product set D² × D¹, h(D² × D¹), (such that the part S¹ × D¹ of its boundary lies in the boundary of D³) and taking the closure of the remaining manifold D³\h(D² × D¹), which is two copies of D³. See Fig 5(b₂) where the interior is supposed to be filled in. Those processes are equivalent to performing 2-dimensional surgeries on the whole continuum of concentric spheres included in D³. More precisely, and introducing at the same time dynamics, we define:

Definition 3 Start with the 3-ball of radius 1 with polar layering: where r the radius of a 2-sphere and P the limit point of the spheres, that is, the center of the ball. Solid 2-dimensional 0-surgery on D³ is the topological procedure shown in Fig 16(a): on all spheres colinear pairs of antipodal points are specified, all on the same diameter, on which the same colinear attracting forces act. The poles have disc neighborhoods of analogous areas. Then, 2-dimensional 0-surgeries are performed on the whole continuum of the concentric spheres using the same homeomorphism h. Moreover, 2-dimensional 0-surgery on the limit point P is defined to be the limit circle of the nested tori resulting from the continuum of 2-dimensional surgeries. That is, the effect of 2-dimensional 0-surgery on a point is the creation of a circle. The process is characterized on one hand by the 1-dimensional core L of the removed solid cylinder joining the antipodal points on the outer shell and intersecting each spherical layer in the two antipodal points and, on the other hand, by the homeomorphism h, resulting in the whole continuum of layered tori. The process can be viewed as drilling out a tunnel along L according to h. For a twisted embedding h, this agrees with our intuition that, for opening a hole, drilling with twisting seems to be the easiest way.

On the other hand, solid 2-dimensional 1-surgery on D³ is the topological procedure where: on all spheres nested annular peels of the solid annulus of analogous areas are specified and the same coplanar attracting forces act on all spheres, see Fig 16(b). Then, 2-dimensional 1-surgeries are performed on the whole continuum of the concentric spheres using the same homeomorphism h. Moreover, 2-dimensional 1-surgery on the limit point P is defined to be the two limit points of the nested pairs of 2-spheres resulting from the continuum of 2-dimensional surgeries. That is, the effect of 2-dimensional 1-surgery on a point is the creation of two new points. The process is characterized by the 2-dimensional central disc of the solid annulus and the homeomorphism h, and it can be viewed as squeezing the central disc D or, equivalently, as pulling apart the upper and lower hemispheres with possible twists if h is a twisted embedding. This agrees with our intuition that for cutting a solid object apart, pulling with twisting seems to be the easiest way.

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Fig 16. Solid 2-dimensional surgery on the 3-ball.

(a) 2-dimensional 0-surgery with the standard embedding (b) 2-dimensional 1-surgery with the standard embedding.

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For both types, the above process is the same as: first removing the center P from D³, performing the 2-dimensional surgeries and then taking the closure of the resulting space. Namely we obtain: which is a solid torus in the case of solid 2-dimensional 0-surgery and two copies of D³ in the case of solid 2-dimensional 1-surgery.

As seen in Fig 16, we also have the two dual solid 2-dimensional surgeries, which represent the reverse processes. As already mentioned in in Section 3.3 the dual case of 2-dimensional 0-surgery is the 2-dimensional 1-surgery and vice versa. More precisely:

Definition 4 The dual case of solid 2-dimensional 0-surgery on D³ is the solid 2-dimensional 1-surgery on a solid torus D² × S¹ whereby a solid cylinder D¹ × D² filling the hole is added, such that the closure of the resulting manifold comprises one 3-ball D³. This is the reverse process shown in Fig 16(a) which results from the orange forces and attracting center. It only remain to define the solid 2-dimensional 1-surgery on the limit circle to be the limit point P of the resulting surgeries. That is, the effect of solid 2-dimensional 1-surgery on the core circle is that it collapses into one point. The above process is the same as first removing the core circle from D² × S¹, doing the 2-dimensional 1-surgeries on the nested tori, with the same coplanar acting forces, and then taking the closure of the resulting space. Given that the solid torus can be written as a union of nested tori together with the core circle: , the resulting manifold is which comprises one copy of D³.

Further, the dual case of solid 2-dimensional 1-surgery on D³ is the solid 2-dimensional 0-surgery on two 3-balls D³ whereby a solid cylinder D² × D¹ joining the balls is added, such that the closure of the resulting manifold comprise of one 3-ball D³. This is the reverse process shown in Fig 16(b) which results from the blue forces and attracting center. We only need to define the solid 2-dimensional 0-surgery on two limit points to be the limit point P of the resulting surgeries. That is, as in solid 1-dimensional surgery, the effect of solid 2-dimensional 0-surgery on two points is their merging into one point. The above process is the same as first removing the centers from the D³ × S⁰, doing the 2-dimensional 0-surgeries on the nested spheres, with the same colinear forces, and then taking the closure of the resulting space. The resulting manifold is which comprises one copy of D³.

Remark 2 The notions of 2-dimensional (resp. solid 2-dimensional) surgery, can be generalized from S² (resp. D³) to a surface (resp. a handlebody) of genus g creating a surface (resp. a handlebody) of genus g + 1.

5.3 Natural phenomena exhibiting solid 2-dimensional 0-surgery

Solid 2-dimensional 0-surgery is often present in natural phenomena where attracting forces between two poles are present, such as the formation of tornadoes, the formation of Falaco solitons, the formation of black holes, gene transfer in bacteria and drop coalescence. We shall discuss these phenomena in some detail pinning down their exhibiting of topological surgery.

Regarding tornadoes: except from their shape (see Fig 17) which fits the cylinder D¹ × S¹ that gets attached in the definition of 2-dimensional 0-surgery, the process by which they are formed also follows the dynamics introduced in Section 5.2. Namely, if certain meteorological conditions are met, an attracting force between the cloud and the earth beneath is created and funnel-shaped clouds start descending toward the ground. Once they reach it, they become tornadoes. In analogy to solid 2-dimensional 0-surgery, first the poles are chosen, one on the tip of the cloud and the other on the ground, and they seem to be joined through an invisible line. Then, starting from the first point, the wind revolves in a helicoidal motion toward the second point, resembling ‘hole drilling’ along the line until the hole is drilled. Therefore, tornado formation undergoes the process of solid 2-dimensional 0-surgery with a twisted embedding, as in Fig 14(b). The initial manifold can be considered as M = D³ × S⁰, that is, one 3-ball on the cloud and one on the ground. Note that in this realization of solid 2-dimensional 0-surgery, the attracting center coincides with the ground and we only see helicoidal motion in one direction.

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Fig 17. (a) Funnel clouds drilling and tornado formation (b) waterspout.

An example of solid 2-dimensional 0-surgery.

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Another natural phenomenon exhibiting solid 2-dimensional 0-surgery is the formation of Falaco solitons, see Fig 18 (for photos of pairs of Falaco solitons in a swimming pool, see [16]). Note that the term ‘Falaco Soliton’ appears in 2001 in [17]. Each Falaco Soliton consists of a pair of locally unstable but globally stabilized contra-rotating identations in the water-air discontinuity surface of a swimming pool. These pairs of singular surfaces (poles) are connected by means of a stabilizing thread. This thread corresponds to the ‘invisible line’ mentioned in the process of tornado formation which is visible in this case. The two poles get connected and their rotation propagates below the water surface along the joining thread and the tubular neighborhood around it. This process is a solid 2-dimensional 0-surgery with a twisted embedding (see Fig 14(b)) where the initial manifold is the water contained in the volume of the pool where the process happens, which is homeomorphic to a 3-ball, that is M = D³. Two differences compared to tornadoes are: here the helicoidal motion is present in both poles and the attracting center is not located on the ground but between the poles, on the topological thread joining them.

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Fig 18. Pairs of Falaco solitons.

An example of solid 2-dimensional 0-surgery.

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It is also worth mentioning that the creation of Falaco solitons is immediate and does not allow us to see whether the transitions of the 2-dimensional 0-surgery shown in Fig 14(b) are followed or not. However, these dynamics are certainly visible during the annihilation of Falaco solitons. Namely, when the topological thread joining the poles is cut, the tube tears apart and slowly degenerates to the poles until they both stops spinning and vanish. Therefore, the continuity of our dynamic model is clearly present during the reverse process which corresponds to a solid 2-dimensional 1-surgery on a pair of Falaco solitons, that is, a solid torus D² × S¹ degenerating into a still swimming pool D³.

Note that it is conjectured in [16] that the coherent topological features of the Falaco solitons and, by extension, the process of solid 2-dimensional 0-surgery appear in both macroscopic level (for example in the Wheeler’s wormholes) and microscopic level (for example in the spin pairing mechanism in the microscopic Fermi surface). For more details see [16].

Another phenomenon undergoing solid 2-dimensional 0-surgery is the formation of a black hole. Most black holes form from the remnants of a large star that dies in a supernova explosion and have a gravitational field so strong that not even light can escape. In the simulation of a black hole formation (see [18]), the density distribution at the core of a collapsing massive star is shown. In Fig 19 matter performs solid 2-dimensional 0-surgery as it collapses into a black hole. Matter collapses at the center of attraction of the initial manifold M = D³ creating the singularity, that is, the center of the black hole, which is surrounded by the toroidal accretion disc (shown in white in Fig 19(c)).

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Fig 19. The formation of a black hole.

An example of solid 2-dimensional 0-surgery.

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Solid 2-dimensional 0-surgery is also found in the mechanism of gene transfer in bacteria. See Fig 20 (also, for description and instructive illustrations see [19]). The donor cell produces a connecting tube called a ‘pilus’ which attaches to the recipient cell, brings the two cells together and transfers the donor’s DNA. This process is similar to the one shown earlier in Fig 16(b) as two copies of D³ merge into one, but here the attracting center is located on the recipient cell. This process is a solid 2-dimensional 0-surgery on two 3-balls M = D³ × S⁰.

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Fig 20. Gene transfer in bacteria.

An example of solid 2-dimensional 0-surgery.

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Finally, drop coalescence is the merging of two dispersed drops into one. As gene transfer in bacteria, this process is also a solid 2-dimensional 0-surgery on two 3-balls M = D³ × S⁰, see Fig 16(b). The process of drop coalescence also exhibits the forces of our model. Namely, the surfaces of two drops must be in contact for coalescence to occur. This surface contact is dependent on both the van der Waals attraction and the surface repulsion forces between two drops. When the van der Waals forces cause rupture of the film, the two surface films are able to fuse together, an event more likely to occur in areas where the surface film is weak. The liquid inside each drop is now in direct contact, and the two drops are able to merge into one.

Remark 3 Although in this Section some natural processes were viewed as a solid 2-dimensional topological surgery on M = D³ × S⁰, we could also consider the initial manifold as being a 3-ball surrounding the phenomena and view it as a surgery on M = D³. Concerning the process of tornado formation, this approach also has a physical meaning. Namely, as the process is triggered by the difference in the conditions of the lower and upper atmosphere, the initial manifold can be considered as the 3-ball containing this air cycle.

5.4 Natural phenomena exhibiting solid 2-dimensional 1-surgery

As already mentioned, the collapsing of the central disc of the sphere caused by the orange attracting forces in Fig 16(b) can also be caused by pulling apart the upper and lower hemispheres of the 3-ball D³, that is, the causal forces can also be repelling. For example, during fracture of metal specimens under tensile forces, solid 2-dimensional 1-surgery is caused by forces that pull apart each end of the specimen. On the other hand, in the biological process of mitosis, both attracting and repelling forces forces are present.

When the tension applied on metal specimens by tensile forces results in necking and then fracture, the process exhibits solid 2-dimensional 1-surgery. More precisely, in experiments in mechanics, tensile forces (or loading) are applied on a cylindrical specimen made of dactyle material (steel, aluminium, etc.). Up to some critical value of the force the deformation is homogeneous (the cross-sections have the same area). At the critical value the deformation is localized within a very small area where the cross-section is reduced drastically, while the sections of the remaining portions increase slightly. This is the ‘necking phenomenon’. Shortly after, the specimen is fractured (view [20] for details). In Fig 21 are the the basic steps of the process: void formation, void coalescence (also known as crack formation), crack propagation, and failure. Here, the process is not as smooth as our theoretical model and the tensile forces applied on the specimen are equivalent to repelling forces. The specimen is homeomorphic to the sphere shown in Fig 16(b) hence the initial manifold is M = D³.

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Fig 21. Tension and the necking phenomenon.

An example of solid 2-dimensional 1-surgery.

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Solid 2-dimensional 1-surgery on M = D³ also happens in the biological process of mitosis, where a cell splits into two new cells. See Fig 22 (for description and instructive illustrations see for example [21]). We will see that both aforementioned forces are present here. During mitosis, the chromosomes, which have already duplicated, condense and attach to fibers that pull one copy of each chromosome to opposite sides of the cell (this pulling is equivalent to repelling forces). The cell pinches in the middle and then divides by cytokinesis. The structure that accomplishes cytokinesis is the contractile ring, a dynamic assembly of filaments and proteins which assembles just beneath the plasma membrane and contracts to constrict the cell into two (this contraction is equivalent to attracting forces). In the end, two genetically-identical daughter cells are produced.

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Fig 22. The process of mitosis.

An example of solid 2-dimensional 1-surgery.

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Remark 4 It is worth noting that the splitting of the cell into two coincide with the fact that 2-dimensional 1-surgery on a point is the creation of two new points (see Definition 3).

7.1 Descriptions of S³

In dimension 3, the simplest c.c.o. 3-manifolds are: the 3-sphere S³ and the lens spaces L(p, q). In this paper however, we will focus on S³. We start by recalling its three most common descriptions:

7.1.1 Via .

S³ can be viewed as with all points at infinity compactified to one single point: . See Fig 25(b). can be viewed as an unbounded continuum of nested 2-spheres centered at the origin, together with the point at the origin, see Fig 25(a), and also as the de-compactification of S³. So, S³ minus the point at the origin and the point at infinity can be viewed as a continuous nesting of 2-spheres.

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Fig 25. S³ is the compactification of .

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7.1.2 Via two 3-balls.

S³ can be viewed as the union of two 3-balls: S³ = B³ ∪ D³, see Fig 26(a). This second description of S³ is clearly related to the first one, since a (closed) neighborhood of the point at infinity can stand for one of the two 3-balls. Note that, when removing the point at infinity in Fig 26(a) we can see the concentric spheres of the 3-ball B³ (in red) wrapping around the concentric spheres of the 3-ball D³, see Fig 26(b). This is another way of viewing as the de-compactification of S³. This picture is the analogue of the stereographic projection of S² on the plane , whereby the projections of the concentric circles of the south hemisphere together with the projections of the concentric circles of the north hemisphere form the well-known polar description of with the unbounded continuum of concentric circles.

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Fig 26. S³ is the result of gluing two 3-balls.

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7.1.3 Via two solid tori.

The third well-known representation of S³ is as the union of two solid tori, S³ = V₁ ∪_ϑ V₂, via the torus homeomorphism ϑ along the common boundary. ϑ maps a meridian of V₂ to a longitude of V₁ which has linking number zero with the core curve c of V₁. The illustration in Fig 27(a) gives an idea of this splitting of S³. In the figure, the core curve of V₁ is in dashed green. So, the complement of a solid torus V₁ in S³ is another solid torus V₂ whose core curve l (in dashed red) may be assumed to pass by the point at infinity. Note that, S³ minus the core curves c and l of V₁ and V₂ (the green and red curves in Fig 27) can be viewed as a continuum of nested tori. When removing the point at infinity in the representation of S³ as a union of two solid tori, the core of the solid torus V₂ becomes an infinite line l and the nested tori of V₂ can now be seen wrapping around the nested tori of V₁. See Fig 27(b). Therefore, can be viewed as an unbounded continuum of nested tori, together with the core curve c of V₁ and the infinite line l. This line l joins pairs of antipodal points of all concentric spheres of the first description. Note that in the nested spheres description (Fig 25) the line l pierces all spheres while in the nested tori description the line l is the ‘untouched’ limit circle of all tori.

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Fig 27. (a) S³ as a union of two solid tori (b) De-compactificated view.

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Remark 5 It is worth observing the resemblance of Fig 27(b) with the well-known representation of the Earth magnetic field. A numerical simulation of the Earth magnetic field via the Glatzmaier-Roberts geodynamo model was made in [23], see Fig 28. The magnetic field lines are lying on nested tori and comprise a visualization of the decompactified view of S³ as two solid tori.

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Fig 28. The Earth magnetic field as a decompactified view of S³ as two solid tori.

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Remark 6 It is also worth mentioning that another way to visualize S³ as two solid tori is the Hopf fibration, which is a map of S³ into S². The parallels of S² correspond to the nested tori of S³, the north pole of S² correspond to the core curve l of V₂ while the south pole of S² corresponds to the core curve c of V₁. An insightful animation of the Hopf fibration can be found in [24].

7.2 Connecting the descriptions of S³

7.2.1 Via corking.

The connection between the first two descriptions of S³ was already discussed in previous Section. The third description is a bit harder to connect with the first two. We shall do this here. A way to see this connection is the following. Consider the description of S³ as the union of two 3-balls, B³ and D³ (Fig 26). Combining with the third description of S³ (Fig 27) we notice that both 3-balls are pierced by the core curve l of the solid torus V₂. Therefore, D³ can be viewed as the solid torus V₁ to which a solid cylinder D¹ × D² is attached via the homeomorphism ϑ: This solid cylinder is part of the solid torus V₂, a ‘cork’ filling the hole of V₁. Its core curve is an arc L, part of the core curve l of V₂. View Fig 29. The second ball B³ (Fig 26) can be viewed as the remaining of V₂ after removing the ‘cork’ D¹ × D²: In other words the solid torus V₂ is cut into two solid cylinders, one comprising the ‘cork’ of V₁ and the other comprising the 3-ball B³.

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Fig 29. Passing from (a) S³ as two solid tori to (b) S³ as two balls.

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Remark 7 If we remove a whole neighborhood B³ of the point at infinity and focus on the remaining 3-ball D³, the line l of the previous picture is truncated to the arc L and the solid cylinder V₂ is truncated to the ‘cork’ of D³.

7.2.2 Via surgery.

We will now examine how we can pass from the two-ball description to the two-tori description of S³ via solid 2-dimensional 0-surgery. We start with two points that have a distance L between them. Let M = D³ be the solid ball having arc L as a diameter. We define this 3-ball as the ‘truncated’ space on which we will focus. When the center of D³ becomes attracting, forces are induced on the two points of D³ and solid 2-dimensional 0-surgery is initiated. The complement space is the other solid ball B³ containing the point at infinity, recall Fig 26. This joining arc L is seen as part of a simple closed curve l passing by the point at infinity. In Fig 30(1) this is shown in S³ while Fig 30 (1′) shows the corresponding decompactified view in .

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Fig 30. Passing from the two balls description to the two solid tori description of S³ via solid 2-dimensional 0-surgery.

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In Fig 30(2), we see the ‘drilling’ along L as a result of the attracting forces. This is exactly the same process as in Fig 16 if we restrict it to D³. But since we are in S³, the complement space B³ participates in the process and, in fact, it is also undergoing solid 2-dimensional 0-surgery. In Fig 30(3), we can see that, as surgery transforms the solid ball D³ into the solid torus V₁, B³ is transformed into V₂. That is, the nesting of concentric spheres of D³ (respectively B³) is transformed into the nesting of concentric tori in the interior of V₁ (respectively V₂). This is a double surgery with one attracting center which is inside the first 3-ball D³ (in grey) and outside the second 3-ball B³ (in red). By Definition 3, the point at the origin (in green) turns into the core curve c of V₁ (in green). Fig 30(3) is exactly the decompactified view of S³ as two solid tori as shown in Fig 27(b) while Fig 30 (3′) is the corresponding view in S³ as shown in Fig 27(1).

Fig 30 shows that one can pass from the second description of S³ to the third by performing solid 2-dimensional 0-surgery (with the standard embedding homeomorphism) along the arc L of D³. It is worth mentioning that this connection between the descriptions of S³ and solid 2-dimensional 0-surgery is a dynamic way to visualize the connection established in Section 7.2.1.

8.1 Defining embedded m-dimensional n-surgery

We will now concretely define the notion of embedded m-dimensional n-surgery in some sphere S^d and we will then focus on the case d = 3.

Definition 5 An embedded m-dimensional n-surgery is a m-dimensional n-surgery where the initial manifold is an m-embedding e: M ↪ S^d, d ≥ m of some m-manifold M. Namely, according to Definition 1:

From now on we fix d = 3. Embedding surgery allows to view it as a process happening in 3-space instead of abstractly. In the case of embedded 1-dimensional 0-surgery on a circle M = S¹, the ambient space gives enough ‘room’ for the initial 1-manifold to become any type of knot. Hence, embedding allows the initial manifold to assume a more complicated homeomorphic configuration. This will be analyzed further in Section 8.2.

Passing now to 2-dimensional surgeries, let us first note that embedded 2-dimensional surgery is often used a theoretical tool in various proofs in low dimensional topology. Further, an embedding of a sphere M = S² in S³ presents no knotting because knots require embeddings of codimension 2. However, in this case the ambient space plays a different role. Namely, embedding 2-dimension surgeries allows the complementary space of the initial manifold to participate actively in the process. Indeed, while some natural phenomena undergoing surgery can be viewed as ‘local’, in the sense that they can be considered independently from the surrounding space, some others are intrinsically related to the surrounding space. This relation can be both causal, in the sense that the ambient space is involved in the triggering of the forces causing surgery, and consequential, in the sense that the forces causing surgery, can have an impact on the ambient space in which they take place. This will be analyzed in Sections 8.3 and 8.4.

8.2 Embedded 1-dimensional 0-surgery and related phenomena

We will now get back to site-specific DNA recombination (see Section 4.2), in order to better define this type of surgery. As seen in this process (recall Fig 8) the initial manifold of 1-dimensional 0-surgery can be a knot, in other words, an embedding of the circle M = S¹ in 3-space. We therefore introduce the notion of embedded 1-dimensional 0-surgery whereby the initial manifold M is embedded in the 3-space. This notion allows the topological modeling of phenomena with more complicated initial 1-manifolds. As mentioned, for our purposes, we will consider S³ as our standard 3-space. For details on the descriptions of S³, see Section 7.1. Since a knot is by definition an embedding of M = S¹ in S³ or , in this case embedded 1-dimensional surgery is the so-called knot surgery. It is worth mentioning that there are infinitely many knot types and that 1-dimensional surgery on a knot may change the knot type or even result in a two-component link. A good introductory book on knot theory is [25] among many other.

Looking back to the process of DNA recombination which exhibits embedded 1-dimensional 0-surgery, a DNA knot is the self-entanglement of a single circular DNA molecule. With the help of certain enzymes, site-specific recombination can transform supercoiled circular DNA into a knot or link. The first electron microscope picture of knotted DNA was presented in [26]. In this experimental study, we see how genetically engineered circular DNA molecules can form DNA knots and links through the action of a certain recombination enzyme. A similar picture is presented in Fig 8, where site-specific recombination of a DNA molecule produces the Hopf link.

Another theoretical example of knot surgery comprises the knot or link diagrams involved in the skein relations satisfied by knot polynomials, such as the Jones polynomial [27] and the Kauffman bracket polynomial [28]. For example, the illustration in Fig 31 represents a so-called ‘Conway triple’, that is, three knot or link diagrams L₊, L₋ and L₀ which are identical everywhere except in the region of a crossing and the polynomials of these three links satisfy a given linear relation.

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Fig 31. One can pass from one of these three links to another via knot surgery.

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Remark 8 In analogy to embedded 1-dimensional 0-surgery, we also have the notion of embedded solid 1-dimensional 0-surgery. As S¹ is the boundary of D², any knot is the boundary of a, so-called, Seifert surface, so embedded solid 1-dimensional 0-surgery could be extended to a Seifert surface.

8.3 Embedded solid 2-dimensional 0-surgery and related phenomena

In Section 7.2.2 we showed how we can pass from the two-ball description to the two-tori description of S³. Although we had not yet defined it at that point, the process we described is, of course, an embedded solid 2-dimensional 0-surgery in S³ on an initial manifold M = D³. It is worth mentioning that all natural processes undergoing embedded solid 2-dimensional 0-surgery on an initial manifold M = D³ can be also viewed in this context. For example, if one looks at the formation of black holes and examines it as an independent event in space, this process shows a decompactified view of the passage from a two 3-ball description of S³, that is, the core of the star and the surrounding space, to a two torus description, that is, the accretion disc surrounding the black hole (shown in white in the third instance of Fig 19) and the surrounding space. In this Section, we will see how some natural phenomena undergoing solid 2-dimensional 0-surgery exhibit the causal or consequential relation to the ambient space mentioned in Section 8.1 and are therefore better described by considering them as embedded in S³.

For example, during the formation of tornados, recall Fig 17(a), the process of solid 2-dimensional 0-surgery is triggered by the difference in the conditions of the lower and upper atmosphere. Although the air cycle lies in the complement space of the initial manifold M = D³ × S⁰, it is involved in the creation of funnel-shaped clouds that will join the two spherical neighborhood (one in the cloud and one in the ground). Therefore the cause of the phenomenon extends beyond its initial manifold and surgery is the outcome of global changes.

We will now discuss phenomena where the outcome of the surgery process propagates beyond the final manifold. A first example are waterspouts. After their formation, the tornado’s cylindrical ‘cork’, that is, the solid cylinder homeomorphic to the product set D¹ × D², has altered the whole surface of the sea (recall Fig 17(b)). In other words, the spiral pattern on the water surface extends beyond the initial spherical neighborhood of the sea, which is represented by one of the two 3-balls of the initial manifold.

As another example, during the formation of black holes, the strong gravitational forces have altered the space surrounding the initial star and the singularity is created outside the final solid torus. In all these phenomena, the process of surgery alters matter outside the manifold in which it occurs. In other words, the effect of the forces causing surgery propagates to the complement space, thus causing a more global change in 3-space.

Remark 9 Looking back at Fig 30, it is worth pinning down the following duality of embedded solid 2-dimensional 0-surgery for M = D³: the attraction of two points lying on the boundary of segment L by the center of D³ can be equivalently viewed in the complement space as the repulsion of these points by the center of B³ (that is, the point at infinity) on the boundary of curve (or line, if viewed in ) l − L.

8.4 Embedded solid 2-dimensional 1-surgery and related phenomena

We will now discuss the process of embedded solid 2-dimensional 1-surgery in S³ in the same way we did for the embedded solid 2-dimensional 0-surgery in S³, recall Fig 30. Taking again M = D³ as the initial manifold, embedded solid 2-dimensional 1-surgery is illustrated in Fig 32. The process begins with disc D in the 3-ball D³ on which colinear attracting forces act, see instances (1) and (1′) for the decompactified view. In (3), the initial 3-ball D³ is split in two new 3-balls and . By Definition 3, the point at the origin (in green) evolves into the two centers of and (in green). This is exactly the same process as in Fig 16 if we restrict it to D³, but since we are in S³, the complement space B³ is also undergoing, by symmetry, solid 2-dimensional 1-surgery. Again, this is a double surgery with one attracting center which is inside the first 3-ball (in yellow) and outside the second 3-ball (in red). This process squeezes the central disc D of D³ while the central disc d of B³ engulfs disc D and becomes the separating plane d ∪ D.

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Fig 32. Embedded solid 2-dimensional 1-surgery.

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As seen in instance (3) of Fig 32, the process alters the existing complement space B³ to and creates a new space which can be considered as the ‘void’ between and . By viewing the process in this way, we pass from a two 3-balls description of S³ to another one, that is, from S³ = B³ ∪ D³ to .

Remark 10 The duality described in 2-dimensional 0-surgery is also present in 2-dimensional 1-surgery. Namely, the attracting forces from the circular boundary of the central disc D to the center of D³ can be equivalently viewed in the complement space as repelling forces from the center of B³ (that is, the point at infinity) to the boundary of the central disc d, which coincides with the boundary of D.

All natural phenomena undergoing embedded solid 2-dimensional 1-surgery take place in the ambient 3-space. However, we do not have many examples of such phenomena which demonstrate the causal or consequential effects discussed in Section 8.1. Yet one could, for example, imagine taking a solid material specimen that has started necking and immerse it in some liquid until its pressure causes fracture to the specimen. In this case the complement space is the liquid and it triggers the process of surgery.

Finally, the annihilation of Falaco solitons is also a case of embedded solid 2-dimensional 1-surgery. The topological thread can be cut by many factors but in all cases these are related to the complement space.

9.1 The dynamical system and its steady state points

In [4], N.Samardzija and L.Greller study the behavior of the following dynamical system (Σ) that generalizes the classical Lotka–Volterra problem [30, 31] into three dimensions: In subsequent work [32], the authors present a slightly different model, provide additional numerical simulations and deepen the qualitative analysis done in [4]. Since both models coincide in the parametric region we are interested in, we will use the original model and notation and will briefly present some key features of the analyses done in [4] and [32].

The system (Σ) is a two-predator and one-prey model, where the predators Y, Z do not interact directly with one another but compete for prey X. As X, Y, Z are populations, only the positive solutions are considered in this analysis. It is worth mentioning that, apart from a population model, (Σ) may also serve as a biological model and a chemical model, for more details see [4].

The parameters A, B, C are analyzed in order to determine the bifurcation properties of the system, that is, to study the changes in the qualitative or topological structure of the family of differential equations (Σ). As parameters A, B, C affect the dynamics of constituents X, Y, Z, the authors were able to determine conditions for which the ecosystem of the three species results in steady, periodic or chaotic behavior. More precisely, the authors derive five steady state solutions for the system but only the three positive ones are taken into consideration. These points are:

It is worth reminding here that a steady state (or singular) point of a dynamical system is a solution that does not change with time.

9.2 Local behavior and numerical simulations

Let, now, J(S_i) be the Jacobian of (Σ) evaluated at S_i for i = 1, 2, 3 and let the sets Γ{J(S_i)} and W{J(S_i)} to be, respectively, the eigenvalues and the corresponding associated eigenvectors of J(S_i). These are as follows:

Using the sets of eigenvalues and eigenvectors presented above, the authors characterize in [4, 32] the local behavior of the dynamical system around these three points using the Hartman-Grobman (or linearization) Theorem. Since 1 > 0 and −1, −B < 0, S₁ is a saddle point for all values of parameters A, B, C. However, the behavior around S₂ and S₃ changes as parameters A, B, C are varied. The authors show that the various stability conditions can be determined by only two parameters: C and B/A. It is also shown in [4] that stable solutions are generated left of and including the line B/A = 1 while chaotic/periodic regions appear on the right of the line B/A = 1. We are interested in the behavior of (Σ) as it passes from stable to chaotic/periodic regions. Therefore we will focus and analyze the local behavior around S₂ and S₃ and present numerical simulations for: stable region (a) where B/A = 1 and and chaotic/periodic region (b) where B/A > 1 and .

• Region (a)

Setting B/A = 1 and equating the right side of (Σ) to zero, one finds as solution the one-dimensional singular manifold: that passes through the points S₂ and S₃. Since all points on L are steady state points, there is no motion along it. For , S₂ is an unstable center while S₃ is a stable center (for a complete analysis of all parametric regions see [4]). This means that if λ₁, λ₂, λ₃ denote the eigenvalues of either S₂ or S₃ with λ₁ ∈ ℝ and λ₂, λ₃ ∈ ℂ, then λ₁ = 0 and Re(λ₂) = Re(λ₃) > 0 for S₂ while λ₁ = 0 and Re(λ₂) = Re(λ₃) < 0 for S₃. Moreover, the point (X, Y, Z) = (1, 1, C/A) is the center of L. The line segment X = 1, 0 < Y < 1 and (1 + C)/A < Z < C/A supports attracting type singularities (and includes S₃) while the line segment defined by X = 1, 1 < Y < 1 + C and 0 < Z < C/A supports unstable singularities (and includes S₂), for details see [32]. More precisely, each attracting point corresponds to an antipodal repelling point, the only exception being the center of L which can be viewed as the spheroid of 0-diameter. The local behavior of (Σ) around S₂ and S₃ in this region together with line L are shown in Fig 33(a). A trajectory (or solution) initiated near L in the repelling segment expands until it gets trapped by the attracting segment, forming the upper and lower hemisphere of a distinct sphere. Hence, a nest of spherical shells surrounding line L is formed, see Fig 34(a). Moreover, the nest fills the entire positive space with stable solutions.

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Fig 33. Local behavior.

Flow induced along L by changing parameter space from (a) B/A = 1 to (b) B/A > 1. Indices 1,2 and 3 indicate the first, second and third component in W(J(S₂)) and W(J(S₃)).

https://doi.org/10.1371/journal.pone.0183993.g033

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Fig 34. Embedded solid 2-dimensional 0-surgery by changing parameter space from (a) B/A = 1 to (b) B/A > 1.

https://doi.org/10.1371/journal.pone.0183993.g034

• Region (b)

For B/A > 1 and , S₂ is an inward unstable vortex and S₃ is an outward stable vortex. This means that in both cases they must satisfy the conditions λ₁ ∈ ℝ and λ₂, λ₃ ∈ ℂ with , the conjugate of λ₂. The eigenvalues of S₂ must further satisfy λ₁ < 0 and Re(λ₂) = Re(λ₃) > 0, while the eigenvalues of S₃ must further satisfy λ₁ > 0 and Re(λ₂) = Re(λ₃) < 0. The local behaviors around S₂ and S₃ for this parametric region are shown in Fig 33(b). It is worth mentioning that Fig 33(b) reproduces Fig 1 of [4] with a change of the axes so that the local behaviors of S₂ and S₃ visually correspond to the local behaviors of the trajectories in Fig 34(b) around the north and the south pole.

Note now that the point S₂ as well as the eigenvectors corresponding to its two complex eigenvalues, all lie in the xy–plane. On the other hand, the point S₃ and also the eigenvectors corresponding to its two complex eigenvalues all lie in the xz–plane. The flow along line L produced by the actions of these eigenvectors forces trajectories initiated near S₂ to wrap around L and move toward S₃ in a motion reminiscent of hole drilling. The connecting manifold L is also called the ‘slow manifold’ in [4] due to the fact that trajectories move slower when passing near it. As trajectories reach S₃, the eigenvector corresponding to the real eigenvalue of S₃ breaks out of the xz–plane and redirects the flow toward S₂. As shown in Fig 34(a) and 34(b), as B/A = 1 moves to B/A > 1, this process transforms each spherical shell to a toroidal shell. The solutions scroll down the toroidal surfaces until a limit cycle (shown in green in Fig 34(b)) is reached. It is worth pointing out that this limit cycle is a torus of 0-diameter and corresponds to the sphere of 0-diameter, namely, the central steady point of L also shown in green in Fig 34(a).

However, as the authors elaborate in [32], while for B/A = 1 the entire positive space is filled with nested spheres, when B/A > 1, only spheres up to a certain volume become tori. More specifically, quoting the authors: “to preserve uniqueness of solutions, the connections through the slow manifold L are made in a way that higher volume shells require slower, or higher resolution, trajectories within the bundle”. As they further explain, to connect all shells through L, (Σ) would need to possess an infinite resolution. As this is never the case, the solutions evolving on shells of higher volume are ‘choked’ by the slow manifold. This generates solution indetermination, which forces higher volume shells to rapidly collapse or dissipate. The behavior stabilizes when trajectories enter the region where the choking becomes weak and weak chaos appears. As shown in both [4] and [32], the outermost shell of the toroidal nesting is a fractal torus. Note that in Fig 34(b) we do not show the fractal torus because we are interested in the interior of the fractal torus which supports a topology stratified with toroidal surfaces. Hence, all trajectories are deliberately initiated in its interior where no chaos is present.

It is worth pointing out that Fig 34 reproduces the numerical simulations done in [32]. More precisely, Fig 34(a) represents solutions of (Σ) for A = B = C = 3 and trajectories initiated at points [1, 1.59, 0.81], [1, 1.3, 0.89], [1, 1.18; 0.95], [1, 1.08, 0.98] and [1, 1, 1]. Fig 34(b) represents solutions of (Σ) for A = 2.9851, B = C = 3 and trajectories initiated at points [1.1075, 1, 1], [1, 1, 0.95], [1, 1, 0.9] and [1, 1, 1].

As already mentioned, as B/A = 1 changes to B/A > 1, S₂ changes from an unstable center to an inward unstable vortex and S₃ changes from a stable center to an outward stable vortex. It is worth reminding that this change in local behavior is true not only for the specific parametrical region simulated in Fig 34, but applies to all cases satisfying . For details we refer the reader to Tables II and III in [4] that recapitulate the extensive diagrammatic analysis done therein.

Finally, it is worth observing the changing of the local behavior around S₂ and S₃ in our numerical simulations. In Fig 34(a), for B/A = 1 we have: while in Fig 34(b), for B/A > 1, both centers change to vortices (inward unstable and outward stable) through the birth of the first eigenvalue shown in bold (negative and positive respectively):

Remark 11 The use of different numerical methods may affect the shape of the attractor. For example, as mentioned in [32], higher resolution produces a larger fractal torus and a finer connecting manifold. However, the ‘hole drilling’ process and the creation of a toroidal nesting is always a common feature.

9.3 Connecting the dynamical system with embedded solid 2-dimensional 0-surgery

In this section, we will focus on the process of embedded solid 2-dimensional 0-surgery on a 3-ball D³ viewed as a continuum of concentric spheres together with their common center: . Recall from Section 5.2 that the process is defined as the union of 2-dimensional 0-surgeries on the whole continuum of concentric spheres and on the limit point P. For each spherical layer, the process starts with attracting forces acting between S⁰ × D², i.e two points, or poles, centers of two discs. In natural phenomena undergoing solid 2-dimensional 0-surgery, such as tornadoes (recall Fig 17) or Falaco solitons (recall Fig 18), these forces often induce a helicoidal motion from one pole to the other along the line L joining them.

Having presented the dynamical system (Σ) in Section 9.1 and its local behavior in Section 9.2, its connection with embedded solid 2-dimensional 0-surgery on a 3-ball is now straightforward. To be precise, surgery is performed on the manifold formed by the trajectories of (Σ). Indeed, as seen in Fig 34(a) and 34(b), with a slight perturbation of parameters, trajectories pass from spherical to toroidal shape through a ‘hole drilling’ process along a slow manifold L which pierces all concentric spheres. The spherical and toroidal nestings in Figs 16(a) and 34 are analogous. The attracting forces acting between the two poles shown in blue in the first instance of Fig 16(a) are realized by the flow along L (also shown in blue in Fig 33(b)). When B/A > 1, the action of the eigenvectors is an attracting force between S₂ and S₃ acting along L, which drills each spherical shell and transforms it to a toroidal shell.

Furthermore, in order to introduce solid 2-dimensional 0-surgery on D³ as a new topological notion, we had to define that 2-dimensional 0-surgery on a point is the creation of a circle. The same behavior is seen in (Σ). Namely, surgery on the limit point P, which is a steady state point, creates the limit cycle which is the limit of the tori nesting. As mentioned in [32], this type of bifurcation is a ‘Hopf bifurcation’, so we can say that we see surgery creating a Hopf bifurcation.

Hence, instead of viewing surgery as an abstract topological process, we may now view it as a property of a dynamical system. Moreover, natural phenomena exhibiting 2-dimensional topological surgery through a ‘hole-drilling’ process, such as the creation of Falaco solitons, the formation of tornadoes, of whirls, of wormholes, etc, may be modeled mathematically by the dynamical system (Σ). For more details, see also [22]. This system enhances the schematic topological model presented in Fig 16(a) with analytical formulation of the underlying dynamics. Indeed, if we link the three time-dependent quantities X, Y, Z to physical parameters of related phenomena undergoing 2-dimensional 0-surgery, system (Σ) can provide time forecasts for these phenomena.

Remark 12 It is worth pointing out that (Σ) is also connected with the 3-sphere S³. We can view the spherical nesting of Fig 34(a) as the 3-ball D³ shown in Fig 30(1) and 30(1′). Surgery on its central point creates the limit cycle which is the core curve c of V₁ shown in Fig 30(3) and 30(3′). If we extend the spherical shells of Fig 34 to all of and assume that the entire nest resolves to a toroidal nest, then the slow manifold L becomes the infinite line l. In the two-ball description of S³, l pierces all spheres, recall Fig 30(1′), while in the two-tori description, it is the core curve of V₂ or the ‘untouched’ limit circle of all tori, recall Fig 30(3) and 30(3′).

Remark 13 In [16] R.M. Kiehn studies how the Navier-Stokes equations admit bifurcations to Falaco solitons. In other words, the author looks at another dynamical system modeling this natural phenomenon which, as we showed in Section 5.3, exhibits solid 2-dimensional 0-surgery. To quote the author: “It is a system that is globally stabilized by the presence of the connecting 1-dimensional string” and “The result is extraordinary for it demonstrates a global stabilization is possible for a system with one contracting direction and two expanding directions coupled with rotation”. It is also worth quoting Langford [33] which states that computer simulations indicate that “the trajectories can be confined internally to a sphere-like surface, and that Falaco Soliton minimal surfaces are visually formed at the North and South pole”. One possible future research direction would be to investigate the similarities between this system and (Σ) in relation to surgery.