The Structure and Enumeration of Link Projections

Martin Bridgeman^{1}

October 15, 1994

Abstract

We define a decomposition of link projections whose pieces we call atoroidal graphs. We describe a surgery operation on these graphs and show that all atoroidal graphs can be generated by performing surgery repeatedly on a family of well known link projections. This gives a method of enumerating atoroidal graphs and hence link projections by recomposing the pieces of the decomposition.

1 Introduction

The problem of enumeration of knots and links has always interested knot theorists. In this paper we introduce a method of enumerating link projections by first decomposing them into pieces called atoroidal graphs. We define surgery on these atoroidal graphs and show how they can be enumerated by performing surgery on a well known family of link projections. By recomposing these atoroidal graphs we can thus enumerate link projections. I have included an enumeration of atoroidal graphs to 12 crossings.

A link projection is given by a 4-valent planar graph G. To form a link we can replace each vertex of G by a crossing. To enumerate links in this way we must first enumerate link projections. It was Kirkman’s success in enumerating link projections or polyhedra as he called them([K1],[K2]) that formed the basis of the knot tables of both Tait([T]) and Little([L1],[L2]). In [C], Conway introduced a notation which made it possible for him to enumerate knots to 11 crossings and links to 10 crossings in a single afternoon where before it had taken years. In his paper Conway defined a basic polyhedron to be a polyhedron with no bigon regions and showed that every link is obtained by replacing each vertex of a basic polyhedron by a rational tangle. These basic polyhedra are closely related to the atoroidal graphs defined in this paper and can be enumerated using the enumeration of atoroidal graphs described.

1Research at MSRI is supported in part by NSF grant no.DMS-9022140.

The decomposition of a link projection into atoroidal graphs is achieved by cutting the projection along certain non-trivial curves. We then define surgery on an atoroidal graph giving a new atoroidal graph with one more vertex. This gives a partial ordering on atoroidal graphs where G_{1} ≺ G_{2} if a surgery on G_{1} results in G_{2} and we show that a graph is initial if and only if it has no vertices of a given type. Using this we can list the initial objects and thus enumerate all atoroidal graphs by repeatedly performing surgery on these initial objects.

The motivation for the paper comes from orbifold theory and hyperbolic geometry but a background in these is not necessary here. For a reference see [Th]. For readers interested, these aspects are laid out in the section on orbifolds.

I would like to thank Curt McMullen, Joe Christy, Rich Schwartz and especially my advisor Bill Thurston.

2 Decomposition

Link Projections

Given a link L a general position projection of L is a 4-valent graph G embedded in S^{2}. As we are only considering such graphs we will use graph to mean a 4-valent graph embedded in S^{2}. Let G be a graph with vertex set V.

Definition: 1 An n-curve of G is a simple closed curve in S^{2}−V intersecting G n times.

Let α be an n-curve(n = 0, 2, 4) of G. Then α splits S^{2} into two disks D_{1}, D_{2}. We say a component D_{i} is trivial if D_{i} ∩ G is either empty, a simple

Figure 1: Trivial 0,2,4-curves

arc, two disjoint simple arcs or two arcs crossing at a single vertex(figure 1). If α has a trivial component then α is trivial. Otherwise α is called non-trivial.

Definition: 2 A graph G is irreducible all n-curves(n = 0, 2) are trivial.

Definition: 3 A graph G is atoroidal if all n-curves(n = 0, 2, 4) are trivial.

Decomposition

If α is a non-trivial n-curve(n = 0, 2, 4) of G then we can decompose G along α into graphs G_{1} and G_{2} as follows. First cut along α, this splits the sphere into two disks D_{1}, D_{2}. To obtain the graph G_{i} from D_{i} we identify the boundary of D_{i} to a single point. We say that G decomposes into G_{1} and G_{2} along α.

We now describe the decomposition of a link projection G into atoroidal graphs. If all n-curves(n = 0, 2, 4) in G are trivial the the decomposition is done. Otherwise decompose G into G_{1} and G_{2} along a non-trivial ncurve(n = 0, 2, 4) where n is chosen to be as small as possible. Now repeat the decomposition on the resultant graphs G_{1} and G_{2}. It is obvious that this decomposition terminates.

The decomposition along non-trivial 0-curves is especially simple, corresponding to splitting a graph into its connected components and thus when dealing with connected graphs we only need concern ourselves with nontrivial 2-curves and 4-curves.

3 Structure of Atoroidal Graphs

Almost all Atoroidal Graphs are Hyperbolic

To investigate the type of atoroidal graphs possible we consider the cell division of S^{2} given by G. We call the cells, faces of G and a face F is called an n-gon if it has n vertices of G on its boundary. Any n-gon F of G gives a 2n-curve α_{F} by taking the boundary of a small neighborhood N_{F} of F.

If G has a 0-gon F then α_{F} is a trivial 0-curve (i.e. it bounds a disk D in S^{2} − G) as G is irreducible. Attach a disk to the boundary of N_{F} ⇒ G is just a jordan curve or the unknot projection(figure 2).

If G has a 1-gon(monogon) F then α_{F} is a trivial 2-curve as G is irreducible. Attach a neighborhood of an arc to N_{F} ⇒ G is the graph having the form of the number 8 (figure 2).

If G has a 2-gon(bigon) F then α_{F} is a trivial 4-curve as G is atoroidal. Attaching a neighborhood of a vertex to N_{F} ⇒ G is the projection of the trefoil and attaching a neighborhood of two parallel arcs of G to N_{F} can be done in 2 ways to give two possible graphs but as can be seen in figure 2 only one is also irreducible. Thus G is either the trefoil projection or the Hopf link projection.

Apart from these 4 exceptional graphs all other atoroidal graphs have faces that are at least triangular and are called hyperbolic graphs.

Surgery

Given any atoroidal graph G which has a face F with greater than 3 vertices then we can perform surgery on G to give another atoroidal graph G′. We choose edges e, e_{1}, e_{2} of F with e_{1}, e_{2} adjacent to e. G′ is obtained by pinching together e_{1}, e_{2}, i.e. take a simple arc α ⊂ F with endpoints in the interior of e_{1}, e_{2} resp. and homotope α to a single point. The graph G′ obtained by performing surgery on G has one more vertex than G(figure 3).

Lemma: 1 G′ is atoroidal.

Proof :Let α′ be an n-curve in G′ which decomposes S^{2} into disks D′_{1}, D′_{2}. We can get G back by splitting open the new crossing v and as this splitting can be done in a small neighborhood of v and α′ is outside such a neighborhood, we get an n-curve α in G which splits S^{2} into disks D_{1}, D_{2}. Note that D′_{i} and Figure 2: Exceptions Figure 3: Surgery D_{i} are either the same or the former is obtained from the latter by pinching two edges together.

As G is atoroidal then α is trivial for n = 0, 2, 4 and we can assume D_{1} is trivial. n = 0, 2 ⇒ D′_{1} = D_{1} as D_{1} has at most one edge intersecting it so we haven’t enough edges to pinch ⇒ D′_{1} trivial. ⇒ α′ trivial. n = 4 ⇒ D_{1} either neighborhood of vertex or neighborhood of two nonintersecting arcs of G. If D_{1} is neighborhood of vertex then the only edges that can be pinched are adjacent ⇒ D′_{1} = D_{1} ⇒ α′ trivial. If D_{1} is neighborhood of two parallel arcs ⇒ D′_{1} = D_{1} or D′_{1} is obtained by pinching the parallel arcs of D_{1} together ⇒ D′_{1} is neighborhood of a vertex ⇒ α′ trivial ⇒ G′ is atoroidal □

4 Orbifolds

This section explains how the work arises out of considering certain orbifolds associated with a link projection. In this setting the decomposition and surgery we define are the torus decomposition and dehn surgery on these orbifolds. An orbifold is a generalization of a manifold in which the space is locally modeled on R^{n} modulo the action of a finite group. For example if a group G acts properly discontinuously on a space M then M/G is an orbifold and is a manifold if the action is also free. For a reference on orbifolds see chapter 13 of [Th].

Associated Orbifolds

We associate two orbifolds O_{G} and O′_{G} to a graph G as follows. We consider G as a graph siting on S^{2} in S^{3}. Let B be a ball in S^{3} with boundary S^{2} and V be the vertex set of G. O_{G} is a polyhedral orbifold with underlying space X_{O_{G}} = B −V, singular locus Σ_{O_{G}} = S^{2} −V and 1-dimensional singular locus Σ_{O_{G}}^{1} = G−V. The 1-dimensional singular locus is marked with D_{2} indicating that any point on it is modeled by D^{3}/D_{2} where D_{2} acts by two reflections in planes meeting in right angles. O′_{G} has underlying space X_{O′_{G}} = S^{3} − V and singular locus Σ_{O′_{G}} = G − V. Here the singular locus is 1-dimensional and is marked with Z_{2} to indicate any point on it is modeled on D^{3}/Z_{2} where Z_{2} acts by rotation of order two.

O′_{G} is the double of O_{G} in the sense of orbifolds.

Decomposition and Surgery

In [B] we show that the torus decomposition on the orbifolds O_{G} and O′_{G} is the decomposition we’ve described on G. By Andreev’s theorem(see [Th]) if G is a hyperbolic graph then O_{G} can be realized as an ideal hyperbolic polyhedron with all dihedral angles right angles. Taking the subgroup of orientation preserving elements of π_{1}(O_{G}) shows us that O′_{G} can also be realized as a hyperbolic orbifold.

We show([B]) that for any graph G the double cover of O′_{G} is a link compliment denoted by L_{G} with one component for each vertex of G and is a hyperbolic link complement iff G is a hyperbolic graph. Also if G′ is obtained by surgery on G then link complement L_{G′} is obtained from L_{G} by removing a simple closed curve, i.e. by dehn drilling.

5 Partial Ordering

Surgery gives atoroidal graphs a partial ordering ≺ by defining G_{1} ≺ G_{2} iff G_{2} is obtained by performing r successive surgeries on G_{1}, r = 0, 1, 2, … Note that surgery cannot be performed on any of the four exceptional atoroidal graphs and they are never the resultant graph of surgery, therefore they are isolated objects(both initial and final). Thus ≺ restricts to a p.o. on hyperbolic graphs. To study ≺ we show that the initial objects are a well known family of graphs and we can generate all atoroidal graphs by performing surgery on these initial objects.

After surgery has been performed on a graph to give a graph G with a new vertex v. The vertex v is a vertex of a triangle T which has adjacent faces F_{1}, F_{2} meeting at v each being greater than triangular(figure 4). A vertex with this local structure we call simple. To find a Ḡ s.t. Ḡ ≺ G we might just look for a simple vertex v and cut open at v(there is a unique way to cut open a simple vertex) but this doesn’t necessarily give an atoroidal graph as the resultant may have non-trivial 4-curves(figure 5). What we will show is that if a graph has a simple vertex v belonging to a triangle T then the graph

Figure 4: Simple vertex

Figure 5: Cutting open at a simple vertex doesn’t work

can be cut open at some vertex of T to give an atoroidal graph. This implies that an initial object cannot have the any simple vertices. Before proving the stated result we will use it to show what the initial objects are.

Since the exceptions are isolated, they never arise in a sequence of surgeries and all other initial objects are hyperbolic. Let G be a hyperbolic initial object(not one of the exceptions), calculating the euler number of the cell division of S^{2} into faces of G we see that G has a triangular face T_{1}. As G is initial, T_{1} has two adjacent triangular faces T_{2}^{l}, T_{2}^{r}(figure 6). Again using the fact that G is initial we have that T_{2}^{l}, T_{2}^{r} both have a neighboring triangular face other than T_{1} labeled T_{3}^{l}, T_{3}^{r} respectively.

If T_{3}^{l} = T_{3}^{r} then G atoroidal ⇒ G is borromian ring projection. Also if T_{3}^{l}, T_{3}^{r} have a common vertex then G irreducible ⇒ G is again the Borromian ring projection which we call T_{3} (figure 6).

If T_{3}^{l}, T_{3}^{r} are disjoint then each has a neighboring triangular face T_{4}^{l}, T_{4}^{r} other than the previous faces T_{2}^{l}, T_{2}^{r}. These are unique as the face adjacent Figure 6: Initial Setup and the resulting Borromian Rings

Figure 7: Next Stage and the resulting graph T_{4}

to both T_{3}^{l} and T_{3}^{r} is at least 4 sided. If now T_{4}^{l}, T_{4}^{r} have a common vertex then using G atoroidal ⇒ G is of the form given in figure 7 which we call T_{4}.

Continuing this we get the collection of graphs {T_{n}}_{n≥3}(figure 8) which together with the exceptional atoroidal graphs are the initial objects of ≺ of which only {T_{n}}_{n≥4} are non-terminal. Knowing the initial objects allows us enumerate all atoroidal graphs by performing surgery repeatedly.

Lemma: 2 Let G be an atoroidal graph with a simple vertex v of triangle T and let G′ be the graph obtained by cutting G open at v. Let e′_{1}, e′_{2} be the two edges of face F′ in G′ pinched to get G then

1. G′ is irreducible.

2. Any non-trivial 4-curve α′ of G′ intersects the face F′ in a single arc β′ which separates e′_{1}, e′_{2}.

Figure 8: T_{3} T_{4} T_{5} T_{6} T_{7} …

Proof :As above let e′_{1}, e′_{2} be the two edges of F′ pinched together to give G with e′ the edge adjacent to both. If ∃ neighborhood N_{e′} of e′ s.t. N_{e′} ∩ α′ is empty ⇒ can pinch e′_{1}, e′_{2} in N_{e′} with α′ giving an n-curve α in G. If n = 0, 2, 4 then α is trivial and splits S^{2} into two disks D_{1}, D_{2} with D_{1} trivial. Similarly α′ splits S^{2} into D′_{1}, D′_{2} with either D′_{1} = D_{1} or D′_{1} obtained from D_{1} by cutting open a crossing. In either case this gives D′_{1} trivial ⇒ α′ trivial. Therefore every non-trivial n-curve (n = 0, 2, 4) must intersect F′ in an arc β′ that has one endpoint on e′ and the other on another edge e′_{3} of F′ with e′_{3} ≠ e′, e′_{1}, e′_{2}.

If n = 0 then α′ doesn’t intersect G′ ⇒ α′ trivial.

If n = 2 then α′ only intersects G′ at the two endpoints of β′. If we pinch e′_{1}, e′_{2} together to get G we can do so by either leaving e′_{1} fixed and pulling e′_{2} through α′ or vise-versa. This gives 4-curves α^{r}, α^{l} resp. in G which are identical with α′ outside a neighborhood of the new vertex v and either go right or left around v as the curves approach v from inside T (figure 9). G atoroidal ⇒ α^{r} is trivial ⇒ α^{r} splits S^{2} into D_{1}, D_{2} s.t. D_{1} trivial. The region containing v also contains another vertex of T so it can’t be trivial. Therefore D_{1} is the region containing vertex v_{2} of T and is neighborhood of v_{2} ⇒ v not simple as one of faces is bigon. This contradiction implies that G′ has no non-trivial 2-curves ⇒ G′ is irreducible.

If n = 4 ⇒ α′ ∩ F′ consists of either 1 or 2 arcs. If it is 2 arcs β′_{1}, β′_{2}

Figure 9: Pinch through α′

Figure 10: Non-Trivial curve can’t intersect in more than one arc

then traversing around α′ we have 4 connected arcs β′_{1}, γ′_{1}, β′_{2}, γ′_{2}. We can join endpoints of γ′_{1} by another arc δ′_{1} in F′ s.t. γ′_{1} ∪ δ′_{1} is a 2-curve in G′ ⇒ trivial. If endpoints of γ′_{1} belong to different edges then both components of S^{2}−γ′_{1}∪δ′_{1} contain vertices which contradicts it being trivial. Therefore both γ′_{1}, γ′_{2} are contained in adjacent faces to F′ and α′ is neighborhood of two parallel arcs of G′ ⇒ α′ trivial(figure 10). Therefore any non-trivial 4-curve in G′ intersects F′ in a single arc β′ separating the two edges pinched in G′ to obtain G □

Definition: 4 An n-curve α (n ≥ 4) in G is trivial iff either

• α is the boundary of a neighborhood of a vertex of G or

• ∃ arc β intersecting G at most once s.t. β ∩ α = ∂β and ∂β splits α into α_{1}, α_{2} each containing at least two points of G. β is called a compression of α.

Figure 11: Only trivial 6-curves in atoroidal graph

If G is atoroidal then the only trivial 6-curves can easily shown to be those curves that split S^{2} into two disks one of which has one of four types given in figure 11. Disks of this kind bounding a 6-curve are called trivial.

Lemma: 3 If α′ is a non-trivial 4-curve in G′ then associated with it are two non trivial 6-curves α^{l}, α^{r} in G.

Proof :α′ intersects F′ in a single arc β′ separating edges e′_{1}, e′_{2} and as before we can pinch e′_{1}, e′_{2} together to get graph G. This can be done in two ways, either by fixing e′_{1} and pushing e′_{2} across β′ or vise-versa. We get two 6-curves α^{r}, α^{l} in G both identical to α′ outside a neighborhood of the new vertex v and either going right or left around the vertex inside the neighborhood of v as before(figure 9).

α^{r} splits S^{2} into disks D_{1}, D_{2} with D_{1} containing vertices v, v_{1} of T and D_{2} containing vertex v_{2}. Therefore if D_{1} is trivial then it must be the same type as the fourth disk described in figure 11. But then α′ would bound a neighborhood of a vertex of G which contradicts α′ non-trivial. If D_{2} is trivial then it is either the same type as the third or fourth disk in figure 11. If its the third type then as before α′ would bound a neighborhood of a vertex of G which contradicts α′ non-trivial. If it is the fourth type then adjacent face F_{2} of T would be a triangle contradicting v being simple. Therefore α^{r} is non-trivial and similarly α^{l} □

Theorem 1 If G has a simple vertex v of a triangle T then either

• splitting open at v gives an atoroidal graph G′ or

• both other vertices v_{1}, v_{2} of T are simple and splitting at either gives an atoroidal graph.

Proof :If G′ is not atoroidal ⇒ ∃α′ non-trivial 4-curve in G′ and α^{r}, α^{l} non-trivial 6-curves in G. Triangle T has adjacent faces F_{1}, F_{2}, F_{3} with both

Figure 12: All vertices of T are simple

F_{1}, F_{2} non-triangular as v is simple. If F_{3} was triangular then α^{r} splits F_{3} in two, one piece containing just one vertex say v_{1} and the other containing two. Therefore α^{r} takes a clockwise path about v_{1} from T through F_{3}. If instead we take an anticlockwise path α′ gives us another 4-curve in G′ called ᾱ′(figure 12). Since ᾱ′ doesn’t intersect e′ then it is trivial. Therefore it splits S^{2} into disks D̄′_{1}, D̄′_{2} with D̄′_{1} trivial. α′ splits S^{2} into D′_{1}, D′_{2} with D′_{1} obtained from D̄′_{1} by crossing two adjacent ends so if D̄′_{1} is neighborhood of two parallel arcs then D′_{1} is neighborhood of vertex and if D̄′_{1} is neighborhood of vertex then G′ would contain a bigon. Therefore F_{3} must be non-triangular and both v_{1} and v_{2} are simple.

If splitting at v_{1} doesn’t give an atoroidal graph then there is a nontrivial 4-curve α′_{1} in G′_{1} and non-trivial 6-curves α_{1}^{r}, α_{1}^{l} in G. Considering the 6-curves α, α_{1}, where α = α^{l} and α_{1} = α_{1}^{r} we will show that they can be isotoped to intersect in only two points.Then by showing that they cannot intersect in the given way(figure 13) the theorem is proven.

Firstly we will show that α, α_{1} can be isotoped to only intersect twice. If they intersect any more then S^{2} − α ∪ α_{1} contains at least four regions that are disks with boundary consisting of one arc of α and α_{1}. As α ∪ α_{1} has 12 intersections with G then each of these four regions cannot have boundaries being n-curves n ≥ 4. Therefore one of these regions D has γ = ∂D either a trivial 0 or 2-curve and γ = β ∪ β_{1} where β, β_{1} are arcs of α, α_{1} resp.. If γ is a 0-curve then either D or D^{c} is a trivial disk. If D is trivial then can isotope to remove two intersections of α and α_{1} by pulling β through β_{1}. If D^{c} is trivial then any of the other 3 disks with boundary consisting of one arc of α and α_{1} are trivial and hence can reduce the number of intersections as in first case.

Figure 13: Two intersecting Non-trivial 6-curves

If γ is a 2-curve then each of β, β_{1} intersect G as if say β didn’t then it would be a compression for α_{1} which contradicts α_{1} being non-trivial. Therefore either D or D^{c} is trivial i.e. a neighborhood of an arc of G. If D is trivial then can isotope by pulling β through β_{1} reducing number of intersections of α and α_{1}. If D^{c} is trivial then other 3 disks with boundary consisting of one arc of α and α_{1} cannot have boundaries being n-curves n ≥ 4 as they can have a maximum of 10 intersections with G between them. Therefore there is region D̄ which either doesn’t intersect G and thus we can isotope as before to reduce the number of intersections of α and α_{1} or is neighborhood of an arc of G which can also be isotoped as before.

So we can assume α and α_{1} intersect twice and divide S^{2} into 4 disks. We label these disks D_{i}, i = 1, …, 4 where D_{1}, D_{2}, D_{3} contain v_{1}, v_{2}, v resp. and γ^{i} = ∂D_{i}. γ^{i} is an n_{i}-curve where ∑ n_{i} = 24 and n_{i} ≥ 4. Also γ^{i} = β^{i} ∪ β_{1}^{i} where β^{i}, β_{1}^{i} are arcs of α, α_{1} resp.(figure 13). Note that the arcs β^{i}, β_{1}^{i} have duplication with each of two arcs that α or α_{1} is divided repeated twice. This is for ease of labeling and can be thought of as the two sides of the same arc on α or α_{1}.

Case 1:If n_{2} = 4 then D_{2} is a neighborhood of v_{2} and both β^{2} and β_{1}^{2} intersect G twice otherwise we get a compression of α or α_{1}. Therefore γ^{1} is a 6-curve and inside D_{1} is a 4-curve γ̃^{1}(figure 14). If it is boundary of neighborhood of two parallel arcs of G then this implies either F_{3} is a bigon or α_{1} is a trivial 6-curve. If it is boundary of neighborhood of a vertex then this would imply that F_{3} was triangular. Therefore n_{2} ≠ 4.

Case 2:If n_{1} = 4 then β_{1}^{1} intersects G only twice as otherwise β^{1} is

Figure 14: Cases

a compression of α_{1}. Therefore γ^{4} is a 6-curve and D_{4} contains a 4-curve γ̃^{4}(figure 14). If it is boundary of neighborhood of two parallel arcs of G then this implies that either F_{1} is a bigon or α is trivial 6-curve. If it is the boundary of neighborhood of vertex then this implies that F_{1} is triangular. Therefore n_{1} ≠ 4 and by symmetry n_{3} ≠ 4.

Case 3:If n_{4} = 4 then D_{4} must be neighborhood of parallel arcs of G which implies F_{1} is a bigon(figure 14). Therefore n_{4} ≠ 4.

Case 4:Therefore n_{i} = 6 and each arc β^{i}, β_{1}^{i} intersects G exactly 3 times. Therefore γ̃^{1}, γ̃^{4} are both 4-curves. If γ̃^{4} is boundary of neighborhood of two parallel arcs of G then this implies either F_{1} is a bigon or both α and α_{1} have compressions, contradicting them being non-trivial(figure 14). If γ̃^{4} is boundary of a neighborhood of a vertex then F_{1} would be triangular. Therefore we have shown that there cannot exist 6-curves intersecting as α and α_{1} do ⇒ if α exists (i.e. G′ isn’t atoroidal) ⇒ α_{1} can’t exist ⇒ G′_{1} is atoroidal. Similarly G′_{2} is atoroidal also □

We have shown that the initial objects of ≺ are {T_{n}}_{n≥3} along with the 4 exceptions. T_{n} is the projection of the (3,n) torus link with the link having three components if 3 divides n and having one component otherwise. From this we see that T_{n} has symmetries taking any directed edge of one of the non-triangular faces to any other. Therefore any surgery on T_{n} gives the same graph which we call T_{n}^{+}.

Lemma: 4 If T_{n} ≺ G and T_{n} ≠ G (n > 4) then T_{n−1} ≺ G.

Proof :If T_{n} ≺ G and T_{n} ≠ G then T_{n}^{+} ≺ G. As in figure 7 we can pinch together edges of T_{2}^{l} to T_{4}^{r} to get T_{n}^{+}. Only one vertex of T_{1} is simple so we cut open at that vertex first. This reduces the pinched T_{2}^{l} to a triangle which has only one simple vertex which we now cut open(figure 15). This resulting graph is T_{n−1}^{+} so we have that T_{n−1} ≺ T_{n−1}^{+} ≺ T_{n}^{+} ≺ G □

Therefore if C_{n} is the set of proper descendants of T_{n}(i.e. T_{n} ∉ C_{n}) then

C_{4} ⊇ C_{5} ⊇ C_{6} ⊇ ⋯ ⊇ C_{n} ⋯

6 Enumeration

We now have a way to enumerate atoroidal graphs up to any prescribed crossing number by performing surgery on the initial objects. Figure 16 is the enumeration of atoroidal graphs up to 12 crossings. To enumerate prime link projections we need only recombine the atoroidal graphs as follows. We choose two atoroidal graphs G_{1} and G_{2} with vertices v_{1} and v_{2} respectively. Now take the compliment of a neighborhood of each vertex and attach their boundaries, making sure to match up the strands of the graphs. In recombining we do not use the first 3 exceptions as either they have no vertices or the compliment of a neighborhood of a vertex is trivial. To enumerate the basic polyhedra of Conway the trefoil projection is also not used as the compliment of a vertex is a bigon.

References

[B] M. Bridgeman Volume Increase under Dehn Drilling Operations. Phd. thesis Princeton June 1994

[C] J. H. Conway An enumeration of knots and links, and some of their algebraic properties. Computational Problems in Abstract Algebra. (1970)

[K1] T. P. Kirkman The enumeration, description and construction of knots of fewer than ten crossings. Trans. Roy. Soc. Edinburgh 32(1885)

Figure 15: T_{n−1}^{+} ≺ T_{n}^{+}

[K2] T. P. Kirkman The 364 unifilar knots of ten crossings enumerated and defined. Trans. Roy. Soc. Edinburgh 32(1885)

[L1] C. N. Little Non-alternate ± knots, of order eight and nine. Trans. Roy. Soc. Edinburgh 35(1889)

[L2] C. N. Little Alternate ± knots of order 11. Trans. Roy. Soc. Edinburgh 36(1890)

[T] P. G. Tait On knots I, II, III (1887, 1884, 1885). Scientfic Papers I

[Th] W. P. Thurston The geometry and topology of three manifolds. Princeton Lecture Notes (1979)

Figure 16: Atoroidal Graphs of 12 crossings or less 18