Quotients of the Gordian and H(2)-Gordian graphs
aa r X i v : . [ m a t h . G T ] F e b QUOTIENTS OF THE GORDIAN AND H(2)-GORDIAN GRAPHS
CHRISTOPHER FLIPPEN, ALLISON H. MOORE, AND ESSAK SEDDIQ
Abstract.
The Gordian graph and H(2)-Gordian graphs of knots are abstract graphs whosevertex sets represent isotopy classes of unoriented knots, and whose edge sets record whetherpairs of knots are related by crossing changes or H(2)-moves, respectively. We investigate quo-tients of these graphs under equivalence relations defined by several knot invariants including thedeterminant, the span of the Jones polynomial, and an invariant related to tricolorability. Weshow, in all cases considered, that the quotient graphs are Gromov hyperbolic. We then prove acollection of results about the graph isomorphism type of the quotient graphs. In particular, wefind that the H(2)-Gordian graph of links modulo the relation induced by the span of the Jonespolynomial is isomorphic with the complete graph on infinitely many vertices. Introduction
The Gordian graph is an abstract graph that organizes the set of knots related by crossing changes.More specifically, the vertex set of the Gordian graph K corresponds with isotopy classes ofknots in the three-sphere. A pair of vertices [ K ] , [ J ] determines an edge whenever knots K and J are related by a crossing change. The Gordian graph is immensely complex. It is a countablyinfinite graph of infinite diameter with infinite valence at every vertex. Despite the abundance ofedges in this graph, measuring the Gordian distance between an arbitrary pair of vertices remainsan intractable problem. By Gordian distance, we mean the minimal number of crossing changesrequired to transform a knot K into a knot J in any sequence of knot diagrams. Gordian distancegeneralizes the unknotting number of a knot, a well-known knot invariant that is easily defined butdifficult to calculate.In [JLM20], the authors defined and studied a general class of knot graphs , which includes boththe Gordian graph, the H(2)-Gordian graph (see section 2.1 for a definition) and other graphsconstructed with different local unknotting operations. The class of knot graphs also includes quo-tients of the Gordian graph and its relatives under equivalence relations defined by knot invariants.Although the global structure of such knot graphs is not well-understood, there are a few sporadicresults. Hirasawa and Uchida showed that every vertex is contained in the complete graph oncountably many vertices [HU02]. Zhang, Yang and Lei adapted this result to the H(n)-Gordiangraph [ZY18, ZYL17]. Gambaudo and Ghys constructed a quasi-isometric embedding of a Eu-clidean lattice in the Gordian graph [GG16], and Jabuka, Liu and Moore proved that knot graphsin general fail to be Gromov hyperbolic [JLM20]. Other articles that have considered the structureand quotients of knot graphs include [BK20, BCJ +
17, IJ11, NO09, Ohy06], to name a few.In this article, we further investigate quotients of the Gordian graph and H(2)-Gordian graphsunder a collection of knot invariants, all of which can be derived from the Jones polynomial. Inparticular, we will consider the Gordian graph K and H(2)-Gordian graph K H , and their quotients Mathematics Subject Classification. under the equivalence relations defined by the span of the Jones polynomial, the determinant, andan invariant β that is related to the tricoloring number of the knot. The first question we consideris whether these quotient graphs are hyperbolic. Recall that in a metric graph, a geodesic triangleis δ -thin if each edge of the triangle is contained in a closed δ -neighborhood of the union of theremaining two. A graph is called δ -hyperbolic (or Gromov hyperbolic) if every geodesic triangle is δ -thin for some parameter δ ≥
0. (See section 4.1 for details.) We show:
Theorem 1.1.
The quotients of the Gordian and H(2)-Gordian graphs by the span of the Jonespolynomial, the determinant, and the invariant β are all Gromov hyperbolic. After establishing the hyperbolicity of these graphs, we turn to the question of their graph isomor-phism types. We first observe that in the case of the invariant β , the quotient graphs are quicklyseen to be isomorphic with infinite path graphs. Theorem 1.2.
The quotient of the Gordian and H(2)-Gordian graphs with respect to the invariant β are both isomorphic to the infinite path graph P N . In [JLM20, Theorem 1.4], the authors constructed various hyperbolic quotients of the Gordian graphand observed that these contrasted greatly with the usual Gordian and H(2)-Gordian graphs, whichare not hyperbolic. All of the hyperbolic quotient graphs constructed by [JLM20] are isometric toa subset of R with the Euclidean metric and in particular, are graphs of infinite diameter. Thequotient graphs with respect to β in Theorem 1.2 also have this property. However, unlike with β , the quotient graphs with respect to the determinant and the span of the Jones polynomial aregraphs of finite diameter (see the proofs of Theorems 4.5 – 4.7). To our knowledge, these are thefirst examples of hyperbolic quotient knot graphs that do not arise from the compatibility condition,as defined in [JLM20].Given the existence of induced subgraphs isomorphic to K ∞ at every vertex in the Gordian andH(2)-Gordian graphs [HU02, ZY18, ZYL17], it is natural to ask to what extent we may identifycomplete graphs in their quotients as well. In the case of the determinant we show: Theorem 1.3.
In the quotient of both the Gordian and H(2)-Gordian graphs by the determinant,the subgraph induced by the set of vertices { [ n k ] , k ≥ } is isomorphic to K ∞ for all odd n ≥ . We also observe the existence of numerous other classes of edges in the quotient graphs with respectto the determinant (see Example 5.1. In the case of span of the Jones polynomial, we prove thatthe quotient of the Gordian graph is nearly isomorphic to K ∞ (see Proposition 5.3 for a precisestatement), but were unable to identify edges of the form { [ n ] , [ n + 1] } for all n ∈ N . However, whenwe consider the quotient of the H(2)-Gordian graph of links with respect to the span of the Jonespolynomial, we find that the graph is indeed isomorphic to K ∞ , the complete graph on infinitelymany vertices. Theorem 1.4.
The quotient of the H(2)-Gordian graph of links by the span of the Jones polynomialis isomorphic to K ∞ . A more precise statement of the graph isomorphism types found is given in Section 5. Theorem 1.3,Proposition 5.3 and 1.4 are all proved by constructing edges in these graphs. This is accomplished
UOTIENTS OF THE GORDIAN AND H(2)-GORDIAN GRAPHS 3 in many cases with explicit calculations of the span of the Jones polynomial and other invariantsfor several special classes of knots.
Organization.
In section 2, we review knots, introduce quotients of knot graphs, and define theknot invariants that will be relevant to our discussion. Section 3 is devoted to the Kauffman bracketmethod for calculating the Jones polynomial, and we give explicit calculations for specific familiesof knots. In section 4 we prove hyperbolicity for the knot graphs, and in section 5 we prove variousresults about the graph isomorphism type of the knot graphs.2.
Knots and knot invariants A knot is a smooth embedding of a circle in the three-sphere, where two knots are consideredequivalent if they agree up to ambient isotopy. A link is a collection of knots, possibly linkedtogether. In this article, we will not make a distinction between an oriented knot and its reverseorientation, but we do distinguish between a knot K and its mirror image − K . Links are consideredup to orientation, however for the particular quotient graphs that we construct, this choice oforientation will not turn out to matter.2.1. The Gordian and H(2)-Gordian graphs and their quotients.
Any knot can be trans-formed into the unknot by a finite sequence of crossing changes ! / . The unknotting number u ( K ) is the minimum number of crossing changes to unknot a knot, minimized over all sequencesof diagrams. The Gordian distance d ( K, J ) is the minimum number of crossing changes required totransform a knot K into a knot J . Unknotting number is a basic, fundamental knot invariant, yetit is extremely difficult to calculate in general. An H(2)-move is another important operation onknots and links, originally defined in [HNT90]. In a diagram, an H(2)-move is realized by the tanglereplacement H ! , or by the resolution of a crossing as a vertical or horizontal smoothing. Aswith crossing changes, any knot can be unknotted with a finite sequence of (component-preserving)H(2)-moves. The H(2)-Gordian distance , d ( K, J ), is the minimum number of component pre-serving H(2)-moves required to transform K into J through some finite sequence of knots. In thelast section (and in some figures), we will consider H(2)-moves along links that are not componentpreserving. In particular, the H(2)-distance allowing for H(2)-moves that may change the numberof link components (called SH(2)-moves by [HNT90]) will occur in this article in the statement ofTheorem 1.4.The following definitions are special cases of knot graphs , as defined by Jabuka-Liu-Moore [JLM20]. Definition 2.1.
Let denote the crossing change unknotting operation, and let H denote anH(2)-unknotting operation, which is assumed to be component-preserving unless stated otherwise.Let p ( K ) be an integer-valued knot invariant.(1) The Gordian knot graph K is the graph whose vertices are isotopy classes of unorientedknots, in which a pair of vertices [ K ] , [ K ′ ] span an edge if there exist a pair of knots K , K ′ that possess diagrams related by a crossing change. Similarly, the H(2)-Gordian knot graph K H is the graph whose vertices are isotopy classes of unoriented knots, and in which a pair ofvertices [ K ] , [ K ′ ] span an edge if there exist knots K, K ′ related by a component-preservingH(2)-move.(2) The quotient graph QK p is a graph whose vertices are equivalence classes [ K ] p of knots K .A pair of knots K and L are equivalent if p ( K ) = p ( L ), and two equivalence classes [ K ] p CHRISTOPHER FLIPPEN, ALLISON H. MOORE, AND ESSAK SEDDIQ n (a) T (2 , n ) torus knots and links . . . . . . nm (b) Generalized twist knots K ( m, n ). p q r (c) Pretzel knots K ( p, q, r ). Figure 1.
Knot diagrams of T (2 , n ) torus links, generalized twist knots K ( m, n ),and K ( p, q, r ) pretzel knots.and [ K ′ ] p span an edge if there exist knots L and L ′ , equivalent to K and K ′ respectively,that possess diagrams related by an a crossing change. For H(2)-moves, the quotient graph QK p H is analogously defined.In all of the knot graphs defined above, there exist self-loops at all vertices. These can be realizedby crossing changes at nugatory crossings or trivial H(2)-moves. For simplicity, we will ignoresuch self-loops. For the remainder of the article, we will assume that knot graphs are simple andundirected.2.2. Special families of knots.
Several standard families of knots and links will be relevant forour arguments. The first of these are the T (2 , n ) torus links. When n is odd, T (2 , n ) is a knotand when n is even, T (2 , n ) is a link of two components. The next are the generalized twist links K ( m, n ). When either m or n or both m and n are even, K ( m, n ) is a knot, and when both m, n are odd, K ( m, n ) is a link of two components. Both T (2 , n ) torus links and generalized twist linksare alternating. A third class of knots that we consider are pretzel knots P ( p, q, r ). When at mostone parameter is even, these are knots, otherwise they are links. See Figure 1.It is well known that the twist knots K (2 , n ) have unknotting number one. More generally, the con-nected sum of K (2 , n ) and a knot J is related to J by a crossing change, so that d ( J K (2 , n ) , J ) = 1for any knot J . Similarly, the T (2 , n ) torus links are the simplest links that can be related to theunknot by a single H(2)-move, and likewise d ( J T (2 , n ) , J ) = 1 for any knot J . There are severalother easily found H(2)-moves that relate the T (2 , n ) torus links and generalized twist links. Thesemoves are shown in Figure 2.In order to prove Theorem 1.4, we will also need to construct two infinite families of knots withspecific Jones polynomials. These are the classes of knots K q , where q ≥ K q,r , where q ≥ r ≥ Knot invariants.
We will consider the quotient graphs QK p and QK p H for several knot andlink invariants that can be derived from the Jones polynomial. The Jones polynomial V L ( t ) is aninvariant of oriented links that takes the form of a Laurent polynomial over the integers. If thelink has an odd number of components, then V L ( t ) is in Z [ t, t − ], and if it has an even number ofcomponents, √ t · V L ( t ) is in Z [ t, t − ] [Jon85]. The span of the Jones polynomial, span( L ), is thedifference between the highest and lowest exponents of the Jones polynomial. UOTIENTS OF THE GORDIAN AND H(2)-GORDIAN GRAPHS 5 (a) d ( T (2 , n ) , U ) = 1. (b) d ( K ( m, n ) , K ( m − , n )) = 1. (c) d ( K ( m, n ) , T (2 , m )) = 1. Figure 2.
Examples of H(2) moves relating generalized twist knots and torusknots.The determinant is an integer valued link invariant which can be obtained by evaluating the Jonespolynomial at t = − | V L ( − | = det( L ) . For connected sums of knots, det( K K ′ ) = det( K ) · det( K ′ ). For knots, the determinant isalways odd, whereas for links the determinant is even. Another invariant we will consider, β ( L ),is an integer-valued knot invariant that can be obtained both from the tricolorability of the knotand with another evaluation of the Jones polynomial, as described below. Although the Jonespolynomial is an invariant of oriented links that is sensitive to mirroring, for the case of knots,none of the invariants of span , det, nor β distinguish a knot from its reverse orientation nor fromits mirror image. Example 2.2.
The Jones polynomial of the T (2 , n ) torus link is given (up to sign) by(1) V T (2 ,n ) ( t ) = t ( n − / (cid:18) t + t + ( − n t n +1 t (cid:19) = t ( n − / t + 1 − t n − X k =0 ( − t ) k ! . We verify this formula in Lemma 3.3 (see also [Jon05]). Here, we have assumed that when n iseven, the T (2 , n ) torus link is oriented in the parallel manner, with both strands pointing up. From(1), it is immediately clear that det( T (2 , n )) = | V T (2 ,n ) ( − | = n and that the span is n .Another invariant we consider is related to tricolorability. The tricolorability of a link originateswith Fox [CF77]; see also the definition in [Ada04]. The tricoloring number of L , tri( L ), is theminimum number of proper, possibly trivial, tricolorings of a link. It can be obtained by evaluatingthe Jones polynomial at t = e πi/ . Proposition 2.3. [Prz98, Theorem 1.13]
Let c ( L ) be the number of link components. The tricol-oring number tri( L ) is given by (2) tri( L ) = 3 | ( V L ( e πi ) | . In [LM86], Lickorish and Millet previously showed that this evaluation of the Jones polynomimal isrelated to the dimension of the first homology group of the branched double cover Σ( L ) of L withcoefficients in Z ,(3) V L ( e πi ) = ± i c ( L ) − ( i √ dim H (Σ( L ); Z ) . CHRISTOPHER FLIPPEN, ALLISON H. MOORE, AND ESSAK SEDDIQ
Example 2.4.
The tricoloring number of the unknot is tri( U ) = 3. For the trefoil knot T (2 , L (3 , H (Σ( L ); Z ) = Z , and so dim H (Σ( L ); Z ) =1. By equations (2) and (3), or by exhaustive enumeration, tri( T (2 , | ( ± ( i √ | = 9.The following properties of tri will also be useful: Lemma 2.5. [Prz98, Lemma 1.4, 1.5] (1) tri( L ) = 3 β for some β ≥ .(2) tri( L ) tri( L ) = 3 tri( L L ) . Notice that Property (1) allows us to consider the positive integer-valued knot invariant β , whichby definition is β ( L ) = log (tri( L )) = 1 + log | V L ( e πi ) | . Notice that for a knot K , equation (3) gives β ( L ) = 1 + dim H (Σ( L ); Z ). By the following result,the tricoloring number is known to give a lower bound on both the Gordian distance and theH(2)-Gordian distance. Proposition 2.6. [AK14, Theorem 5.5] , [Miy11, Proposition 4.2] Let K and K ′ be a pair of knotsrelated by a single H(2)-move or a crossing change. Then (cid:12)(cid:12)(cid:12) V K ( e πi ) /V K ′ ( e πi ) (cid:12)(cid:12)(cid:12) ∈ { , √ , / √ } A similar statement is proven in [Prz98, Lemma 1.5]. We can repackage Proposition 2.6 as a lowerbound on Gordian and H(2)-Gordian distance in terms of β as follows. Proposition 2.7.
Let K and K ′ be a pair of knots. Then | β ( K ) − β ( K ′ ) |≤ d ( K, K ′ ) and | β ( K ) − β ( K ′ ) |≤ d ( K, K ′ ) . Proof.
Suppose that K and K ′ are related by a single crossing change. Then Proposition 2.6 impliesthat (cid:12)(cid:12)(cid:12) V K ( e πi ) /V K ′ ( e πi ) (cid:12)(cid:12)(cid:12) ∈ { , , / } , and so | tri( K ) / tri( K ′ ) |∈ { , , / } . Applying log base 3, we equivalently have that | β ( K ) − β ( K ′ ) |≤
1. Thus for any pair of knots
K, K ′ , | β ( K ) − β ( K ′ ) |≤ d ( K, K ′ ). The proof for d ( K, K ′ ) isidentical. (cid:3) The Kauffman bracket and Jones polynomial
In this section, we will review the definition of the Kauffman bracket [Kau87] and state some ofits properties, most of which can be found in [Kau87] and [EKT03]. We then use the bracket tocalculate the span of the Jones polynomial for several families of knots in a series of lemmas at theend of the section.The
Kauffman bracket h L i ∈ Z [ a, a − ] of an unoriented link diagram D L is a Laurent polynomialdefined by the following axioms:(1) h(cid:13) ⊔ D L i = δ h D L i , where δ = ( − a − a − ), UOTIENTS OF THE GORDIAN AND H(2)-GORDIAN GRAPHS 7 T D T T N T T ∗ UTU T + UT U
Figure 3.
Denominator closure, numerator closure, vertical tangle product andhorizontal tangle sum.(2) h i = a h i + a − h H i ,(3) h(cid:13)i = 1.Recall that the bracket is not an invariant of links because it fails to be invariant under the thirdReidemeister move. The deficiency of the bracket is corrected by a multiplicative factor that recordsthe writhe w ( D L ) of the diagram. This yields the polynomial X L ( a ) = ( − a ) − w ( D L ) h D L i which is indeed a topological invariant of the link L . The polynomial X L ( a ) is equivalent to theJones polynomial V L ( t ) after the change of variable t = a − [Jon85, Kau87].Let T be a two-string tangle diagram. There are two standard closures of T , called the numeratorand denominator closures T N and T D . We denote the horizontal tangle sum by T + U , and thevertical tangle product by T ∗ U . These operations are shown in Figure 3. The mirror of a tangle,obtained by changing all of the crossings, will be denoted − T . The zero tangle is [0] = , and theinfinity tangle is [ ∞ ] = H .By applying the axioms (1) and (2), we can write any tangle as an element in the bracket skeinmodule of a disk with four marked points with basis {h i , h∞i} , i.e. as a linear combination h T i = f T h i + g T h∞i , where f T , g T are polynomials in the ring Z [ a, a − ]. The bracket vector br ( T ) of the tangle T isdefined as br ( T ) = [ f T , g T ] T . The following properties are well known and easy to verify (see forexample, [EKT03]). Proposition 3.1. [EKT03, Proposition 2.2] (1) (cid:20) h T N ih T D i (cid:21) = (cid:20) δ δ (cid:21) br ( T ) .(2) br ( T + U ) = (cid:20) f U g U f U + δg U (cid:21) br ( T ) .(3) br ( T ∗ U ) = (cid:20) δf U + g U f U g U (cid:21) br ( T ) . CHRISTOPHER FLIPPEN, ALLISON H. MOORE, AND ESSAK SEDDIQ [ n ] [ − n ] [ n ] [ − n ] Figure 4.
Vertical and horizontal twist tangles are denoted [1 /n ] and [ n ], respec-tively.Given a tangle decomposition of a link diagram as D L = ( T + U ) N , Proposition 3.1 implies thatthe Kauffman bracket of D L can be expressed as the evaluation of a bilinear form ( T, U )
7→ h D L i in Z [ a, a − ], made explicit by(4) h D L i = br ( T ) (cid:20) δ δ (cid:21) br ( U ) T . Notice that because the span of a polynomial is preserved under multiplication by a monomial,the span of the Kauffman bracket h D L i agrees with the span of X L ( a ). In particular, the spanof the Kauffman bracket and the span of the Jones polynomial are both link invariants related byspan( h D L i ) = 4 span( V L ) . Kauffman [Kau87], Murasugi [Mur87], and Thistlethwaite [Thi87] provedthat for nonsplit alternating links, the span of the Jones polynomial equals the minimal crossingnumber of the link. We will make use of these properties to calculate the bracket polynomialfor certain tangles and the span of the Jones polynomial for several classes of knots. The firstcalculation, giving the bracket vectors for tangles consisting of horizontal or vertical twists, as inFigure 4, is proved in [KW19, Lemma 2.2].
Lemma 3.2. [KW19, Lemma 2.2]
Let n be a positive integer. Then(1) h [ n ] i = a n [0] + a n − P n − k =0 ( − a − ) k [ ∞ ] = a n − (cid:16) − ( − a − ) n a − (cid:17) [ ∞ ] + a n [0] (2) h [1 /n ] i = a − n +2 P n − k =0 ( − a ) k [0] + a − n [ ∞ ] = a − n [ ∞ ] + a − n +2 (cid:16) − ( − a ) n a (cid:17) [0]Using the calculations for horizontal and vertical twists, we can now calculate the Jones polynomi-als for T (2 , n ) torus links. This formula is well-known, but we include a calculation for complete-ness. Lemma 3.3.
Let T (2 , n ) be a torus link for n = 0 that is assumed to have parallel strand orienta-tions when n is even. Then the Jones polynomial of T (2 , n ) is V T (2 ,n ) ( t ) = ( − n +1 t ( n − / (cid:18) t + t + ( − n t n +1 t (cid:19) = ( − n +1 t ( n − / t + 1 − t n − X k =0 ( − t ) k ! . Proof.
By our conventions, the torus link T (2 , − n ) with n > n )-twist, denoted [1 /n ]. By Lemma 3.2(2) together withLemma 3.1(1),(5) h (cid:20) n (cid:21) D i = δa − n + a − n +2 (cid:18) − ( − a ) n a (cid:19) . UOTIENTS OF THE GORDIAN AND H(2)-GORDIAN GRAPHS 9 or alternatively,(6) h (cid:20) n (cid:21) D i = δa − n + a − n +2 n − X k =0 (cid:0) − a (cid:1) k . Let us first consider h [1 /n ] D i written with a geometric series as in (5). Multiplying through by δ gives h (cid:20) n (cid:21) D i = − a − n +2 − a − n − + a − n +2 (cid:18) − ( − a ) n a (cid:19) . Since we assumed that a torus link has parallel strand orientations when n is even, we have thatthe writhe w ( T (2 , − n )) = − n , which gives us the Kauffman polynomial X T (2 , − n ) ( a ) = ( − n +1 a n − a + 1 − a − (cid:0) − a (cid:1) n a !! . Next, we mirror the diagram of T (2 , − n ) to obtain T (2 , n ). Mirroring induces the change of variable t → t − . Following this with a change of variables a → t − , we obtain the Jones polynomial, V T (2 ,n ) ( t ) = ( − n +1 t ( n − / (cid:18) t + t + ( − n t n +1 t (cid:19) . Stated for the case of a knot when n is odd we have V T (2 ,n ) ( t ) = t ( n − / (cid:18) − t − t n +1 + t n +2 − t (cid:19) . Next we will consider the the version of the formula with the summation (6). This proceeds similarlyto the first case and gives the Kauffman polynomial X T (2 , − n ) ( a ) = ( − n +1 a n − a + 1 − a n − X k =0 ( − a ) k ! . After mirroring and applying t → t − , and the change of variables a → t − , we obtain V T (2 , − n ) = ( − n +1 t ( n − / t + 1 − t n − X k =0 ( − t ) k ! . (cid:3) For example, in the case of n = 3, we obtain the polynomial V T (2 , ( t ) = t + t − t . Corollary 3.4.
When n > the span of the Jones polynomial of the T (2 , n ) torus link is n .Proof. This follows because for n >
1, the T (2 , n ) torus links are non-split and alternating. Notethat when n = 0, T (2 ,
0) is the two-component unlink and when n = 1, T (2 ,
1) is the unknot, andthese have spans 1 and 0, respectively. (cid:3)
Lemma 3.5.
The span of the Jones polynomial of the generalized twist link K ( q, p ) is q + p .Proof. This follows because the generalized twist links K ( q, p ) are non-split alternating links ofminimal crossing number q + p . (cid:3) /q − / Figure 5.
The tangle T q := [ − /
3] + [1 /q ]) ∗ [1]. In this example, q = 5.For several statements, we will need to calculate the entire Kauffman bracket vector of a specialtangle T q := ([ − /
3] + [1 /q ]) ∗ [1]) that is shown in Figure 5. Lemma 3.6.
The Kauffman bracket vector of T q := ([ − /
3] + [1 /q ]) ∗ [1]) is br ( T q ) T = " a − q − − a − q − − a − q +6 + a q − a − q − − a − q − + a q − P q − k =1 ( − a ) k , q ≥ " a − a − + a − − a − + a − , q = 1 Proof.
By Lemma 3.12, when q ≥
3, we calculate br ([ − /
3] + [1 /q ]) = (cid:20) f q g q f q + δg q (cid:21) (cid:20) f − / g − / (cid:21) = (cid:20) a − q +2 P q − k =0 ( − a ) k a − q a − q +2 P q − k =0 ( − a ) k + δa − q (cid:21) (cid:20) a − a − + a − a (cid:21) = (cid:20) ( a − q +3 − a − q − + a − q − ) P q − k =0 ( − a ) k a − q − − a − q − + a − q +5 P q − k =1 ( − a ) k (cid:21) , where the indexing of the last sum takes into account cancellation of terms. Applying next Lemma3.13, we obtain br (([ − /
3] + [1 /q ]) ∗ [1]) = (cid:20) − a a a − (cid:21) br ( T q ) T = (cid:20) − a a a − (cid:21) (cid:20) a − q − − a − q − − a − q +6 + a q − a − q − − a − q − + a q − P q − k =1 ( − a ) k (cid:21) = (cid:20) a − q − − a − q − − a − q +6 + a q − a − q − − a − q − + a q − P q − k =1 ( − a ) k (cid:21) , where the first entry of the matrix product is simplified after cancelling numerous terms.Next, consider the special case that q = 1. By Lemma 3.12, we have br ([ − /
3] + [1 /q ]) = (cid:20) a a − − a − (cid:21) (cid:20) a − a − + a − a (cid:21) = (cid:20) a − a − + a − − a − + a − (cid:21) . (cid:3) UOTIENTS OF THE GORDIAN AND H(2)-GORDIAN GRAPHS 11 q (a) The knot K q := T Dq = ([ − / /q ]) ∗ [1]) D q r (b) The knot K q,r := N ( T q + [ r ]) Figure 6. (Left) The knot K q := T Dq = ([ − /
3] + [1 /q ]) ∗ [1]) D . The isotopydemonstrates that K q is an alternating knot. (Right) The knot K q,r . The shadedband indicates where an H(2)-move relates K q,r to K q .In the following two lemmas, we determine the span of the Kauffman bracket for two families ofknots that are built from the special tangle T q . The two classes of knots, K q and K q,r , are shownin Figure 6. We will assume that q ≥ r ≥ q = r = 3 gives the knot K , = 10 . Taking q = 5 and r = 3 gives K , = 12 n ,and taking q = 3 and r = 5 gives K , = 12 n [LM20]. Lemma 3.7.
The span of the Jones polynomial of the knot K q is q + 3 .Proof. The statement follows because for all q , K q is an alternating knot of q + 3 crossings, asshown in Figure 6. (cid:3) Lemma 3.8.
The span of the Jones polynomial of the knot K q,r is q + r + 2 .Proof. The family of knots K q,r is obtained as the numerator closure of the tangle sum T q + [ r ].Therefore equation (4) and Lemma 3.6 may be applied to calculate the bracket of K q,r as it appearsin the diagram of N ( T q + [ r ]). When q ≥ N ( T q + [ r ]) = br ( T q ) (cid:20) δ δ (cid:21) br ([ r ]) T = (cid:20) a − q − − a − q − − a − q +6 + a q − a − q − − a − q − + a q − P q − k =1 ( − a ) k (cid:21) T (cid:20) δ δ (cid:21) (cid:20) a r a r − P r − k =0 ( − a − ) k (cid:21) = (cid:20) a − q − − a − q − − a − q +6 + a q − a − q − − a − q − + a q − P q − k =1 ( − a ) k (cid:21) T (cid:20) δa r + a r − P r − k =0 ( − a − ) k a r + δa r − P r − k =0 ( − a − ) k (cid:21) After noticing that the second entry of the last vector is a telescoping sum that reduces to − a − r ,we obtain the product(7) N ( T q + [ r ]) = (cid:0) a − q − − a − q − − a − q +6 + a q − (cid:1) δa r + a r − r − X k =0 ( − a − ) k ! + a − q − − a − q − + a q − q − X k =1 ( − a ) k ! (cid:0) − a − r (cid:1) . We need only extract the terms of highest and lowest exponents from equation (7). The polynomialreduces to − a q + r + (interior terms) − a − r − q − , from which it is clear that the span of the bracket is 4( q + r +2) and the span of the Jones polynomialis q + r + 2.When q = 1, N ( T q + [ r ]) = br ( T q ) (cid:20) δ δ (cid:21) br ([ r ]) T = (cid:20) a − a − + a − − a − + a − (cid:21) T (cid:20) δ δ (cid:21) (cid:20) a r a r − P r − k =0 ( − a − ) k (cid:21) = (cid:20) a − a − + a − − a − + a − (cid:21) T (cid:20) δa r + a r − P r − k =0 ( − a − ) k − a − r (cid:21) = − a r +4 + (interior terms) − a − r − . The span is then 4( r + 3) = 4( r + q + 2), and the span of the Jones polynomial is q + r + 2. (cid:3) Finally, we state a lemma that we will require when considering the spans of the Jones polynomialsof knots.
Lemma 3.9.
The span of the Jones polynomial of a knot cannot be one or two.Proof.
We will show, equivalently, that there are no knots such that the normalized bracket poly-nomial X K ( a ) has span 4 or 8. Take K ′ to be the unknot, and let K be any other knot. Ganzellshowed that for any pair of knots K, K ′ , the difference of the polynomials X K ( a ) − X K ′ ( a ) isdivisible by a − X K ( a ) is of the form X K ( a ) = ( c M a M + · · · c m a m )( a −
1) + 1 , which implies that the span of X K ( a ) is max {
12 + M, } − min { m, } . There are four cases. Ifspan X K ( a ) = 12 + M − m ≥
12 or if span X K ( a ) = 0, then clearly span X K ( a ) cannot be 4 or 8.In the case that span X K ( a ) = 11 + M is equal to 4 or 8, then M = − −
3, which contradictsthat M ≥ m ≥
0. In the case that span X K ( a ) = 1 − m is equal to 4 or 8, then m = − −
7. Yetbecause max {
12 + M, } = 1, we have M ≤ −
11, which contradicts that M ≥ m ≥ − (cid:3) Hyperbolicity of quotient knot graphs
Hyperbolic graphs.
Let G be a graph with vertex set V ( G ) and edge set E ( G ). We onlyconsider graphs that are connected, undirected, simple, and either finite or countably infinite. Aconnected graph G can be endowed with a metric, making it into a metric space ( X G , d ) as follows(here, we follow the notation and conventions of [JLM20, Section 2.1]). We first define a metric d ( v, w ) on vertices v, w ∈ V ( G ) as the minimum number of edges needed to connect v to w , andimplicitly assume that every edge in the graph has length 1. We then extend the distance to anypair of points x, y on an edge e ∈ E ( G ) by d ( x, y ) = | x − y | . The metric space ( X G , d ) can beconsidered a geodesic metric space . In particular, the distance between any pair of points in ( X , d )is realized by at least one rectifiable shortest path. Given three distinct points x, y, z in ( X G , d ), a geodesic triangle { [ xy ] , [ yz ] , [ xz ] } is a triple of geodesics connecting the vertices of the triangle. For UOTIENTS OF THE GORDIAN AND H(2)-GORDIAN GRAPHS 13 δ ≥
0, we say that a geodesic triangle is δ -thin if each edge is contained in a closed δ -neighborhoodof the union of the remaining two. The metric space X G is δ -hyperbolic if every geodesic trianglein X G is δ -thin, and we call X G Gromov hyperbolic (or just hyperbolic) if it is δ -hyperbolic forsome δ ≥
0. We will generally abuse notation and let G (or K ) refer to both an abstract graph anda metric graph where the distance function is understood.Graphs provide many examples of hyperbolic metric spaces. Example 4.1.
For example, any connected acyclic graph (i.e. a tree) is δ -hyperbolic for all δ ≥ Lemma 4.2. If diam( G ) = d , then G is d -hyperbolic.Proof. Take a geodesic triangle with edges [ xy ] , [ yz ] , [ xz ]. Without loss of generality any point p ∈ [ xy ] is contained in N d/ ([ xz ] ∪ [ yz ]). (cid:3) Proofs of hyperbolicity for quotients of the Gordian and H(2)-Gordian graphs.
Inorder to prove Theorem 4.4, we first observe that the indexing set will correspond with all naturalnumbers.
Lemma 4.3.
The invariant β takes all natural number values.Proof. First notice that β = 1 for the unknot. Next, let n T (2 ,
3) be the connected sum of n trefoil knots. Since tri( T (2 , n T (2 , n +1 . So β = n + 1for n T (2 , (cid:3) In section 2.3 we discussed that the tricoloring invariant β is related to the first homology of thebranched double cover with Z / Z -coefficients, and that these invariants give a lower bound on H(2)-Gordian distance. In particular, the H(2)-move and β are compatible in the sense of Definition 1.2of [JLM20]. The following statement can therefore be seen as a corollary of Theorem 5.1 of [JLM20].However, we give a direct argument. Theorem 4.4.
The quotient graphs QK β and QK β H are δ -hyperbolic for all δ .Proof. We will prove that QK β is an infinite path graph P N . From Lemma 4.3, we see that theinvariant β realizes all natural numbers, and so the vertex set of QK β is N . Using the same family ofconnected sums of trefoils and the fact that the trefoil is related to the unknot by a crossing change,then there exists an edge between n T (2 ,
3) and n +1 ( T (2 ,
3) for all n ≥
1. By Proposition 2.7,we also know that a crossing change cannot change β by more than one, therefore no other edgescan exist in this graph. Thus QK β is an infinite path graph P N . Since P N is connected and acyclic, QK β is δ -hyperbolic for all δ .The proof for QK β H is identical after noticing that the trefoil knot is also unknotted with a singleH(2)-move. (cid:3) Theorem 4.5.
The quotient graphs QK det and QK det H are δ -hyperbolic for all δ ≥ .Proof. We will first show that the quotient graph QK det has diam( QK det ) ≤
2. Note that theequivalence classes span the odd natural numbers, and that there exists a representative twist knot K ( n,
2) for each equivalence class [2 n +1] for n ≥
1. Then every equivalence class [2 n +1] is adjacentto [1], because twist knots K ( n,
2) have unknotting number one. Therefore diam( QK det ) ≤
2. ByLemma 4.2 it is δ -hyperbolic for all δ ≥ QK det H , we will similarly show that the quotient graph has diameter less than or equal two. Here,there exists a representative torus knot T (2 , n + 1) for each equivalence class [2 n + 1] for n ≥ n + 1] is adjacent to [1], because T (2 , n + 1) torus knots haveH(2)-unknotting number one. Therefore diam( QK det ) ≤
2. Again by Lemma 4.2, it is δ -hyperbolicfor all δ ≥ (cid:3) Recall from Lemma 3.9 that there are no knots whose Jones polynomials have span 1 or span 2.Thus we define g QK span to be QK span − { [1] , [2] } . Theorem 4.6.
The quotient graph g QK span is δ -hyperbolic for all δ ≥ .Proof. We claim that diam( g QK span ) ≤
2. This follows because every equivalence class [ n ], for n ≥
3, is adjacent to [0]. This adjacency is realized by the twist knots K ( m, m + 2. The result then follows from Lemma 4.2. (cid:3) We similarly define g QK span H to be QK span H − { [1] , [2] } . Theorem 4.7.
The quotient graph e K span H is δ -hyperbolic for all δ ≥ .Proof. We claim that diam( e K span H ) ≤
2. To see this, first observe that every equivalence class[2 n + 1], for odd n ≥
1, is adjacent to [0]. This adjacency is realized by the T (2 , n + 1) torus knots,which have span 2 n + 1. Additionally, for every n ≥
3, the class [ n ] is adjacent to [ n + 1]. Thisadjacency is realized by the twist knots K ( n,
1) and K ( n + 1 , (cid:3) Remark 4.8.
It is possible to construct distinct pairs of knots related by a crossing change thatshare the invariants considered here. For example, this could be done with pretzel knots of the form P ( q, − , − q ) for q ≥ P ( q, − , − q ) is related to P ( q, , − q ) by a crossing change.Because P ( q, , − q ) ≃ P ( − q, , q ) is the mirror of P ( q, − , − q ), their Jones polynomials have thesame span, and the values of β and det will agree. However, here we consider only simple knotgraphs (ignoring self edges). 5. Graph isomorphism type
In this section we study the possible isomorphism types of the graphs QK p and QK p H for p = β, detand span. In the case of the invariant β , the isomorphism types are easily identified. Theorem 1.2.
Each of QK β and QK β H are isomorphic to the path graph P N . UOTIENTS OF THE GORDIAN AND H(2)-GORDIAN GRAPHS 15
Proof.
It follows immediately from the proof of Theorem 4.4 that the vertex sets V ( QK β ) = N and V ( QK β H ) = N , and that the edge sets E ( QK β ) = { [ n ] , [ n + 1] } and E ( QK β H ) = { [ n ] , [ n + 1] } for all n ∈ N . (cid:3) Unlike the quotient knot graphs for β , the graphs QK det and QK det H have finite diameter. We canexploit the fact that the determinant is multiplicative over connected sums in order to obtain thefollowing two statements. By K ∞ , we mean the complete graph on countably many vertices. Theorem 1.3.
In both QK det and QK det H , the subgraph induced by the set of vertices { [ n k ] , k ≥ } is isomorphic to K ∞ for all odd n ≥ .Proof. Let n be positive and odd. In QK det , any vertex [ n i ] may be realized by the twist knot K (2 , n i − ) because det( K (2 , n i − )) = n i . Any such twist knot is related to the unknot, which hasdet( U ) = 1, establishing the edges { [ n ] , [ n i ] } . Next for n ≥
3, consider any two vertices [ n i ] , [ n j ],with i < j , in the set { [ n k ] , k ≥ } . The vertex [ n j ] is realized by a connected sum of twist knots,with det( K (2 , n i −
12 ) K (2 , n j − i −
12 )) = n i · n j − i = n j . The connected sum K (2 , n i − ) K (2 , n j − i − ) is related to K (2 , n i − ) by a single crossing changein the clasp of K (2 , n j − i − ) which unknots that summand. This gives the edge { [ n i ] , [ n j ] } . Thusall vertices { [ n k ]; n ≥ , k ≥ } are adjacent, and so the induced subgraph is isomorphic to K ∞ .In QK det H , the proof is similar, except that we use T (2 , n + 1) torus knots rather than twist knots.Since det( T (2 , n + 1)) = 2 n + 1 and T (2 , n + 1) is related to the unknot by a single H(2)-move, thisgives the edges { [ n ] , [ n i ] } . For the edges { [ n i ] , [ n j ] } , with n ≥ , j > i ≥
1, we take the connectedsum T (2 , n i −
12 ) T (2 , n j − i −
12 ) , which has determinant n j , and is related to T (2 , n i − ) of determinant n i by the single H(2)-movewhich unknots the second summand. (cid:3) Example 5.1.
It is certainly possible to construct additional edges in the graphs QK det and QK det H ,and we describe here three examples.(1) By an argument similar to that of Theorem 1.3, there exist edges in QK det and QK det H forany pair of vertices [ n ] , [ m ], with n, m odd and where m divides n . In this case, let n = dm .Then for QK det , the relevant crossing change is found in the second summand of K (2 , m −
12 ) K (2 , d −
12 ) , and for QK det H the relevant H(2)-move is found in the second summand T (2 , m −
12 ) T (2 , d −
12 ) . (2) In QK det , the edges { [2 n + 1] , [2 n + 5] } for n ≥ K (2 , n ), K (2 , n + 2). In the graph QK det H , the edges { [2 n + 1] , [2 n + 3] } for n ≥ K (2 , n ), K (2 , n + 1).(3) Consider the P ( p, q, r ) pretzel knots, where in order to obtain a knot we assume that atmost one of p, q, r is even. Is well known that the P ( p, q, r ) pretzel knots have determinant | pq + pr + qr | . It is also easy to see that the knot P ( p, q, r ) is related to P ( p, q − , r )via a crossing change and is related to P ( p, q − , r ) via an H(2)-move. In particular,for any p ≥
1, we have that det( P ( p, , p + 3 and det( P ( p, , p + 2 anddet( P ( p, , p + 1. Hence, there exist edges in QK det relating 2 p + 1 and 4 p + 3 forall p ≥ QK det H relating 2 p + 1 and 3 p + 2 for all p ≥ { [2 n + 1] , [2 m + 1] } for any n, m . Therefore, we ask: Question 5.2.
Are the graphs QK det and QK det H isomorphic to K ∞ ? Next we turn our focus to the graphs g QK span and g QK span H . Recall from Lemma 3.9 that for knotsthe invariant span takes values in { , , , . . . } , therefore we remove n = 1 , g QK span and g QK span H . Proposition 5.3.
The edges { [0] , [ n ] } are in E ( g QK span ) for all n ≥ and the edges { [ m ] , [ n ] } arein E ( g QK span ) for all m, n , where n − ≥ m ≥ .Proof. The twist knots K (2 , n −
2) have span n for all n ≥
3. Because these all have unknottingnumber one, they realize the edges { [0] , [ n ] } . Next, consider an arbitrary pair [ m ] , [ n ], where m ≥ n ≥ m + 2. If n = m + 2, then the twist knots K (2 , m ) and K (2 , m −
2) are related by a crossingchange and have spans n and m , respectively. If n > m + 2, then K (2 , m −
2) and the connectedsum K (2 , m − K (2 , n − m −
2) have spans m and n , respectively, because span is additive overconnected sum. An edge relates this pair because there is a crossing change that unknots the secondsummand. These knots realize the edges { [ m ] , [ n ] } when m ≥ n ≥ m + 2. (cid:3) Notice that Proposition 5.3 nearly shows that the graph g QK span is isomorphic with K ∞ . The onlymissing edges in the proof are those of the form { [ n ] , [ n + 1] } . We are in fact aware of some sporadicpairs of knots related by a crossing change whose spans differ by one (see for example [Moo10].)These pairs are listed below and shown in Figure 7. • The twist knot K (2 ,
3) = 5 (span 5) and the pretzel knot P (3 , − ,
2) = 8 (span 6). • The pretzel P (3 , , −
2) = T (3 ,
4) = 8 (span 5) and the pretzel knot P (5 , , −
2) (span 6). • The pretzel P (5 , , −
2) = T (3 ,
5) = 10 (span 6) and T (2 ,
7) = 7 (span 7). • The knot 7 (span 7) and the pretzel knot P ( − , ,
2) = 10 (span 8).However, we know of no general construction of knots related by a crossing change that alters thespan by one. Therefore we ask:
UOTIENTS OF THE GORDIAN AND H(2)-GORDIAN GRAPHS 17
Figure 7.
Each grey disk indicates where a crossing change relates the knot pic-tured to another knot whose span of the Jones polynomial differs by one.
Question 5.4.
Is there an edge between [ n ] and [ n + 1] for all n ≥ in g QK span ? For the following statement, we will consider a quotient of the H(2)-Gordian graph of links , denoted L H , rather than the version for knots. In particular, the vertex set of L H consists of isotopy classesof unoriented links. An H(2)-move may or may not be component preserving, and an edge inthis graph exists whenever a pair of links L, L ′ is related by any H(2)-move. We remark thatwhile the Jones polynomial of links is sensitive to orientation, and the H(2)-move may or may notrespect strand orientation, the span of the Jones polynomial (or Kauffman bracket) is insensitiveto orientation. Thus the quotient graph QL span H is well-defined for unoriented links. Theorem 1.4.
The graph QL span H is isomorphic to K ∞ .Proof. We take our indexing set for K ∞ to be n ≥
0. For every pair n, m , we will construct anedge in QL span H . There are seven cases in our analysis; the first three are specific to the vertices[0] , [1] , [2], and the latter cases are more general. We will use knots to construct edges wheneverpossible (see Remark 5.5 below), even if a simpler construction can be made with links.Case 1. Recall that span T (2 ,
0) = 1, span T (2 ,
1) = 0, and span T (2 , n ) = n for all n ≥
2. Edgesincident to [0] are realized by an H(2)-move that unknots the T (2 , n ) torus knot or link, as seen inFigure 2(a).Case 2. Consider edges incident to [1]. The edge { [1] , [2] } is realized by an H(2)-move relating thetwo-component and three-component unlinks. If n >
2, the edge { [1] , [ n ] } is found by taking thedisjoint union U ⊔ T (2 , n − n , andis related to the two-component unlink (of span 1) by the banding that unties the torus link. SeeFigure 8.Case 3. Consider edges incident with [2]. The edge { [2] , [3] } is realized by the Hopf link and trefoilknot, and the edges { [2] , [ n ] } for n ≥ T (2 , T (2 , n − { [ n ] , [ n + 1] } for n ≥
2. These edges are realized by torus knots and links because T (2 , n ) is related to T (2 , n + 1) by an H(2)-move. Alternatively, these edges can be realized bypairs of twist knots K (2 , n −
2) and K (2 , n − K (2 , n −
2) has span n , and thesepairs are related by a single H(2)-move in the twist region, as in Figure 2(b).Case 5. Consider { [ m ] , [ n ] } when n − m > n > m ≥ K q and K q,r where q = m − r = n − m + 1. Here, m ≥ q ≥ n ≥ r ≥ K q Figure 8.
The connected sums and bandings of links in cases (2), and (3) ofTheorem 1.4.is q + 3 = m and by Lemma 3.8, the span of K q,r is q + r + 2 = n . These knots are related by theshaded band shown in Figure 6.Case 6. Consider { [ m ] , [ n ] } when n − m > n > m ≥ n − m is even, thegeneralized twist link K ( m, n − m ) is a knot. An H(2)-move relates the torus knot T (2 , m ), whichhas span m , to K ( m, n − m ), which has span n . This is shown in Figure 2(c).Case 7. Consider { [ m ] , [ n ] } when n − m > n > m ≥ n − m is odd, and so T (2 , n − m ) is a knot. The connected sum T (2 , n − m ) K (2 , m −
2) has span n ,and is related to K (2 , m − m , by the H(2)-move that unknots the torus knotsummand.This case analysis establishes that { [ n ] , [ m ] } ∈ V ( QL span H ) for all n, m ∈ N ∪ { } . (cid:3) Remark 5.5.
Notice that most of the edges constructed in the proof above are also contained inthe graph g QK span H . The only exception are the edges from [0] to [2 n ], realized by the torus links T (2 , n ), and the edges that are incident with the vertices [1] and [2]. This prompts us to ask: Question 5.6.
Are there edges { [0] , [2 n ] } , for n ≥ , realized by pairs of knots? Is the graph g QK span H isomorphic to K ∞ ? Acknowledgements.
The authors appreciate the support of the Honors Summer UndergraduateResearch Program (HSURP) at Virginia Commonwealth University.
References [Ada04] Colin C. Adams.
The knot book . American Mathematical Society, Providence, RI, 2004. An elementaryintroduction to the mathematical theory of knots, Revised reprint of the 1994 original.[AK14] Tetsuya Abe and Taizo Kanenobu. Unoriented band surgery on knots and links.
Kobe J. Math. , 31(1-2):21–44, 2014.[BCJ +
17] Ryan Blair, Marion Campisi, Jesse Johnson, Scott A. Taylor, and Maggy Tomova. Neighbors of knots inthe gordian graph.
The American Mathematical Monthly , 124(1):4–23, 2017.[BK20] Sebastian Baader and Alexandra Kjuchukova. Symmetric quotients of knot groups and a filtration of theGordian graph.
Math. Proc. Cambridge Philos. Soc. , 169(1):141–148, 2020.[CF77] Richard H. Crowell and Ralph H. Fox.
Introduction to knot theory . Springer-Verlag, New York-Heidelberg,1977. Reprint of the 1963 original, Graduate Texts in Mathematics, No. 57.[EKT03] Shalom Eliahou, Louis H. Kauffman, and Morwen B. Thistlethwaite. Infinite families of links with trivialJones polynomial.
Topology , 42(1):155–169, 2003.[Gan14] Sandy Ganzell. Local moves and restrictions on the Jones polynomial.
J. Knot Theory Ramifications ,23(2):1450011, 8, 2014.
UOTIENTS OF THE GORDIAN AND H(2)-GORDIAN GRAPHS 19 [GG16] Jean-Marc Gambaudo and ´Etienne Ghys. Braids and signatures. In
Six papers on signatures, braids andSeifert surfaces , volume 30 of
Ensaios Mat. , pages 174–216. Soc. Brasil. Mat., Rio de Janeiro, 2016.Reprinted from Bull. Soc. Math. France
33 (2005), no. 4, 541–579 [ MR2233695].[HNT90] Jim Hoste, Yasutaka Nakanishi, and Kouki Taniyama. Unknotting operations involving trivial tangles.
Osaka J. Math. , 27(3):555–566, 1990.[HU02] Mikami Hirasawa and Yoshiaki Uchida. The Gordian complex of knots.
J. Knot Theory Ramifications ,11(3):363–368, 2002. Knots 2000 Korea, Vol. 1 (Yongpyong).[IJ11] Kazuhiro Ichihara and In Dae Jong. Gromov hyperbolicity and a variation of the gordian complex.
Pro-ceedings of the Japan Academy, Series A, Mathematical Sciences , 87(2):17–21, 2011.[JLM20] Stanislav Jabuka, Beibei Liu, and Allison H. Moore. Knot graphs and Gromov hyperbolicity.arXiv:1912.03766v1 [math.GT], 2020.[Jon85] Vaughan F. R. Jones. A polynomial invariant for knots via von Neumann algebras.
Bull. Amer. Math.Soc. (N.S.) , 12(1):103–111, 1985.[Jon05] Vaughan F. R. Jones. The Jones polynomial. Unpublished note, 8 2005.[Kau87] Louis H. Kauffman. State models and the Jones polynomial.
Topology , 26(3):395–407, 1987.[KW19] Takeyoshi Kogiso and Michihisa Wakui. A bridge between Conway–Coxeter friezes and rational tanglesthrough the Kauffman bracket polynomials.
J. Knot Theory Ramifications , 28(14):1950083, 40, 2019.[LM86] W. B. R. Lickorish and K. C. Millett. Some evaluations of link polynomials.
Comment. Math. Helv. ,61(3):349–359, 1986.[LM20] Charles Livingston and Allison H. Moore. Knotinfo: Table of knot invariants. URL: knotinfo.math.indiana.edu , June 2020.[Miy11] Yasuyuki Miyazawa. Gordian distance and polynomial invariants.
J. Knot Theory Ramifications ,20(6):895–907, 2011.[Moo10] Hyeyoung Moon.
Calculating knot distances and solving tangle equations involving Montesinos links . PhDthesis, The University of Iowa, 7 2010.[Mur87] Kunio Murasugi. Jones polynomials and classical conjectures in knot theory.
Topology , 26(2):187–194,1987.[NO09] Yasutaka Nakanishi and Yoshiyuki Ohyama. The Gordian complex with pass moves is not homogeneouswith respect to Conway polynomials.
Hiroshima Math. J. , 39(3):443–450, 2009.[Ohy06] Yoshiyuki Ohyama. The C k -Gordian complex of knots. J. Knot Theory Ramifications , 15(1):73–80, 2006.[Prz98] J´ozef H. Przytycki. 3-coloring and other elementary invariants of knots. In
Knot theory (Warsaw, 1995) ,volume 42 of
Banach Center Publ. , pages 275–295. Polish Acad. Sci. Inst. Math., Warsaw, 1998.[Thi87] Morwen B. Thistlethwaite. A spanning tree expansion of the Jones polynomial.
Topology , 26(3):297–309,1987.[ZY18] Kai Zhang and Zhiqing Yang. A note on the Gordian complexes of some local moves on knots.
J. KnotTheory Ramifications , 27(9):1842002, 6, 2018.[ZYL17] Kai Zhang, Zhiqing Yang, and Fengchun Lei. The H ( n )-Gordian complex of knots. J. Knot TheoryRamifications , 26(13):1750088, 7, 2017., 26(13):1750088, 7, 2017.