Foundations of the Colored Jones Polynomial of singular knots
aa r X i v : . [ m a t h . G T ] S e p FOUNDATIONS OF THE COLORED JONES POLYNOMIAL OF SINGULARKNOTS
MOHAMED ELHAMDADI AND MUSTAFA HAJIJ
Abstract.
This article gives the foundations of the colored Jones polynomial for singular knots.We extend Masbum and Vogel’s algorithm [30] to compute the colored Jones polynomial for anysingular knot. We also introduce the tail of the colored Jones polynomial of singular knots and useits stability properties to prove a false theta function identity that goes back to Ramanujan.
Contents
1. Introduction 12. The Kauffman Bracket Skein Module 22.1. The Jones-Wenzl Idempotents 32.2. The Colored Jones Polynomial 43. Singular Knots and the Colored Jones Polynomial 43.1. Computing the Colored Jones Polynomial of Singular Links 64. The tail of the Colored Jones Polynomial for Singular Knots 125. Conclusion 155.1. Acknowledgment 15References 161.
Introduction
The colored Jones polynomial J n,L ( q ) of a link L is a sequence of Laurent polynomials in thevariable q . The label n is a positive integer which usually stands for the color. The study ofthe Jones polynomials and, in general, quantum invariants have attracted much attention in thepast 30 years. Particularly over the past decade, a growing interest has appeared in regards ofthe coefficients of the colored Jones polynomial. The interest stems mainly from certain stabilitybehavior these coefficients have for adequate knots and links. The stability of the colored Jonespolynomial was first observed by Dasbach and Lin in [7], where they showed that for an alternatinglink L the absolute values of the first and last three leading coefficients of J n,L ( q ) are independentof the color n for sufficiently large values of n . This finding was used to derive an upper boundfor the volume of the complement of alternating prime non-torus knots in terms of the leading twoand last two coefficients of J ,K ( q ). In the same article, Dasbach and Lin conjectured that thefirst n coefficients of J n +1 ,L ( q ) agree with the last n coefficients of J n,L ( q ) for any alternating link L . Thus, the tail of the colored Jones polynomial for such a stable sequence is a q − series whosefirst n coefficients agree with the first n coefficients of J n,L ( q ). This stability was then proven foradequate knots by Armond in [2]. Furthermore, Garoufalidis and Lˆe [13] gave another proof foralternating knots using different techniques. They also proved that higher order stability occursfor alternating knots. Lee [19] extended Armond’s result to all links and showed that the tail of alink L is trivial if and only if L is non A-adequate. Recently, Lee gave categorified version of herresult in [19] proving a conjecture of Rozansky [38] stating that the categorification of the coloredJones polynomial of a non A-adequate link has a trivial tail homology. Other work on the stability of the colored Jones polynomial can be found in [15] where the stability is shown using simple skeintheoretic techniques. The work of Armond and Dasbach was then extended to the quantum spinnetwork in [17] and [16].One of the primary interests of these coefficients seems to be driven from their relation to thefamous Ramanujan-type q -series. One of the earliest connections with the Ramanujan type q -serieswas observed in [18], in which the author studied the asymptotic behaviors of the colored Jonespolynomials. The q -series associated with the colored Jones polynomial exhibits many interestingproperties. In fact, for many knots with small crossings, these q -series are equal to theta functionsor false theta functions. More interestingly, the study of the tail has been used to prove Andrews-Gordon identities for the two-variable, Ramanujan theta function in [3] and corresponding identitiesfor the false theta function in [16]. These two families of q -series identities were obtained frominvestigating (2 , p )-torus knots. For q -series techniques proving these identities, refer to [27]. Thecolored Jones polynomial was extended to singular knots using skein theory in [4]. In this paper,extended Masbum and Vogel’s algorithm [30] to compute the colored Jones polynomial for anysingular link. Furthermore, we investigate the stability of these coefficients for certain singulartorus knots and show how they can be used to prove natural Ramanujan-type identities.Since we want this article to be self contained we include reviews of the necessary material forthe convenience of the reader. The paper is organized as follows: In section 2 we give the basicsof the Kauffman bracket skein module and the colored Jones polynomial. In section 3 we give thebasics of the colored Jones polynomial for singular links and extend Masbum and Vogel’s algorithm[30] to compute the colored Jones polynomial for singular links. In Section 4 we compute the tailof the colored Jones polynomial of singular torus knots and we show that this tail gives a naturalRamanujan-type q -series identity.2. The Kauffman Bracket Skein Module
Let M be an oriented 3-manifold. A framed link in M is an oriented embedding of a disjointunion of oriented annuli in M . A framed point in the boundary ∂M of M is a closed interval in ∂M . A band in M is an oriented embedding of I × I into M that meets ∂M orthogonally at twoframed points x and y in ∂M . Definition 2.1. [33] Let M be an oriented 3-manifold and let A be an invertible element in acommutative ring R with a unit. Let L M denotes the set of all isotopy classes of unoriented framedlinks in M . The empty link is considered to be an element of L M . Let RL M be the free R -modulegenerated by L M . Let R ( M ) is the submodule of RL M generated by all expressions of the form(1) − A − A − , (2) L ⊔ + ( A + A − ) L, where L ⊔ is the disjoint union of a framed link L in M and the trivial framed knot . The Kauffman bracket skein module of the 3-manifold M is the quotient module S ( M, R , A ) = RL M /R ( M ) , (2.1)When the context is clear, we will write S ( M ) instead of S ( M, R , A ). The definition of theKauffman bracket skein module can be extended to 3-manifolds with boundaries. Let x , · · · , x n be a set, possibly empty, of designated framed points on ∂M . Let L M be the set of all surfaces in M decomposed into a union of finite number of framed links and bands joining the points { x i } ni =1 .The relative Kauffman bracket skein module is defined to be the quotient module S ( M, R , A, { x i } ni =1 ) = RL M /R ( M ) . (2.2) OUNDATIONS OF THE COLORED JONES POLYNOMIAL OF SINGULAR KNOTS 3
Note that the construction of the relative Kauffman bracket skein module is functorial in the sensethat an embedding of oriented 3-manifolds with 2 n (framed) points on the boundaries j : ( M, { x i } ni =1 ) ֒ → ( M ′ , { y i } ni =1 ) (2.3)induces a homomorphism of R -modules S ( M, R , A, { x i } ni =1 ) → S ( M ′ , R , A, { y i } ni =1 ) . (2.4)If the 3-manifold M is homeomorphic to F × I where F an oriented surface with a finite set ofpoints (possibly empty) in its boundary ∂F and I is an interval, then one can project framed linksin M to link diagrams in F .It is well known ([33]) that the Kauffman bracket skein module of the 3-sphere S is free on theempty link, that is S ( S ) = R . Now we consider the relative Kauffman bracket skein module of D = I × I × I with 2 n marked points on its boundary ∂D . The first n points are placed on thetop edge of D and the other n points on the bottom edge. Recall that the relative skein moduledoes not depend on the exact position of the points { x i } ni =1 . However, one needs to specify theposition here in order to define an algebra structure on S ( D , R , A, { x i } ni =1 ).Let S and S be two elements in L D such that ∂S j , where j = 1 ,
2, consists of the points { x i } ni =1 that we specified above.Define S × S to be the surface in D obtained by attaching S on the top of S and then compressthe result to D . This multiplication extends to a well-defined multiplication on S ( D , R , A, { x i } ni =1 ).With this multiplication the module S ( D , R , A, { x i } ni =1 ) becomes an associative algebra over R known as the n th Temperley-Lieb algebra
T L n . The ring R will be the field Q ( A ) generated by theindeterminate A over the rational numbers through the rest of the paper.2.1. The Jones-Wenzl Idempotents.
Our invariant is defined in terms of the Jones-Wenzl idem-potent (JWI). This is an element in
T L n denoted by f ( n ) that has played a crucial role in theunderstanding of the Temperley-Lieb algebra and its applications. The idempotent has a centralrole in defining the SU (2) Witten-Reshetikhin-Turaev Invariants [25, 28, 39]. It also has a majorimportance in the colored Jones polynomial and its applications [5, 16, 17, 39, 41], and quantum spinnetworks [30]. The definition of the projector goes back to Jones [21]. The recursive formula wewill use here goes back to Wenzl [44]: n = n − − (cid:16) ∆ n − ∆ n − (cid:17) n − n − n − , = (2.5)where ∆ n = ( − n A n +1) − A − n +1) A − A − . The idempotent satisfies the following characterizing properties:
OUNDATIONS OF THE COLORED JONES POLYNOMIAL OF SINGULAR KNOTS 4 n = n , n − i − in = 0 , (2.6)The first property is called the idempotency property of the JWI and the second property is calledthe annihilation property. The JWI also satisfies the following: n mm + n = m + n , ∆ n = n (2.7)2.2. The Colored Jones Polynomial.
The singular knot invariant that we are studying in thispaper is a generalization of the colored Jones polynomial for classical knots. For this reason wequickly review the basics of the colored Jones polynomial. Given a framed link L in S . Wedecorate every component of L , according to its framing, by the n th Jones-Wenzl idempotent andtake the evaluation of the decorated framed link as an element of S ( S ). Up to a power of ± A , thisdepends on the framing of L , the value of this element is defined to be the n th (unreduced) coloredJones polynomial ˜ J n,L ( A ). One recover the reduced Jones polynomial by a change of variable anda division by ∆ n . Namely, J n +1 ,L ( q ) = ˜ J n,L ( A )∆ n (cid:12)(cid:12)(cid:12)(cid:12) A = q / (2.8)One of the primary focus of this article is the coefficients stability of an extension of the coloredJones polynomial to singular knots. We give more details about the stability properties that thecolored Jones polynomial satisfies in section 4.3. Singular Knots and the Colored Jones Polynomial
We give a quick introduction to the basics of singular knot theory. For more details see [14] and[12]. A singular link on n components is the image of a smooth immersion of n circles in S thathas finitely many double points. The double points are usually called singularities . See Figure 1. Figure 1.
Regular and singular crossingsSingular knots are also called rigid -valent graphs . The double points are then referred toas vertices. Two singular knots are ambient isotopic if there is an orientation preserving self-homeomorphism of S that takes one link to the other and preserves a small rigid disk around eachvertex. In this paper we work with singular link diagrams which are projections of the link on the OUNDATIONS OF THE COLORED JONES POLYNOMIAL OF SINGULAR KNOTS 5 plane such that the information at each crossing is preserved by leaving a little break in the lowerstrand. In this context a version of Reidemeister’s theorem holds for singular links. Namely, twosingular links L and L are ambient isotopic if and only if one can obtain a diagram of L from adiagram of L by a finite sequence of classical and singular Reidemeister moves shown in Figure 2. Figure 2.
Classical and singular Reidemeister moves. The top three moves are RI , RII and
RIII are the classical moves and the bottom moves denoted
RIV (twodiagrams on the left) and RV are the singular Reidemeister.Similar to the case of classical knot theory, if one does not allow the Reidemeister move RI thenone obtains what is called regular isotopy of singular links.Singular knot theory have gained a lot of interest in the past two decades. This was primarilymotivated by Vassiliev invariants [43]. In particular, most classical knot theory invariants havebeen extended to the singular versions. For instance, Fiedler [12] extended the Jones and Alexanderpolynomials to singular knot invariants. In [22] Juyumaya and Lambropoulou constructed Jones-type invariant for singular links using a Markov trace on a version of the Hecke algebra. Gemein [14]studied certain extensions of the Artin and the Burau representations to the singular braid monoid.Bataineh, Elhamdadi and Hajij extended the colored Jones polynomial definition to singular knotsin [4]. Churchill, Elhamdadi, Hajij and Nelson showed that the set of colorings by some algebraicstructures is an invariant of unoriented singular links and used it to distinguish several singularknots and links in [8]. This work was extended to oriented singular knots in [9]. Kauffman and Vogel[26] extended the Dubrovnik polynomial to an invariant of singular knots in R . Kauffman gave aone variable specialization of the Kauffman-Vogel polynomial utilizing the Jones-Wenzl projector.We will denote this invariant by [ . ] . This invariant is defined via the following axioms:(1) " = (2) " =
11 11 22 22 (3) [ ] =In [4] we gave a natural generalization for [ . ] as follows. Definition 3.1.
Let L be a singular link. For an integer n ≥
1, the rational function [ L ] n can bedefined by the following rules: OUNDATIONS OF THE COLORED JONES POLYNOMIAL OF SINGULAR KNOTS 6 (1) " n = n n (2) " n = nn nn n n n n (3) [ ] n = n The proof that the previous three relations gives an invariant for singular links in S can bedone by showing that [ . ] n is invariant under the singular Reidemeister moves. The details of thisproof can be found in [10]. It is clear that the invariant [ . ] n can be viewed as an extension of theunreduced colored Jones polynomial for links in S . Namely, for a zero-framed knot K in S wehave ˜ J n,K = [ K ] n . For this reason, we will denote this invariant by the most common notationof the unreduced colored Jones polynomial, that is ˜ J n . On the other hand, when computing thetail of the colored Jones polynomial of singular links we prefer to work with the normalized versionof the colored Jones polynomial. We define the normalized colored Jones polynomial of a singularlink K by: J n +1 ,K ( q ) = 1∆ n [ K ] n (cid:12)(cid:12)(cid:12)(cid:12) A = q / . This definition can be seen as an extension of the definition of the normalized colored Jones poly-nomial from the classical links.3.1.
Computing the Colored Jones Polynomial of Singular Links.
In [30] Masbum andVogel gave an algorithm to compute the colored Jones polynomial using colored trivalent graphs.We review their algorithm and we show how it can be extended to compute the colored Jonespolynomial of singular knots. We recall first some identities and definitions from [30].Consider the skein module of I × I with a + b + c specified points on the boundary. Partition theset of the a + b + c points on the boundary of the disk into 3 sets of a , b and c points respectivelyand at each cluster of points we place an appropriate idempotent, i.e. the one whose color matchesthe cardinality of this cluster. The skein module constructed this way will be denoted by T a,b,c . The skein module T a,b,c is either zero dimensional or one dimensional. The skein module T a,b,c isone dimensional if and only if the element shown in Figure 3 exists. For this element to exist it isnecessary to find non-negative integers x, y and z such that a = x + y , b = x + z and c = y + z . b ax yz c Figure 3.
The skein element τ a,b,c in the space T a,b,c The following definition characterizes the existence of this skein element in terms of the integers a , b and c . Definition 3.2.
A triple of non-negative integers ( a, b, c ) is admissible if a + b + c is even and a + b ≥ c ≥ | a − b | . OUNDATIONS OF THE COLORED JONES POLYNOMIAL OF SINGULAR KNOTS 7
When the triple ( a, b, c ) is admissible, one can write x = a + b − c , y = a + c − b , and z = b + c − a . Inthis case we will denote the skein element that generates the space by τ a,b,c . We will call the triple( a, b, c ) the interior colors of τ a,b,c and the triple ( x, y, z ) the interior colors of τ a,b,c . Note thatwhen the triple ( a, b, c ) is not admissible then the space T a,b,c is zero dimensional.The fact that the inside colors are determined by the outside colors allows us to replace τ a,b,c bya trivalent graph as follows: b ax yz cb a c Figure 4.
The skein element and its corresponding trivalent vertexA colored trivalent graph in S is an embedded trivalent graph in S with edges labeled bynon-negative integers. One usually uses the word color to refer to a label of the edge of a trivalentgraph. A colored trivalent graph is called admissible if the three edges meeting at a vertex satisfythe admissibility condition of the definition 3.2. If D is an admissible colored trivalent graph thenthe Kauffman bracket evaluation of D is defined to be the evaluation of D as an element in S ( S )after replacing each edge colored n by the projector f ( n ) and each admissible vertex colored ( a, b, c )by the skein element τ a,b,c , as in Figure 4. If a colored trivalent graph has a non-admissible vertexthen we will consider its evaluation in S ( S ) to be zero.We will need the evaluation of the following important colored trivalent graphs shown in Figure5. a bcd efabc Figure 5.
The theta graph on the left and the tetrahedron graph on the right.For an admissible triple ( a, b, c ), an explicit formula for the theta coefficient , denoted θ ( a, b, c ),was computed in [30] and is given by: a b c = ( − x + y + z [ x + y + z + 1]![ x ]![ z ]![ y ]![ x + y ]![ x + z ]![ y + z ]! (3.1) OUNDATIONS OF THE COLORED JONES POLYNOMIAL OF SINGULAR KNOTS 8 where x, y and z are the interior colors of the vertex ( a, b, c ). In terms of the Pochhammer symbolthe previous identity is given by θ ( a, b, c ) = ( − x + y + z q − ( x + y + z ) / ( q ; q ) x ( q ; q ) y ( q ; q ) z ( q ; q ) x + y + z +1 (1 − q )( q ; q ) x + y ( q ; q ) y + z ( q ; q ) x + z , (3.2)where ( q ; q ) n = n − Y i =0 (1 − q i +1 ) . (3.3)The tetrahedron coefficient is defined to be the evaluation of the graph appearing on the righthandside of Figure 5 and a formula of it can be found in [30]. The tetrahedron graph in Figure 5 isdenoted by T et (cid:20) a d ef c b (cid:21) . Besides the previous two coefficients the following two identities holdin T a,b,c : b cad ef = T et (cid:20) a d ef c b (cid:21) θ ( a, b, c ) b ca (3.4)and b ca = λ ab,c b ca , (3.5)where λ ab,c = ( − ( a + b − c ) / A ( a ′ + b ′ − c ′ ) / , and x ′ = x ( x + 2).Define the space T a,b similar to the skein module T a,b,c . Namely, this module is the submodule ofthe skein module of the disk with a + b marked point on the boundary and place the idempotents f a and f b on the appropriate sets of points as we did for T a,b,c . This module is also zero dimensionalor one dimensional. Using the properties of the idempotent one can see that this space is onedimensional if and only if a = b and zero dimensional otherwise. In T a,b the following identity alsoholds: dab c = δ da θ ( a, b, c )∆ a a (3.6)We define the module of the disk D a,bc,d similar to the modules T a,b,c and T a,b . See Figure 6 for anillustration. OUNDATIONS OF THE COLORED JONES POLYNOMIAL OF SINGULAR KNOTS 9 a bc d
Figure 6.
The relative skein module D a,bc,d Now we define the bilinear form <, > : D a,bc,d × D a,bc,d −→ S ( S ) as follows. Let E and F be twodiagrams in D a,bc,d . The diagram < E, F > is an element in S ( S ) defined in Figure 7. a bd c F E Figure 7.
The diagram < E, F > in S ( S ).Let B H = {T i | ( a, c, i ) , ( b, d, i ) ∈ ADM } be the set of tangles defined in Figure 8 (a). It is knownthat this set forms an orthogonal basis for the space D a,bc,d with respect to the bilinear form <, > .For more detail see [28]. iac db (a) iac db (b) Figure 8. (a)The element T i in the B H (b)The element T ′ i in the B V By symmetry, the set B V = {T ′ i | ( a, b, i ) , ( c, d, i ) ∈ ADM } is also a basis. The change of basisbetween these two bases B H and B V is given by: ac dbj = X i (cid:26) a b ic d j (cid:27) iac db (3.7)The previous identity is also called the recoupling identity . The coefficient (cid:26) a b ic d j (cid:27) is usuallycalled the 6 j − symbol. The fusion identity is given by: OUNDATIONS OF THE COLORED JONES POLYNOMIAL OF SINGULAR KNOTS 10 ba = X i ∆ a + b θ ( a, b, i ) aa bbi (3.8)The fusion identity (3.8) and identity (3.5) can be used to obtain the crossing fusion identity : ba = X i ∆ a + b θ ( a, b, i ) λ ia,b aa bbi (3.9)Let L be a singular link in S . We want to compute the value of [ L ] n = ˜ J n,L . Now ˜ J n,L is askein element in the skein module S ( S ) obtained by replacing every crossing by the right hand-sideof rule (1) in Definition 3.1 and every singular crossing by the skein element on the right hand sideof rule (2) in Definition 3.1. In order to evaluate the skein element ˜ J n,L we show how it can berealized as a linear combination of colored trivalent graphs in S ( S ). The evaluation of any coloredtrivalent graph in S ( S ) can then by calculated using the algorithm given in [30]. This gives us amethod of computing the evaluation of ˜ J n,L for any singular link L .The skein element ˜ J n,L can be realized as a Q ( A )-linear combination of colored trivalent graphsin S ( S ) as follows:(1) Use the crossing fusion identity to change every crossing to a Q ( A )-linear combination oftrivalent graphs as in equation (3.9).(2) Now we know that the singular crossings are replaced by the skein element (2) in Definition3.1. We notice that this skein element can be realized as a trivalent graph as follows: nn n n n n = nn nn n n n n The evaluation of any trivalent graph in S can be calculated by using an algorithm that utilizesthe recoupling formula and identities (3.4) and (3.6). The details of this algorithm can be found in[30]. We give an example to illustrate how the invariant ˜ J n,L can be computed for a singular link L using this method. Example 3.3.
To illustrate how the invariant ˜ J n,L can be computed in practice we compute theexample given in Figure 9. We denote this singular knot by ST ( k, l ). When l = 0 we will denotethis knot simply by ST k . OUNDATIONS OF THE COLORED JONES POLYNOMIAL OF SINGULAR KNOTS 11 kl Figure 9.
Singular Torus ST ( k, l ).To compute the value of this invariant we first notice that for positive integers k and n thefollowing skien identity holds: nn n n n n ⊗ k = n X i =0 R n,i i n n n n (3.10)where R n,i = θ (2 n, n, i ) k − θ ( n, n, i ) k ∆ i (3.11)To prove identity (3.10), we apply the fusion identity to obtain: nn n n n n = n X i =0 ∆2 iθ ( n, n, i ) i n n n n = n X i =0 B n,i i n n n n where B n,i = T et (cid:20) i n nn n n (cid:21) θ (2 n, n, i ) ∆ i θ ( n, n, i ) (3.12)But since, T et (cid:20) i n nn n n (cid:21) = θ (2 n, n, i ) . (3.13)One obtains: B n,i = ∆ i θ ( n, n, i )Furthermore, i n n n n ⊗ nn n n n n ⊗ k = ( P n,i ) k i n n n n OUNDATIONS OF THE COLORED JONES POLYNOMIAL OF SINGULAR KNOTS 12 where P n,i = T et (cid:20) i n nn n n (cid:21) θ ( n, n, i )However, T et (cid:20) i n nn n n (cid:21) = T et (cid:20) i n nn n n (cid:21) (3.14)Hence, both equations and (3.13) and (3.14) imply: P n,i = θ (2 n, n, i ) θ ( n, n, i )Thus, nn n k copies 2 n n n nn nn = n X i =0 B n,i i n n n n ⊗ nn n n n n ⊗ k − = n X i =0 B n,i ( P n,i ) k − i n n n n . Hence, 3.10 follows. Thus colored Jones polynomial of ST ( k, l ) is given by : J n +1 ,ST ( k,l ) = 1∆ n n X i =0 θ (2 n, n, i ) k ∆ i θ ( n, n, i ) k ( λ i, n ) l The tail of the Colored Jones Polynomial for Singular Knots
The Study of the properties of the tail of the colored Jones polynomial have attracted attentionrecently (see for instance [6, 16, 17, 19, 35]). One of the main reasons for this is due to the fact thatthis tail have been proved to give rise to Ramanujan theta and false theta identities [3, 16]. In thissection we start the investigation of the properties of the singular colored Jones polynomial and wecompute the tail of the torus singular knot ST k . First, we briefly review the basics of the head andthe tail of the colored Jones polynomial. For more details see [2, 3, 16, 17].If P ( q ) and P ( q ) are elements in Z [ q − ][[ q ]], we write P ( q ) . = n P ( q ) if their first n coefficientsagree up to a sign. It was proven in [3] that the coefficients of the colored Jones polynomial of analternating link L stabilize in the following sense: For every n ≥
2, we have J n +1 ,L ( q ) . = n J n,L ( q ).We give the following example to illustrate this further Example 4.1.
The colored Jones polynomial for the knot 6 , up to multiplication with a suitablepower q ± a n for some integer a n , is given in the following table: OUNDATIONS OF THE COLORED JONES POLYNOMIAL OF SINGULAR KNOTS 13 n = 2 1 − q + 2 q − q + 2 q − q + q n = 3 1 − q + 4 q − q + 6 q − q + 6 q + ...n = 4 1 − q + 2 q + q − q − q + 7 q + ...n = 5 1 − q + 2 q − q + 2 q − q + 2 q + ...n = 6 1 − q + 2 q − q − q + q + 5 q + ...n = 7 1 − q + 2 q − q − q + 4 q − q + 7 q + ...n = 8 1 − q + 2 q − q − q + 2 q + 3 q − q + ... This motivated the authors of [3] to define the tail of the colored Jones polynomial of a link. Moreprecisely, define the q -series series associated with the colored Jones polynomial of an alternatinglink L whose n th coefficient is the n th coefficient of J n,L ( q ). Stated differently, the tail of the coloredJones polynomial of a link L is defined to be a series T L ( q ), that satisfies T L ( q ) . = n J n,L ( q ) for all n ≥
1. Hence from the table above we deduce that the tail of the colored Jones polynomial of theknot 6 is given by : T ( q ) = 1 − q + 0 q + 2 q − q + 0 q − q + 2 q + ... In the same way, the head of the colored Jones polynomial of a link L is defined to be the tail of J n,L ( q − ). In this paper we consider the tail of a sequence of invariants of singular knots. For thisreason we define the tail of a sequence of power series in general. Definition 4.2.
Let P = { P n ( q ) } n ∈ N be a sequence of formal power series in Z [ q − ][[ q ]]. The tailof the sequence P - if it exists - is the formal power series T P in Z [[ q ]] that satisfies T P ( q ) . = n P n ( q )Next we prove and compute the tail of the colored Jones polynomial for the singular knot ST k .This is the link shown in Figure 9 with only singular crossings. Theorem 4.3.
For k ≥
1, we have T ST k ( q ) = ( q ; q ) k ∞ ∞ X i =0 q i ( q ; q ) i (4.1) Proof.
The colored Jones polynomial of the singular knot ST k is given by: J n,ST k = 1∆ n n X i =0 θ (2 n, n, i ) k ∆ i θ ( n, n, i ) k . (4.2)The theorem hence follows by proving that: J n +1 ,ST k = n ( q ; q ) k ∞ ∞ X i =0 q i ( q ; q ) i . First we note that equation 3.2 implies : θ (2 n, n, i ) θ ( n, n, i ) = ( − n q − n ( q ; q ) n ( q ; q ) n − i ( q ; q ) n + i +1 ( q ; q ) n ( q ; q ) n − i ( q ; q ) n + i +1 (4.3)On the other hand, OUNDATIONS OF THE COLORED JONES POLYNOMIAL OF SINGULAR KNOTS 14 ( q ; q ) n ( q ; q ) n = n − Y k =0 (1 − q k +1 ) n − Y k =0 (1 − q k +1 )= 1 n − Y k = n (1 − q k +1 )= n − Y k =0 − q n + k +1 ) . = n . (4.4)Moreover, ( q ; q ) n − i +1 ( q ; q ) n + i +1 = 1 − q n + i +2 + O ( n + i + 3) . = n . (4.5)and ( q ; q ) n + i +1 ( q ; q ) n = 1 − q n +1 + O (2 n + 2) . = n . (4.6)Hence equation 4.2 becomes: J n +1 ,ST k . = n ( q ; q ) kn ∆ n n X i =0 ∆ i ( q ; q ) n − i = ( q ; q ) kn ∆ n n X i =0 ∆ i ( q ; q ) i . = n ( q ; q ) kn n X i =0 q i ( q ; q ) i (4.7)The result follows. (cid:3) The special case when k = 2 can be computed using another method that gives rise to aninteresting false theta function identity. We do this by utilizing another method to evaluate thecolored Jones polynomial of the singular knot ST . Using Definition 3.1 we see that [ ST ] n is equalto the evaluation of following skein element :[ ST ] n = n n n (4.8)This can be used to show the following result. Theorem 4.4. T ST . = n ( q ; q ) n n X i =0 q i + i ( q ; q ) i (4.9) Proof.
We use the bubble skein formula given in [17] on the bubble showing in skein element onthe right hand side of equation (4.8), we obtain
OUNDATIONS OF THE COLORED JONES POLYNOMIAL OF SINGULAR KNOTS 15 [ ST ] n = n X i =0 (cid:24) n nn n (cid:25) i n + in + i n n (4.10)= n X i =0 (cid:24) n nn n (cid:25) i ∆ n ∆ n + i . (4.11)For the definition of the coefficient (cid:24) n nn n (cid:25) i see [16] Theorem 2 .
4. Hence, J n,ST k = n X i =0 (cid:24) n nn n (cid:25) i ∆ n ∆ n + i (4.12)By Lemma 4.10 part (1) of [16] we have n X i =0 (cid:24) n nn n (cid:25) i ∆ n ∆ n + i . = n ( q ; q ) n n X i =0 q i + i ( q ; q ) i (4.13)The result follows. (cid:3) Theorems 4.3 and 4.4 imply immediately the following:
Corollary 4.5.
The following identity holds :( q ; q ) ∞ ∞ X i =0 q i + i ( q ; q ) i = ( q ; q ) ∞ ∞ X i =0 q i ( q ; q ) i (4.14)The identity in Corollary 4.5 is a well-known false theta function identity. In fact both sides areequal to Ψ( q , q ) where, Ψ( a, b ) = ∞ X i =0 a i ( i +1)2 b i ( i − − ∞ X i =1 a i ( i − b i ( i +1)2 (4.15)See for instance page 169 of [1] or see page 200 in [37]. For a recent study of the previous identity,also related to the colored Jones polynomial, see also the work of Bringmann and Milas in [6].5. Conclusion
The existence of the tail of the colored Jones polynomial of singular links is still an open question.The tail of the colored Jones polynomial of non-singular links exists for adequate links. The questionof an analogue of adequate links in the singular case seems to be an interesting question that isworth pursuing.5.1.
Acknowledgment.
We would like to thank Antun Milas for useful conversations and sug-gesting a correction in this paper.
OUNDATIONS OF THE COLORED JONES POLYNOMIAL OF SINGULAR KNOTS 16
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