The head and tail conjecture for alternating knots

Armond, Cody

doi:10.2140/agt.2013.13.2809

Cited by 38 publications

(72 citation statements)

References 8 publications

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“…These two link invariants which take the form of formal q-series with integer coefficients. The existence of these two power series was conjectured by Dasbach and Lin [6] and was proven by Armond in [3]. Higher stability of the coefficients of the colored Jones polynomial of an alternating link was shown by Garoufalidis and Le in [7].…”

Section: Introductionmentioning

confidence: 92%

“…In [3] C. Armond and O. Dasbach defined a product structure on the tail of the color Jones polynomial. In this section we will define a few product structures on the tail of trivalent graphs in S(S 2 ) using similar techniques to the ones in [3].…”

Section: Tail Multiplication Structures On Quantum Spin Networkmentioning

confidence: 99%

“…In this section we will define a few product structures on the tail of trivalent graphs in S(S 2 ) using similar techniques to the ones in [3]. Let Γ 1 and Γ 2 be trivalent graphs in S(S 2 ).…”

Section: Tail Multiplication Structures On Quantum Spin Networkmentioning

confidence: 99%

“…Skein theoretic techniques have been used in [2] and [3] to understand the head and tail of an alternating link. Write S(S 3 ) to denote the Kauffman Bracket Skein Module of S 3 .…”

Section: Introductionmentioning

confidence: 99%

“…Hikami [11] realized that that Rogers-Ramanujan identities appear in the study of the colored Jones polynomial of torus knots. In [3] Armond and Dasbach calculate the head and the tail of the colored Jones polynomial via multiple methods and use these computations to prove number theoretic identities. In this paper we show that the skein theoretic techniques we developed herein can be also used to prove classical identities in number theory.…”

Section: Introductionmentioning

confidence: 99%

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The tail of a quantum spin network

Hajij

2015

Ramanujan J

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Abstract. The tail of a sequence {Pn(q)} n∈N of formal power series in Z [[q]] is the formal power series whose first n coefficients agree up to a common sign with the first n coefficients of Pn. This paper studies the tail of a sequence of admissible trivalent graphs with edges colored n or 2n. We use local skein relations to understand and compute the tail of these graphs. We also give product formulas for the tail of such trivalent graphs. Furthermore, we show that our skein theoretic techniques naturally lead to a proof for the Andrews-Gordon identities for the two variable Ramanujan theta function as well to corresponding identities for the false theta function.

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Section: Introductionmentioning

confidence: 92%

Section: Tail Multiplication Structures On Quantum Spin Networkmentioning

confidence: 99%

Section: Tail Multiplication Structures On Quantum Spin Networkmentioning

confidence: 99%

“…Skein theoretic techniques have been used in [2] and [3] to understand the head and tail of an alternating link. Write S(S 3 ) to denote the Kauffman Bracket Skein Module of S 3 .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The tail of a quantum spin network

Hajij

2015

Ramanujan J

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On Rational Knots and Links in the Solid Torus

2018

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We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize this by giving an infinite family of ambient isotopy invariants of colored diagrams in the Kauffman bracket skein module of an oriented surface.2000 Mathematics Subject Classification. Primary 57M27; Secondary 57M25, 11A55 .

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