Quantum SU(3) Invariant of 3-Manifolds via Linear Skein Theory

Ohtsuki, Tatsuyuki; Yamada, Shǔji

doi:10.1142/s021821659700025x

Cited by 36 publications

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“…The module ‫ށ‬ 2 .F; ∅; R/ is isomorphic to the S U 3 -skein module of F I introduced in Frohman-Zhong [18] and Sikora [62] (see also Ohtsuki-Yamada [46]). Consequently, Corollary 5.2 implies Theorem 9.5 The S U 3 -skein module of F I; S 3 .F I; R/ (in notation of [62]) is a free R-module with a basis given by all A 2 -webs in F with no 0-gons, no true bigons, and no true 4-gons.…”

Section: Application To Knotsmentioning

confidence: 99%

Confluence theory for graphs

Sikora

Westbury

2007

Algebr. Geom. Topol.

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We develop a theory of confluence of graphs. We describe an algorithm for proving that a given system of reduction rules for abstract graphs and graphs in surfaces is locally confluent. We apply this algorithm to show that each simple Lie algebra of rank at most 2, gives rise to a confluent system of reduction rules of graphs (via Kuperberg's spiders) in an arbitrary surface. As a further consequence of this result, we find canonical bases of S U 3 -skein modules of cylinders over orientable surfaces.57M15, 57M27; 05C10, 16S15

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Section: Application To Knotsmentioning

confidence: 99%

Confluence theory for graphs

Sikora

Westbury

2007

Algebr. Geom. Topol.

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“…We also define the A 2 clasp of type (n, m) according to Ohtsuki and Yamada [17]. We review some formulas for clasped A 2 web spaces in [21].…”

Section: Lemma 23 (Properties Of a 2 Clasps) For Any Positive Integmentioning

confidence: 99%

A 𝑞-series identity via the 𝔰𝔩₃ colored Jones polynomials for the (2,2𝔪)-torus link

Yuasa

2018

Proc. Amer. Math. Soc.

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ABSTRACT. The colored Jones polynomial is a q-polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A q-series called a tail is obtained as the limit of the sl 2 colored Jones polynomials {Jn(K; q)}n for some link K, for example, an alternating link. For the sl 3 colored Jones polynomials, the existence of a tail is unknown. We give two explicit formulas of the tail of the sl 3 colored Jones polynomials colored by (n, 0) for the (2, 2m)-torus link. These two expressions of the tail provide an identity of q-series. This is a knot-theoretical generalization of the Andrews-Gordon identities for the Ramanujan false theta function.

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“…Using these clasps, Lickorish first found a quantum sl(2) invariants of 3-manifolds [22]. Ohtsuki and Yamada generalized it for the quantum sl(3) [28] and Yokota did for the quantum sl(n) [45]. A benefit of using the quantum sl(n) representation theory for link invariants is that some nontrivial facts from the original work of [10,32,33] do follow easily such as the integrality [21].…”

Section: The Homfly Polynomials and The Colored Homfly Polynomials Spmentioning

confidence: 99%

“…Using the representation theory of complex simple Lie algebras, Reshetikhin and Turaev found quantized simple Lie algebras invariants of links and 3-manifolds [32,33] and these invariants have been studied extensively [4,5,13,28,16,27,45].…”

Section: Introductionmentioning

confidence: 99%

THE QUANTUM sl(n, ℂ) REPRESENTATION THEORY AND ITS APPLICATIONS

Jeong¹,

Kim²

2012

Journal of the Korean Mathematical Society

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Abstract. In this paper, we study the quantum sl(n) representation category using the web space. Specially, we extend sl(n) web space for n ≥ 4 as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial Pn(q) specialized to a one variable polynomial can be computed by a linear expansion with respect to a presentation of the quantum representation category of sl(n). Moreover, we correct the false conjecture [30] given by Chbili, which addresses the relation between some link polynomials of a periodic link and its factor link such as Alexander polynomial (n = 0) and Jones polynomial (n = 2) and prove the corrected conjecture not only for HOMFLY polynomial but also for the colored HOMFLY polynomial specialized to a one variable polynomial.

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Quantum SU(3) Invariant of 3-Manifolds via Linear Skein Theory

Cited by 36 publications

References 0 publications

Confluence theory for graphs

Confluence theory for graphs

A 𝑞-series identity via the 𝔰𝔩₃ colored Jones polynomials for the (2,2𝔪)-torus link

THE QUANTUM sl(n, ℂ) REPRESENTATION THEORY AND ITS APPLICATIONS

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