Twist Formulas for One-Row Colored A2 Webs and \(\mathfrak {s}\mathfrak {l}_{3}\) Tails of (2, 2m)-Torus Links

Abstract

The \(\mathfrak {s}\mathfrak {l}_{3}\) colored Jones polynomial \(J_{\lambda }^{\mathfrak {s}\mathfrak {l}_{3}}(L)\) is obtained by coloring the link components with two-row Young diagram λ. Although it is difficult to compute \(J_{\lambda }^{\mathfrak {s}\mathfrak {l}_{3}}(L)\) in general, we can calculate it by using Kuperberg’s A2 skein relation. In this paper, we show some formulas for twisted two strands colored by one-row Young diagram in A2 web space and compute \(J_{(n,0)}^{\mathfrak {s}\mathfrak {l}_{3}}(T(2,2m))\) for an oriented (2,2m)-torus link. These explicit formulas derives the \(\mathfrak {s}\mathfrak {l}_{3}\) tail of T(2,2m). They also give explicit descriptions of the \(\mathfrak {s}\mathfrak {l}_{3}\) false theta series with one-row coloring because the \(\mathfrak {s}\mathfrak {l}_{2}\) tail of T(2,2m) is known as the false theta series.

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Funding

This work was supported by Grant-in-Aid for JSPS Fellows Grant Number 19J00252 and Grant-in-Aid for Early-Career Scientists Grant Number 19K14528.

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Correspondence to Wataru Yuasa.

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Yuasa, W. Twist Formulas for One-Row Colored A2 Webs and \(\mathfrak {s}\mathfrak {l}_{3}\) Tails of (2, 2m)-Torus Links. Acta Math Vietnam (2021). https://doi.org/10.1007/s40306-020-00397-9

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Keywords

  • Colored Jones polynomial
  • False theta series

Mathematics Subject Classification (2010)

  • 57K14
  • 57K16
  • 11F27