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2023•Journal of Knot Theory and Its RamificationsOpen access

The colored Jones polynomial of a cable of the figure-eight knot

Hitoshi Murakami, Anh T. Tran

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Abstract

We study the asymptotic behavior of the [Formula: see text]-dimensional colored Jones polynomial of a cable of the figure-eight knot, evaluated at [Formula: see text] for a real number [Formula: see text]. We show that if [Formula: see text] is sufficiently large, the colored Jones polynomial grows exponentially when [Formula: see text] goes to the infinity. Moreover the growth rate is related to the Chern–Simons invariant of the knot exterior associated with an [Formula: see text] representation.

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What this paper is about

We study the asymptotic behavior of the [Formula: see text]-dimensional colored Jones polynomial of a cable of the figure-eight knot, evaluated at [Formula: see text] for a real number [Formula: see text]. We show that if [Formula: see text] is sufficiently large, the colored Jones polynomial grows exponentially when [Formula: see text] goes to the infinity. Moreover the growth rate is related to the Chern–Simons invariant of the knot exterior associated with an [Formula: see text] representation.

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Available abstract

We study the asymptotic behavior of the [Formula: see text]-dimensional colored Jones polynomial of a cable of the figure-eight knot, evaluated at [Formula: see text] for a real number [Formula: see text]. We show that if [Formula: see text] is sufficiently large, the colored Jones polynomial grows exponentially when [Formula: see text] goes to the infinity. Moreover the growth rate is related to the Chern–Simons invariant of the knot exterior associated with an [Formula: see text] representation.

Key concepts: Jones polynomial, Knot polynomial, Mathematics, Alexander polynomial, Colored, Knot (papermaking), Quantum invariant, Knot invariant

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