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2004 Alexander polynomial, finite type invariants and volume of hyperbolic knots
Efstratia Kalfagianni
Algebr. Geom. Topol. 4(2): 1111-1123 (2004). DOI: 10.2140/agt.2004.4.1111

Abstract

We show that given n>0, there exists a hyperbolic knot K with trivial Alexander polynomial, trivial finite type invariants of order n, and such that the volume of the complement of K is larger than n. This contrasts with the known statement that the volume of the complement of a hyperbolic alternating knot is bounded above by a linear function of the coefficients of the Alexander polynomial of the knot. As a corollary to our main result we obtain that, for every m>0, there exists a sequence of hyperbolic knots with trivial finite type invariants of order m but arbitrarily large volume. We discuss how our results fit within the framework of relations between the finite type invariants and the volume of hyperbolic knots, predicted by Kashaev’s hyperbolic volume conjecture.

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Efstratia Kalfagianni. "Alexander polynomial, finite type invariants and volume of hyperbolic knots." Algebr. Geom. Topol. 4 (2) 1111 - 1123, 2004. https://doi.org/10.2140/agt.2004.4.1111

Information

Received: 22 September 2004; Accepted: 15 November 2004; Published: 2004
First available in Project Euclid: 21 December 2017

zbMATH: 1078.57014
MathSciNet: MR2113898
Digital Object Identifier: 10.2140/agt.2004.4.1111

Subjects:
Primary: 57M25
Secondary: 57M27, 57N16

Rights: Copyright © 2004 Mathematical Sciences Publishers

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