Alexander polynomial, finite type invariants and volume of hyperbolic knots

Kalfagianni, Efstratia

doi:10.2140/agt.2004.4.1111

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Alexander polynomial, finite type invariants and volume of hyperbolic knots

Efstratia Kalfagianni

Algebraic & Geometric Topology 4 (2004) 1111–1123

DOI: 10.2140/agt.2004.4.1111

arXiv: math.GT/0411384

Abstract

We show that given $n > 0$ , there exists a hyperbolic knot $K$ with trivial Alexander polynomial, trivial finite type invariants of order $\leq n$ , 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 $\leq m$ 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.

Keywords

Alexander polynomial, finite type invariants, hyperbolic knot, hyperbolic Dehn filling, volume.

Mathematical Subject Classification 2000

Primary: 57M25

Secondary: 57M27, 57N16

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Publication

Received: 22 September 2004

Accepted: 15 November 2004

Published: 25 November 2004

Authors

Efstratia Kalfagianni
Department of Mathematics Michigan State University East Lansing MI 48824 USA	School of Mathematics Institute for Advanced Study Princeton NJ 08540 USA

Volume 4, issue 2 (2004)