Algebraic and Geometric Topology 4 (2004), paper no. 48, pages 1111-1123.

Alexander polynomial, finite type invariants and volume of hyperbolic knots

Efstratia Kalfagianni

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.

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

AMS subject classification. Primary: 57M25. Secondary: 57M27, 57N16.

DOI: 10.2140/agt.2004.4.1111

E-print: arXiv:math.GT/0411384

Submitted: 22 September 2004. Accepted: 15 November 2004. Published: 25 November 2004.

Notes on file formats

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


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