Volume 4, issue 2 (2004)

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Alexander polynomial, finite type invariants and volume of hyperbolic knots

Efstratia Kalfagianni

Algebraic & Geometric Topology 4 (2004) 1111–1123

arXiv: math.GT/0411384


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.

Alexander polynomial, finite type invariants, hyperbolic knot, hyperbolic Dehn filling, volume.
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M27, 57N16
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Received: 22 September 2004
Accepted: 15 November 2004
Published: 25 November 2004
Efstratia Kalfagianni
Department of Mathematics
Michigan State University
East Lansing MI 48824
School of Mathematics
Institute for Advanced Study
Princeton NJ 08540