#### 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
##### Abstract

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