Algebraic and Geometric Topology 4 (2004),
paper no. 48, pages 1111-1123.
Alexander polynomial, finite type invariants and volume of hyperbolic knots
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.
AMS subject classification.
Secondary: 57M27, 57N16.
Submitted: 22 September 2004.
Accepted: 15 November 2004.
Published: 25 November 2004.
Notes on file formats
Department of Mathematics, Michigan State University
E. Lansing, MI 48824, USA
School of Mathematics, Institute for Advanced Study
Princeton, NJ 08540, USA
Email: email@example.com, firstname.lastname@example.org
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