|
|
|
Alexander polynomial, finite type invariants and volume of
hyperbolic knots
Efstratia Kalfagianni
|
|
Algebraic & Geometric Topology 4
(2004) 1111–1123
|
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.
|
Mathematical Subject Classification
Primary: 57M25
Secondary: 57M27, 57N16
|
Publication
Received: 22 September 2004
Accepted: 15 November 2004
Published: 25 November 2004
|
|
