Geometry & Topology, Vol. 5 (2001)
Paper no. 26, pages 831--883.
Concordance and Mutation
P Kirk and C Livingston
Abstract.
We provide a framework for studying the interplay between concordance
and positive mutation and identify some of the basic structures
relating the two.
The fundamental result in understanding knot concordance is the
structure theorem proved by Levine: for n>1 there is an isomorphism
phi from the concordance group C_n of knotted (2n-1)-spheres in
S^{2n+1} to an algebraically defined group G_{+-}; furthermore,
G__{+-} is isomorphic to the infinite direct sum Z^infty direct sum
Z_2^infty direct sum Z_4^infty. It was a startling consequence of the
work of Casson and Gordon that in the classical case the kernel of phi
on C_1 is infinitely generated. Beyond this, little has been
discovered about the pair (C_1,phi).
In this paper we present a new approach to studying C_1 by introducing
a group, M, defined as the quotient of the set of knots by the
equivalence relation generated by concordance and positive mutation,
with group operation induced by connected sum. We prove there is a
factorization of phi, C_1-->M-->G_-. Our main result is that both maps
have infinitely generated kernels.
Among geometric constructions on classical knots, the most subtle is
positive mutation. Positive mutants are indistinguishable using
classical abelian knot invariants as well as by such modern invariants
as the Jones, Homfly or Kauffman polynomials. Distinguishing positive
mutants up to concordance is a far more difficult problem; only one
example has been known until now. The results in this paper provide,
among other results, the first infinite families of knots that are
distinct from their positive mutants, even up to concordance.
Keywords.
Knot theory, concordance, mutation
AMS subject classification.
Primary: 57M25.
Secondary: 57M27.
DOI: 10.2140/gt.2001.5.831
E-print: arXiv:math.GT/9912174
Submitted to GT on 18 December 2000.
(Revised 17 November 2001.)
Paper accepted 30 October 2001.
Paper published 21 November 2001.
Notes on file formats
P Kirk and C Livingston
Department of Mathematics
Indiana University, Bloomington, IN 47405, USA
Email: pkirk@indiana.edu, livingst@indiana.edu
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