Vol. 228, No. 2, 2006

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ISSN: 0030-8730
Knot adjacency, genus and essential tori

Efstratia Kalfagianni and Xiao-Song Lin

Vol. 228 (2006), No. 2, 251–275
Abstract

A knot K is called n-adjacent to another knot Kif K admits a projection containing n generalized crossings such that changing any 0 < m n of them yields a projection of K. We apply techniques from the theory of sutured 3-manifolds, Dehn surgery and the theory of geometric structures of 3-manifolds to study the extent to which nonisotopic knots can be adjacent to each other. A consequence of our main result is that if K is n-adjacent to Kfor all n , then K and Kare isotopic. This provides a partial verification of the conjecture of V. Vassiliev that finite type knot invariants distinguish all knots. We also show that if no twist about a crossing circle L of a knot K changes the isotopy class of K, then L bounds a disc in the complement of K. This leads to a characterization of nugatory crossings on knots.

Keywords
knot adjacency, essential tori, finite type invariants, Dehn surgery, sutured 3-manifolds, Thurston norm, Vassiliev’s conjecture
Mathematical Subject Classification 2000
Primary: 57M25, 57M27, 57M50
Milestones
Received: 6 April 2005
Revised: 26 June 2006
Accepted: 3 July 2006
Published: 1 December 2006
Authors
Efstratia Kalfagianni
Department of Mathematics
Wells Hall
Michigan State University
East Lansing, MI 48824
United States
http://www.math.msu.edu/~kalfagia
†Xiao-Song Lin
Department of Mathematics
University of California, Riverside
Riverside, CA 92521-0135
United States
http://math.ucr.edu/~xl