Knot adjacency, genus and essential tori

Efstratia Kalfagianni and Xiao-Song Lin

Vol. 228 (2006), No. 2, 251–275

DOI: 10.2140/pjm.2006.228.251

Abstract

A knot K is called n-adjacent to another knot K′ if 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 K′ for all n ∈ ℕ, then K and K′ are 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