Mathematics – Geometric Topology
Scientific paper
2004-03-01
Pacific Journal of Math., Vol. 228, No. 2, 251-276 (2006).
Mathematics
Geometric Topology
32 pages, 4 Figures. This version will appear in the Pacific J. of Math. The appendix by Darryl McCullough is now a separate p
Scientific paper
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 \leq 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 answer the question of the extent to which non-isotopic 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, then K and K' are isotopic. This provides a partial verification of the conjecture of V. Vassiliev that the 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 gives a characterization of the nugatory crossings of a knot.
Kalfagianni Efstratia
Lin Xiao-Song
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