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Knot adjacency, genus and essential tori | | Efstratia Kalfagianni
; Xiao-Song Lin
; | | Date: |
1 Mar 2004 | | Subject: | Geometric Topology | math.GT | | 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 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. In the last section we study adjacency to fibred knots and we show the following: A crossing circle of a fibred knot K bounds a disc in the complement of K if and only if there is a crossing change supported on L that doesn’t change the isotopy class of K. This solves a problem in Kirby’s List for fibred knots. | | Source: | arXiv, math.GT/0403024 | | Services: | Forum | Review | PDF | Favorites | |
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