Document Zbl 1123.57007 - zbMATH Open
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Knot adjacency, genus and essential tori. (English) Zbl 1123.57007
Two knots are said to be \(n\)-adjacent if there exists a projection of one of them with \(n\) generalized crossings and a fixed set of generalized changes so that a change at any \(0 < m < n+1\) of them yields a projection of a knot isotopic to the other.
The authors show that knots \(n\)-adjacent for all \(n\) are isotopic.
Since \(n\)-adjacency implies \(n\)-similarity by Y. Ohyama, [Topology Appl. 37, No. 3, 249–255 (1990; Zbl 0724.57006)], and \(n\)-similarity implies \(n\)-equivalence by K. Y. Ng and T. Stanford, [Math. Proc. Cambridge Phil. Soc. 126, No. 1, 63–76 (1999; Zbl 0961.57006)], and by Gusarov, \(n\)-equivalence and equality of finite type invariants of order \(<n\) are the same, the result is evidence for Vassiliev’s conjecture that isotopy is implied by equality of finite type invariants for all \(n\).
Some of the results are a generalization of H. Howards and J. Luecke in [Bull. Lond. Math. Soc. 34, No. 4, 431–437 (2002; Zbl 1027.57004)].
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 57M50 | General geometric structures on low-dimensional manifolds |