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A criterion for detecting inequivalent tunnels for a knot
- Source :
- Mathematical Proceedings of the Cambridge Philosophical Society. 107:483-491
- Publication Year :
- 1990
- Publisher :
- Cambridge University Press (CUP), 1990.
-
Abstract
- Let K be an oriented knot in the 3-sphere S3. An exterior of K is the closure of the complement of a regular neighbourhood of K, and is denoted by E(K). A Seifert surface for K is an oriented surface S( ⊂ S3) without closed components such that ∂S = K. We denote S ∩ E(K) by Ŝ, and we regard S as obtained from Ŝ by a radial extension. S is incompressible if Ŝ is incompressible in E(K). A tunnel for K is an embedded arc τ in S3 such that τ ∪ K = ∂τ. We denote τ ∪ E(K) by τ, and we regard τ as obtained from τ by a radial extension. Let τ1, τ2 be tunnels for K. We say that τ1 and τ2 are homeomorphic if there is a self-homeomorphism f of E(K) such that f(τ1) = τ2. The tunnels τ1 and τ2 are isotopic if τ1 is ambient isotopic to τ2 in E(K). Then the main result of this paper is as follows: Theorem. Let K be a knot in S3, and let τ1, τ2 be tunnels for K. Suppose that there are incompressible Seifert surfaces S1 S2 for K such that S1 ∪ S2 = K, and τi ⊂ Si (i = 1, 2). If τ1 and τ2 are isotopic, then there is an ambient isotopyhτ (0 ≤ t ≤ 1) of S3 such that ht(K) = K, and h1(τ1) = τ2.
- Subjects :
- Combinatorics
Physics
Seifert surface
General Mathematics
Topology
Knot (mathematics)
Subjects
Details
- ISSN :
- 14698064 and 03050041
- Volume :
- 107
- Database :
- OpenAIRE
- Journal :
- Mathematical Proceedings of the Cambridge Philosophical Society
- Accession number :
- edsair.doi...........fcc77a7905825d32e7a435c120f05ac1