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Topology of knot spaces in dimension 3
Author(s) -
Budney Ryan
Publication year - 2010
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdp058
Subject(s) - mathematics , unknot , knot (papermaking) , isotopy , homotopy , combinatorics , pure mathematics , knot complement , topology (electrical circuits) , skein relation , knot theory , chemical engineering , engineering
This paper gives a detailed description of the homotopy type of , the space of long knots in ℝ 3 , the same space of knots studied by Vassiliev via singularity theory. Each component of corresponds to an isotopy class of long knot, and we list the components via the companionship trees associated to knots. The knots with the shortest companionship trees are: the unknot, torus knots, and hyperbolic knots. The homotopy type of these components of were computed by Hatcher. In the case where the companionship tree has more than one vertex, we give a fibre‐bundle description of the corresponding components of , recursively, in terms of the homotopy types of components of corresponding to knots with shorter companionship trees. The primary case studied in this paper is the case of a knot that has a hyperbolic manifold contained in the JSJ‐decomposition of its complement. Moreover, the homotopy type of as an SO 2 ‐space is determined, which gives a detailed description of the homotopy type of the space of embeddings of S 1 in S 3 .