Ryder, Hayley Jane (1993) The structure of racks. PhD thesis, University of Warwick.

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Abstract

In this thesis we look at the structure of racks. Chapter two looks at congruences on racks. We examine operator group equivalence and associated group equivalence in detail. We show that the fundamental quandle of a knot in S3 embeds into the knot group if and only if the knot is prime. In chapter three we look at conditions on the associated group and the operator group of a rack. We prove that G is the associated group of a rack only if the associated group of Conj (G) is isomorphic to G x N, where N is abelian. We also show that any group can be the operator group of a rack. Chapter four looks at expanding and extending racks. We derive necessary and sufficient conditions for rotation blocks to form a rack when used to expand a rack. We also show that any rack, R, can be extended to a complete rack which has the same operator group as R. The work in chapter five is closely connected to the work of Joyce in [ J ]. We define racks which can be used to represent any rack. In chapter six we show that the lattice of congruences on a transitive rack is isomorphic to a sublattice of the lattice of subgroups of the associated group. We generalize this result to non-transitive racks. The last chapter looks at the fundamental rack of a knot in S3.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Knot theory
Date:	August 1993
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Rourke, C. P. (Colin Patrick), 1943-
Sponsors:	Science and Engineering Research Council (Great Britain) (SERC)
Extent:	iv, 105 p.
Language:	eng
URI:	http://wrap.warwick.ac.uk/id/eprint/3645

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