0000000000634273

AUTHOR

Daniel Lines

showing 4 related works from this author

PERIPHERALLY SPECIFIED HOMOMORPHS OF LINK GROUPS

2005

Johnson and Livingston have characterized peripheral structures in homomorphs of knot groups. We extend their approach to the case of links. The main result is an algebraic characterization of all possible peripheral structures in certain homomorphic images of link groups.

Pure mathematicsAlgebra and Number TheoryLink groupGeometric Topology (math.GT)Mathematics::Geometric TopologyMathematics - Geometric Topology57M0557M25FOS: MathematicsAlgebraic Topology (math.AT)57M25; 57M05Mathematics - Algebraic TopologyAlgebraic numberNuclear ExperimentKnot (mathematics)MathematicsJournal of Knot Theory and Its Ramifications
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KNOTS WITH UNKNOTTING NUMBER ONE AND GENERALISED CASSON INVARIANT

1996

We extend the classical notion of unknotting operation to include operations on rational tangles. We recall the “classical” conditions (on the signature, linking form etc.) for a knot to have integral (respectively rational) unknotting number one. We show that the generalised Casson invariant of the twofold branched cover of the knot gives a further necessary condition. We apply these results to some Montesinos knots and to knots with less than nine crossings.

CombinatoricsAlgebra and Number TheoryKnot (unit)Unknotting numberMathematics::Geometric TopologyCasson invariantMathematicsKnot theoryFinite type invariantJournal of Knot Theory and Its Ramifications
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INVOLUTIONS ON KNOT GROUPS AND VARIETIES OF REPRESENTATIONS IN A LIE GROUP

2002

We prove the existence of a rationalisation [Formula: see text] of a classical or high-dimensional knot group Π which admits an involution if the Alexander polynomials of the knot are reciprocal. Using the group [Formula: see text] and its involution, we study the local structure, in the neighbourhood of an abelian representation, of the space of representation of the knot group Π in a a Lie group. We apply these results to the groups of classical prime knots up to 10 crossings.

Knot complementAlgebraPure mathematicsAlgebra and Number TheoryKnot invariantKnot groupQuantum invariantSkein relationTricolorabilityMathematics::Geometric TopologyMathematicsKnot theoryTrefoil knotJournal of Knot Theory and Its Ramifications
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Varieties of representations of virtual knot groups in SL2(C)

2002

Abstract We study the local structure of the variety of representations of a virtual knot group in SL 2 ( C ) near an abelian representation ρ 0 . To such a representation is attached a complex number ω and there are three cases. If ω and ω −1 are not roots of the Alexander polynomial, there are only abelian representations around ρ 0 . If ω is a root and ω −1 is not, there are only reducible representations. If both ω and ω −1 are roots and certain homological conditions hold, there are irreducible representations.

Pure mathematicsInduced representationQuantum invariantAlexander polynomialKnot polynomialVirtual knotKnot theoryAlgebraKnot invariantRepresentation theory of SUVirtual knot groupsRepresentation spacesGeometry and TopologyMathematicsTopology and its Applications
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