20,094 research outputs found
Mutant knots with symmetry
Mutant knots, in the sense of Conway, are known to share the same Homfly
polynomial. Their 2-string satellites also share the same Homfly polynomial,
but in general their m-string satellites can have different Homfly polynomials
for m>2. We show that, under conditions of extra symmetry on the constituent
2-tangles, the directed m-string satellites of mutants share the same Homfly
polynomial for m<6 in general, and for all choices of m when the satellite is
based on a cable knot pattern.
We give examples of mutants with extra symmetry whose Homfly polynomials of
some 6-string satellites are different, by comparing their quantum sl(3)
invariants.Comment: 15 page
A basis for the Birman-Wenzl algebra
An explicit isomorphism is constructed between the Birman-Wenzl algebra,
defined algebraically by J. Birman and H. Wenzl using generators and relations,
and the Kauffman algebra, constructed geometrically by H. R. Morton and P.
Traczyk in terms of tangles. The isomorphism is obtained by constructing an
explicit basis in the Birman-Wenzl algebra, analogous to a basis previously
constructed for the Kauffman algebra using 'Brauer connectors'. The geometric
isotopy arguments for the Kauffman algebra are systematically replaced by
algebraic versions using the Birman-Wenzl relations.Comment: This is a lightly edited version of an article written in 1989 but
never fully completed. It was originally intended as a joint paper with A.
J.Wassermann. 33 pages, 20 figure
The Burau matrix and Fiedler's invariant for a closed braid
It is shown how Fiedler's `small state-sum' invariant for a braid can be
calculated from the 2-variable Alexander polynomial of the link which consists
of the closed braid together with the braid axis.Comment: 7 pages LaTeX2e. To appear in Topology and its Application
Mutants and SU(3)_q invariants
Details of quantum knot invariant calculations using a specific
SU(3)_q-module are given which distinguish the Conway and Kinoshita-Teresaka
pair of mutant knots. Features of Kuperberg's skein-theoretic techniques for
SU(3)_q invariants in the context of mutant knots are also discussed.Comment: 17 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTMon1/paper18.abs.htm
Mutual braiding and the band presentation of braid groups
This paper is concerned with detecting when a closed braid and its axis are
'mutually braided' in the sense of Rudolph. It deals with closed braids which
are fibred links, the simplest case being closed braids which present the
unknot. The geometric condition for mutual braiding refers to the existence of
a close control on the way in which the whole family of fibre surfaces meet the
family of discs spanning the braid axis. We show how such a braid can be
presented naturally as a word in the `band generators' of the braid group
discussed by Birman, Ko and Lee in their recent account of the band
presentation of the braid groups. In this context we are able to convert the
conditions for mutual braiding into the existence of a suitable sequence of
band relations and other moves on the braid word, and derive a combinatorial
method for deciding whether a braid is mutually braided.Comment: 13 pages, 10 figures. To appear in the proceedings of 'Knots in
Hellas 98
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