219 research outputs found
Invariants of genus 2 mutants
Pairs of genus 2 mutant knots can have different Homfly polynomials, for
example some 3-string satellites of Conway mutant pairs. We give examples which
have different Kauffman 3-variable polynomials, answering a question raised by
Dunfield et al in their study of genus 2 mutants. While pairs of genus 2 mutant
knots have the same Jones polynomial, given from the Homfly polynomial by
setting v=s^2, we give examples whose Homfly polynomials differ when v=s^3. We
also give examples which differ in a Vassiliev invariant of degree 7, in
contrast to satellites of Conway mutant knots.Comment: 16 pages, 20 figure
Braids: A Survey
This article is about Artin's braid group and its role in knot theory. We set
ourselves two goals: (i) to provide enough of the essential background so that
our review would be accessible to graduate students, and (ii) to focus on those
parts of the subject in which major progress was made, or interesting new
proofs of known results were discovered, during the past 20 years. A central
theme that we try to develop is to show ways in which structure first
discovered in the braid groups generalizes to structure in Garside groups,
Artin groups and surface mapping class groups. However, the literature is
extensive, and for reasons of space our coverage necessarily omits many very
interesting developments. Open problems are noted and so-labelled, as we
encounter them.Comment: Final version, revised to take account of the comments of readers. A
review article, to appear in the Handbook of Knot Theory, edited by W.
Menasco and M. Thistlethwaite. 91 pages, 24 figure
Automatic Evaluation of the Links-Gould Invariant for all Prime Knots of up to 10 Crossings
This paper describes a method for the automatic evaluation of the Links-Gould
two-variable polynomial link invariant (LG) for any link, given only a braid
presentation. This method is currently feasible for the evaluation of LG for
links for which we have a braid presentation of string index at most 5. Data
are presented for the invariant, for all prime knots of up to 10 crossings and
various other links. LG distinguishes between these links, and also detects the
chirality of those that are chiral. In this sense, it is more sensitive than
the well-known two-variable HOMFLY and Kauffman polynomials. When applied to
examples which defeat the HOMFLY invariant, interestingly, LG `almost' fails.
The automatic method is in fact applicable to the evaluation of any such state
sum invariant for which an appropriate R matrix and cap and cup matrices have
been determined.Comment: 28 pages, 6 figures.
Minor corrections and references added since version
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