9,389 research outputs found
Link invariants via counting surfaces
A Gauss diagram is a simple, combinatorial way to present a knot. It is known
that any Vassiliev invariant may be obtained from a Gauss diagram formula that
involves counting (with signs and multiplicities) subdiagrams of certain
combinatorial types. These formulas generalize the calculation of a linking
number by counting signs of crossings in a link diagram. Until recently,
explicit formulas of this type were known only for few invariants of low
degrees. In this paper we present simple formulas for an infinite family of
invariants in terms of counting surfaces of a certain genus and number of
boundary components in a Gauss diagram. We then identify the resulting
invariants with certain derivatives of the HOMFLYPT polynomial.Comment: This is a revised version, to appear in Geom. Dedicata, 29 pages,
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A refined Jones polynomial for symmetric unions
Motivated by the study of ribbon knots we explore symmetric unions, a
beautiful construction introduced by Kinoshita and Terasaka in 1957. For
symmetric diagrams we develop a two-variable refinement of the Jones
polynomial that is invariant under symmetric Reidemeister moves. Here the two
variables and are associated to the two types of crossings,
respectively on and off the symmetry axis. From sample calculations we deduce
that a ribbon knot can have essentially distinct symmetric union presentations
even if the partial knots are the same.
If is a symmetric union diagram representing a ribbon knot , then the
polynomial nicely reflects the geometric properties of . In
particular it elucidates the connection between the Jones polynomials of
and its partial knots : we obtain and , which has the form of a symmetric product reminiscent of the Alexander polynomial of ribbon knots.Comment: 28 pages; v2: some improvements and corrections suggested by the
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A combinatorial approach to knot recognition
This is a report on our ongoing research on a combinatorial approach to knot
recognition, using coloring of knots by certain algebraic objects called
quandles. The aim of the paper is to summarize the mathematical theory of knot
coloring in a compact, accessible manner, and to show how to use it for
computational purposes. In particular, we address how to determine colorability
of a knot, and propose to use SAT solving to search for colorings. The
computational complexity of the problem, both in theory and in our
implementation, is discussed. In the last part, we explain how coloring can be
utilized in knot recognition
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