9,389 research outputs found

    Link invariants via counting surfaces

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    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, many figure

    A refined Jones polynomial for symmetric unions

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    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 WD(s,t)W_D(s,t) of the Jones polynomial that is invariant under symmetric Reidemeister moves. Here the two variables ss and tt 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 DD is a symmetric union diagram representing a ribbon knot KK, then the polynomial WD(s,t)W_D(s,t) nicely reflects the geometric properties of KK. In particular it elucidates the connection between the Jones polynomials of KK and its partial knots KΒ±K_\pm: we obtain WD(t,t)=VK(t)W_D(t,t) = V_K(t) and WD(βˆ’1,t)=VKβˆ’(t)β‹…VK+(t)W_D(-1,t) = V_{K_-}(t) \cdot V_{K_+}(t), which has the form of a symmetric product f(t)β‹…f(tβˆ’1)f(t) \cdot f(t^{-1}) reminiscent of the Alexander polynomial of ribbon knots.Comment: 28 pages; v2: some improvements and corrections suggested by the refere

    A combinatorial approach to knot recognition

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    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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