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    Quandle cohomology and state-sum invariants of knotted curves and surfaces
    (American Mathematical Society, 2003) Carter, J. Scott; Jelsovsky, Daniel; Kamada, Seiichi; Langford, Laurel; Saito, Masahico
    The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed. Our proof is based on a state-sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles (also known as distributive groupoids). A quandle is a set with a binary operation -- the axioms of which model the Reidemeister moves in classical knot theory. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in 3-space and knotted surfaces in 4-space. Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants. Non-triviality of the invariants is proved for a variety of knots and links, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.
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    Computations of quandle cocycle invariants of knotted curves and surfaces
    (Cornell Tech, 1999) Carter, J. Scott; Jelsovsky, Daniel; Kamada, Seiichi; Saito, Masahico
    State-sum invariants for knotted curves and surfaces using quandle cohomology were introduced by Laurel Langford and the authors in math.GT/9903135 In this paper we present methods to compute the invariants and sample computations. Computer calculations of cohomological dimensions for some quandles are presented. For classical knots, Burau representations together with Maple programs are used to evaluate the invariants for knot table. For knotted surfaces in 4-space, movie methods and surface braid theory are used. Relations between the invariants and symmetries of knots are discussed.
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    Shifting homomorphisms in quandle cohomology and skeins of cocycle knot invariants
    (Cornell Tech, 2000) Carter, J. Scott; Jelsovsky, Daniel; Kamada, Seiichi; Saito, Masahico
    Homomorphisms on quandle cohomology groups that raise the dimensions by one are studied in relation to the cocycle state-sum invariants of knots and knotted surfaces. Skein relations are also studied. Comment: 14 pages; 6 Figures. Minor corrections: The main application of one proposition remains true, but the proposition has been downgraded to a conjecture
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    Asymptotic Translation Length in the Curve Complex
    (New York Journal of Mathematics, 2013) Valdivia, Aaron D.
    We show that when the genus and punctures of a surface are directly proportional by some rational number the minimal asymptotic translation length in the curve complex has behavior inverse to the square of the Euler characteristic. We also show that when the genus is fixed and the number of punctures varies the behavior is inverse to the Euler characteristic.
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    Lipschitz Constants To Curve Complexes For Punctured Surfaces
    (Elsevier B.V, 2014) Valdivia, Aaron D.
    We give asymptotic bounds for the optimal Lipschitz constants for the systole map from the Teichmuller space to the curve complex. We give similar results to those known for closed surfaces in the cases when the genus is fixed or the ratio of genus and punctures is a rational number. Comment: 7 pages, 2 figures