8 research outputs found

    Vanishing of 3-Loop Jacobi Diagrams of Odd Degree

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    We prove the vanishing of the space of 3-loop Jacobi diagrams of odd degree. This implies that no 3-loop finite-type invariant can distinguish between a knot and its inverse.Comment: 13 pages. Section on the even degree case expanded. Various minor correction

    The degree 2 part of the LMO invariant of cyclic branched covers of knots obtained by plumbing the doubles of two knots

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    "Revision" added on March 10, 2023"Revision 2" added on April 12, 2023The LMO invariant is a universal quantum invariant of 3-manifolds. In this paper, we present the degree 2 part of the LMO invariant of cyclic branched covers of knots by using the 3-loop polynomial of knots, and we calculate it concretely for knots obtained by plumbing the doubles of two knots

    Two-loop part of the rational homotopy of spaces of long embeddings

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    Arone and Turchin defined graph-complexes computing the rational homotopy of the spaces of long embeddings. The graph-complexes split into a direct sum by the number of loops in graphs. In this paper we compute the homology of its two-loop part.Comment: 19 pages, 2 figures. (No changes with previous version

    The 3-loop polynomial of knots obtained by plumbing the doubles of two knots

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    The 3-loop polynomial of a knot is a polynomial presenting the 3-loop part of the Kontsevich invariant of knots. In this paper, we calculate the 3-loop polynomial of knots obtained by plumbing the doubles of two knots; this class of knots includes untwisted Whitehead doubles. We construct the 3-loop polynomial by calculating the rational version of the Aarhus integral of a surgery presentation. As a consequence, we obtain an explicit presentation of the 3-loop polynomial for the knots

    On the 3-loop polynomial of genus 1 knots with trivial Alexander polynomial

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    We give a restriction of the set of possible values of the 3-loop polynomials of genus 1 knots with trivial Alexander polynomial. As its special case, we present the 3-loop polynomial of any genus 1 knot with (≤ 2)-loop polynomials by using five Vassiliev invariants of the knot. Further, we give a new example of the calculation of the 3-loop polynomial
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