Vanishing of 3-Loop Jacobi Diagrams of Odd Degree

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

"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

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

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

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