Narrow Results

The 3-loop invariant (or, the 3-loop polynomial) of a knot is a rational form (or, 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 invariant 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.

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