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

Abstract

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.