Some applications of Tristram–Levine signatures and relation to Vassiliev invariants

Abstract

Tristram and Levine introduced a continuous family of signature invariants for knots. We show that any possible value of such an invariant is realized by a knot with given Vassiliev invariants of bounded degree. We also show that one can make a knot prime preserving Alexander polynomial and Vassiliev invariants of bounded degree. Finally, the Tristram–Levine signatures are applied to obtain a condition on (signed) unknotting number.