The unknotting number and classical invariants, I

Abstract

Given a knot $K$ we introduce a new invariant coming from the Blanchfield pairing and we show that it gives a lower bound on the unknotting number of $K$. This lower bound subsumes the lower bounds given by the Levine–Tristram signatures, by the Nakanishi index and it also subsumes the Lickorish obstruction to the unknotting number being equal to one. Our approach in particular allows us to show for $25$ knots with up to $12$ crossings that their unknotting number is at least three, most of which are very difficult to treat otherwise.