Almost positive links have negative signature

TL;DR

This paper investigates the signature of links with diagrams having few negative crossings, establishing bounds and characterizations for various cases, and generalizing previous results in knot theory.

Contribution

It provides new bounds on the signature of links with limited negative crossings and characterizes specific classes of knots and links with zero or negative signatures, extending prior work.

Findings

Links with all positive crossings except one have negative signature.

Links with two negative crossings have nonpositive signature, except certain cases.

Complete classification of trivial links with up to two negative crossings.

Abstract

We analyze properties of links which have diagrams with a small number of negative crossings. We show that if a nontrivial link has a diagram with all crossings positive except possibly one, then the signature of the link is negative. If a link diagram has two negative crossings, we show that the signature of the link is nonpositive with the exception of the left-handed Hopf link (with possible trivial components). We also characterize those links which have signature zero and diagrams with two negative crossings. In particular, we show that if a nontrivial knot has a diagram with two negative crossings then the signature of the knot is negative, unless the knot is a twist knot with negative clasp. We completely determine all trivial link diagrams with two or fewer negative crossings. For a knot diagram with three negative crossings, the signature of the knot is nonpositive except the left-handed trefoil knot. These results generalize those of L. Rudolph, T. Cochran, E. Gompf, P. Traczyk, and J. H. Przytycki, solve Conjecture 5 of [P-2], and give a partial answer to Problem 2.8 of [Co-G] about knots dominating the trefoil knot or the trivial knot. We also describe all unknotting number one positive knots.