Almost positive links have negative signature

Details Almost positive links have negative signature Almost positive links have negative signature

2009-04-27

arxiv.org/abs/0904.4130v1

Mathematics

Geometric Topology

95 pages, 160 figures

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Almost positive links have negative signature does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Almost positive links have negative signature, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Almost positive links have negative signature will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-322611