Almost positive links have negative signature - Mathematics > Geometric TopologyReport as inadecuate

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Abstract: We analyze properties of links which have diagrams with a small number ofnegative crossings. We show that if a nontrivial link has a diagram with allcrossings positive except possibly one, then the signature of the link isnegative. If a link diagram has two negative crossings, we show that thesignature of the link is nonpositive with the exception of the left-handed Hopflink with possible trivial components. We also characterize those links whichhave signature zero and diagrams with two negative crossings. In particular, weshow that if a nontrivial knot has a diagram with two negative crossings thenthe signature of the knot is negative, unless the knot is a twist knot withnegative clasp. We completely determine all trivial link diagrams with two orfewer negative crossings. For a knot diagram with three negative crossings, thesignature of the knot is nonpositive except the left-handed trefoil knot. Theseresults generalize those of L. Rudolph, T. Cochran, E. Gompf, P. Traczyk, andJ. H. Przytycki, solve Conjecture 5 of P-2, and give a partial answer toProblem 2.8 of Co-G about knots dominating the trefoil knot or the trivialknot. We also describe all unknotting number one positive knots.

Author: Jozef H. Przytycki, Kouki Taniyama


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