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ALMOST POSITIVE LINKS HAVE NEGATIVE SIGNATURE.
- Source :
-
Journal of Knot Theory & Its Ramifications . Feb2010, Vol. 19 Issue 2, p187-289. 103p. 155 Diagrams. - Publication Year :
- 2010
-
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 (possibly, with extra 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 for the left-handed trefoil knot. These results generalize those of Rudolph, Cochran, Gompf, Traczyk and Przytycki, solve [27, Conjecture 5], and give a partial answer to [3, Problem 2.8] about knots dominating the trefoil knot or the trivial knot. We also describe all unknotting number one positive knots. [ABSTRACT FROM AUTHOR]
- Subjects :
- *KNOT theory
*POLYNOMIALS
*GRAPHIC methods
*CHARTS, diagrams, etc.
*SIGNATURE (Law)
Subjects
Details
- Language :
- English
- ISSN :
- 02182165
- Volume :
- 19
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Journal of Knot Theory & Its Ramifications
- Publication Type :
- Academic Journal
- Accession number :
- 49168343
- Full Text :
- https://doi.org/10.1142/S0218216510007838