TY - JOUR

T1 - Almost positive links have negative signature

AU - Przytycki, Józef H.

AU - Taniyama, Kouki

N1 - Funding Information:
K. Taniyama was partially supported by Grant-in-Aid for Scientific Research (C) (No. 18540101), Japan Society for the Promotion of Science. The completion of this work has been done while he was visiting George Washington University. He is grateful to the hospitality of the mathematical department of the George
Funding Information:
J. Przytycki was partially supported by the NSA grant (# H98230-08-1-0033), by the Polish Scientific Grant: Nr. N N201387034, by the GWU REF grant, and by the CCAS/UFF award.

PY - 2010/2

Y1 - 2010/2

N2 - 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.

AB - 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.

KW - Almost positive link

KW - Jones polynomial

KW - Positive link

KW - Signature

KW - TristramLevine signature

KW - Twist knot

KW - Unknotting number

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U2 - 10.1142/S0218216510007838

DO - 10.1142/S0218216510007838

M3 - Article

AN - SCOPUS:77951678511

SN - 0218-2165

VL - 19

SP - 187

EP - 289

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

IS - 2

ER -