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The Rasmussen Invariant, Four-genus, and Three-genus of an Almost Positive Knot Are Equal

Published online by Cambridge University Press:  20 November 2018

Keiji Tagami
Keiji Tagami*
Affiliation:
Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan e-mail: tagami.k.aa@m.titech.ac.jp
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Abstract

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An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, 4-genus, and 3-genus of a positive knot are equal. In this paper, we prove that the Rasmussen invariant, 4-genus, and 3-genus of an almost positive knot are equal. Moreover, we determine the Rasmussen invariant of an almost positive knot in terms of its almost positive knot diagram. As corollaries, we prove that all almost positive knots are not homogeneous, and there is no almost positive knot of 4-genus one.

Keywords

57M2757M25almost positive knotfour-genusRasmussen invariant
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 2 , 14 June 2014 , pp. 431 - 438
Copyright
Copyright © Canadian Mathematical Society 2014

Footnotes

This work was supported by JSPS KAKENHI Grant number 25001362.

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