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

Published:2014-02-10
Printed: Jun 2014
• Keiji Tagami,
Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan
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## Abstract

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: almost positive knot, four-genus, Rasmussen invariant
 MSC Classifications: 57M27 - Invariants of knots and 3-manifolds 57M25 - Knots and links in $S^3$ {For higher dimensions, see 57Q45}