Journal of the Mathematical Society of Japan

Simple ribbon fusions and genera of links

Kengo KISHIMOTO, Tetsuo SHIBUYA, and Tatsuya TSUKAMOTO

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Abstract

Let $K$ be the result of a 1-fusion (band sum) of a knot $k$ and a distant trivial knot in $S^3$. From results of D. Gabai and of M. G. Scharlemann, we know that the genus of $K$ is at least that of $k$ and that equality holds if and only if the band sum is, in fact, a connected sum (in which case $K$ is ambient isotopic to $k$). In this paper, we consider a generalization of this result to an $m$-fusion of a link and a distant trivial link with $m$-components.

Article information

Source
J. Math. Soc. Japan Volume 68, Number 3 (2016), 1033-1045.

Dates
First available in Project Euclid: 19 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1468956158

Digital Object Identifier
doi:10.2969/jmsj/06831033

Mathematical Reviews number (MathSciNet)
MR3523537

Zentralblatt MATH identifier
06642403

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Keywords
link fusion genus

Citation

KISHIMOTO, Kengo; SHIBUYA, Tetsuo; TSUKAMOTO, Tatsuya. Simple ribbon fusions and genera of links. J. Math. Soc. Japan 68 (2016), no. 3, 1033--1045. doi:10.2969/jmsj/06831033. https://projecteuclid.org/euclid.jmsj/1468956158.


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References

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  • C. Goldberg, On the genera of links, Ph.D. Thesis of Princeton University, 1970.
  • K. Kobayashi, T. Shibuya and T. Tsukamoto, Simple ribbon moves for links, Osaka J. Math., 51 (2014), 545–571.
  • D. Rolfsen, Knots and links, Publish or Perish, Inc., 1976.
  • M. Scharlemann, Sutured manifolds and generalized Thurston norms, J. Differential Geom., 29 (1989), 557–614.