Hostname: page-component-5cf477f64f-pw477 Total loading time: 0 Render date: 2025-04-06T05:25:13.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Equivariant cobordisms between freely periodic knots

Part of: Differential topology Low-dimensional topology

Published online by Cambridge University Press:  22 June 2022

Keegan Boyle Open the ORCID record for Keegan Boyle [Opens in a new window]  and
Jeffrey Musyt
Keegan Boyle*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada
Jeffrey Musyt
Affiliation:
Department of Mathematics and Statistics, Slippery Rock University, Slippery Rock, PA, USA e-mail: jeffrey.musyt@sru.edu
*
e-mail: kboyle@math.ubc.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider free symmetries on cobordisms between knots, which is equivalent to cobordisms between knots in lens spaces. We classify which freely periodic knots bound equivariant surfaces in the 4-ball in terms of corresponding homology classes in lens spaces. We give a numerical condition determining the free periods for which torus knots bound equivariant surfaces in the 4-ball.

Keywords

Knot theoryequivariant cobordismfreely periodic knots

MSC classification

Secondary: 57R85: Equivariant cobordism 57M60: Group actions in low dimensions
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 2 , June 2023 , pp. 450 - 457
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Atiyah, M. F. and Bott, R., A Lefschetz fixed point formula for elliptic complexes. II. Applications . Ann. Math. 88(1968), 451–491.CrossRefGoogle Scholar
Baker, K. and Etnyre, J., Rational linking and contact geometry . In: Perspectives in analysis, geometry, and topology, Progress in Mathematics, 296, Birkhäuser/Springer, New York, 2012, pp. 19–37.CrossRefGoogle Scholar
Bonahon, F., Difféotopies des espaces lenticulaires . Topology 22(1983), no. 3, 305–314.CrossRefGoogle Scholar
Boyle, K. and Issa, A., Equivariant 4-genera of strongly invertible and periodic knots. Preprint, 2021. https://arxiv.org/abs/2101.05413CrossRefGoogle Scholar
Boyle, K. and Issa, A., Equivariantly slicing strongly negative amphichiral knots. Preprint, 2021. https://arxiv.org/abs/2109.01198 Google Scholar
Brody, E. J., The topological classification of the lens spaces . Ann. Math. 71(1960), 163–184.CrossRefGoogle Scholar
Cha, J. C. and Ko, K. H., On equivariant slice knots . Proc. Amer. Math. Soc. 127(1999), no. 7, 2175–2182.CrossRefGoogle Scholar
Chbili, N., On the invariants of lens knots . In: KNOTS’96 (Tokyo), World Scientific Publishing, River Edge, NJ, 1997, pp. 365–375.Google Scholar
Dai, I., Hedden, M., and Mallick, A., Corks, involutions, and Heegaard Floer homology . J. Eur. Math. Soc.. Published online 21 May 2022.Google Scholar
Davis, J. F. and Naik, S., Alexander polynomials of equivariant slice and ribbon knots in S3 . Trans. Amer. Math. Soc. 358(2006), no. 7, 2949–2964.CrossRefGoogle Scholar
Doig, M. and Wehrli, S., A combinatorial proof of the homology cobordism classification of lens spaces. Preprint, 2015. https://arxiv.org/abs/1505.06970 Google Scholar
Kawauchi, A., A survey of knot theory, Birkhäuser, Basel, 1996, Translated and revised from the 1990 Japanese original by the author.Google Scholar
Meeks, W. H. III and Scott, P., Finite group actions on 3-manifolds . Invent. Math. 86(1986), no. 2, 287–346.CrossRefGoogle Scholar
Milnor, J., Whitehead torsion . Bull. Amer. Math. Soc. 72(1966), 358–426.CrossRefGoogle Scholar
Naik, S., Equivariant concordance of knots in S3 . In: KNOTS’96 (Tokyo), World Scientific Publishing, River Edge, NJ, 1997, pp. 81–89.Google Scholar
Sakuma, M., On strongly invertible knots . In: Algebraic and topological theories (Kinosaki, 1984), Kinokuniya, Tokyo, 1986, pp. 176–196.Google Scholar
Schreier, O., Über die gruppen ${A}^a{B}^b=1$ . Abh. Math. Sem. Univ. Hamburg 3(1924), no. 1, 167–169.CrossRefGoogle Scholar
Smith, P. A., Fixed-point theorems for periodic transformations . Amer. J. Math. 63(1941), 1–8.CrossRefGoogle Scholar