Geometry & Topology

The degree of the Alexander polynomial is an upper bound for the topological slice genus

Peter Feller

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Abstract

We use the famous knot-theoretic consequence of Freedman’s disc theorem — knots with trivial Alexander polynomial bound a locally flat disc in the 4–ball — to prove the following generalization: the degree of the Alexander polynomial of a knot is an upper bound for twice its topological slice genus. We provide examples of knots where this determines the topological slice genus.

Article information

Source
Geom. Topol., Volume 20, Number 3 (2016), 1763-1771.

Dates
Received: 13 April 2015
Revised: 4 September 2015
Accepted: 6 September 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859002

Digital Object Identifier
doi:10.2140/gt.2016.20.1763

Mathematical Reviews number (MathSciNet)
MR3523068

Zentralblatt MATH identifier
06624258

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds

Keywords
topological slice genus Alexander polynomial

Citation

Feller, Peter. The degree of the Alexander polynomial is an upper bound for the topological slice genus. Geom. Topol. 20 (2016), no. 3, 1763--1771. doi:10.2140/gt.2016.20.1763. https://projecteuclid.org/euclid.gt/1510859002


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References

  • J W Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928) 275–306
  • M Borodzik, S Friedl, On the algebraic unknotting number, Trans. London Math. Soc. 1 (2014) 57–84
  • M Borodzik, S Friedl, The unknotting number and classical invariants, I, Algebr. Geom. Topol. 15 (2015) 85–135
  • J C Cha, C Livingston, KnotInfo: Table of knot invariants (2015) Available at \setbox0\makeatletter\@url http://www.indiana.edu/~knotinfo {\unhbox0
  • B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton Univ. Press (2012)
  • M H Freedman, A surgery sequence in dimension four; the relations with knot concordance, Invent. Math. 68 (1982) 195–226
  • M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357–453
  • M H Freedman, F Quinn, Topology of $4$–manifolds, Princeton Mathematical Series 39, Princeton Univ. Press (1990)
  • S Garoufalidis, P Teichner, On knots with trivial Alexander polynomial, J. Differential Geom. 67 (2004) 167–193
  • L H Kauffman, L R Taylor, Signature of links, Trans. Amer. Math. Soc. 216 (1976) 351–365
  • L Rudolph, Constructions of quasipositive knots and links, I, from: “Knots, braids and singularities”, (C Weber, editor), Monogr. Enseign. Math. 31, Enseignement Math., Geneva (1983) 233–245
  • L Rudolph, Some topologically locally-flat surfaces in the complex projective plane, Comment. Math. Helv. 59 (1984) 592–599
  • L Rudolph, Quasipositivity and new knot invariants, Rev. Mat. Univ. Complut. Madrid 2 (1989) 85–109
  • L Rudolph, Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. 29 (1993) 51–59
  • H F Trotter, Homology of group systems with applications to knot theory, Ann. of Math. 76 (1962) 464–498