publication . Preprint . Article . 2013

On the Concordance Genus of Topologically Slice Knots

Jennifer Hom;
Open Access English
  • Published: 27 Nov 2013
The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of K coming from the knot Floer complex of K. As an application, we prove that there are topologically slice knots with 4-ball genus equal to one and arbitrarily large concordance genus.
arXiv: Mathematics::Geometric TopologyMathematics::Algebraic GeometryAstrophysics::Cosmology and Extragalactic Astrophysics
free text keywords: Mathematics - Geometric Topology, 57M25, 57N70, 57R58, General Mathematics

[Fre82] M. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no. 3, 357-453.

[Hed07] M. Hedden, Knot Floer homology of Whitehead doubles, Geom. Topol. 11 (2007), 2277-2338. [OpenAIRE]

[Hed09] , On knot Floer homology and cabling II, Int. Math. Res. Not. 12 (2009), 2248-2274.

[Hom11] J. Hom, The knot Floer complex and the smooth concordance group, preprint (2011), arXiv:1111.6635v1.

[Hom12] , Bordered Heegaard Floer homology and the tau-invariant of cable knots, preprint (2012), arXiv:1202.1463v1.

[Lev10] A. Levine, Knot doubling operators and bordered Heegaard Floer homology, preprint (2010), arXiv:1008.3349v1.

[Liv04] C. Livingston, The concordance genus of knots, Algebr. Geom. Topol. 4 (2004), 1-22.

[LOT08] R. Lipshitz, P. S. Ozsv´ath, and D. Thurston, Bordered Heegaard Floer homology: Invariance and pairing, preprint (2008), arXiv:0810.0687v4. [OpenAIRE]

[OS03] P. S. Ozsv´ath and Z. Szab´o, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003), 615-639.

[OS04a] , Holomorphic disks and genus bounds, Geom. Topol. 8 (2004).

[OS04b] , Holomorphic disks and knot invariants, Adv. Math. 186 (2004), no. 1, 58-116.

[OS06] , Heegaard diagrams and Floer homology, preprint (2006), arXiv:0602232v1.

[Pet09] I. Petkova, Cables of thin knots and bordered Heegaard Floer homology, preprint (2009), arXiv:0911.2679v1.

[Ras03] J. Rasmussen, Floer homology and knot complements, Ph.D. thesis, Harvard University, 2003, arXiv:0306378v1.

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