A note on the concordance invariants epsilon and upsilon

Hom, Jennifer

doi:10.1090/proc/12706

A note on the concordance invariants epsilon and upsilon
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by Jennifer Hom

Proc. Amer. Math. Soc. 144 (2016), 897-902

DOI: https://doi.org/10.1090/proc/12706

Published electronically: May 28, 2015

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Abstract:

Ozsváth-Stipsicz-Szabó recently defined a one-parameter family $\Upsilon _K(t)$ of concordance invariants associated to the knot Floer complex. We compare their invariant to the $\{ -1, 0, 1\}$-valued concordance invariant $\varepsilon (K)$, which is also associated to the knot Floer complex. In particular, we give an example of a knot $K$ with $\Upsilon _K(t) \equiv 0$ but $\varepsilon (K) \neq 0$.

References

Matthew Hedden, On knot Floer homology and cabling. II, Int. Math. Res. Not. IMRN 12 (2009), 2248–2274. MR 2511910, DOI 10.1093/imrn/rnp015
Stephen Hancock, Jennifer Hom, and Michael Newman, On the knot Floer filtration of the concordance group, J. Knot Theory Ramifications 22 (2013), no. 14, 1350084, 30. MR 3190122, DOI 10.1142/S0218216513500843
Jennifer Hom, The knot Floer complex and the smooth concordance group, Comment. Math. Helv. 89 (2014), no. 3, 537–570. MR 3260841, DOI 10.4171/CMH/326
Jennifer Hom, A note on cabling and $L$-space surgeries, Algebr. Geom. Topol. 11 (2011), no. 1, 219–223. MR 2764041, DOI 10.2140/agt.2011.11.219
Jennifer Hom, On the concordance genus of topologically slice knots, preprint (2012), to appear in Int. Math. Res. Not. IMRN, available at arXiv:1203.4594v1.
Jennifer Hom, An infinite rank summand of topologically slice knots, preprint (2013), to appear in Geom. Topol., available at arXiv:1310.4476v1.
Jennifer Hom, Bordered Heegaard Floer homology and the tau-invariant of cable knots, J. Topol. 7 (2014), no. 2, 287–326. MR 3217622, DOI 10.1112/jtopol/jtt030
Matthew Hedden and Liam Watson, On the geography and botany of knot Floer homology, preprint (2014), arXiv:1404.6913v2.
Peter Ozsváth and Zoltán Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003), 615–639. MR 2026543, DOI 10.2140/gt.2003.7.615
Peter Ozsváth and Zoltán Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004), no. 1, 58–116. MR 2065507, DOI 10.1016/j.aim.2003.05.001
Peter Ozsváth and Zoltán Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005), no. 6, 1281–1300. MR 2168576, DOI 10.1016/j.top.2005.05.001
Peter Ozsváth, András Stipsicz, and Zoltán Szabó, Concordance homomorphisms from knot Floer homology, preprint (2014), arXiv:1407.1795.
Jacob Andrew Rasmussen, Floer homology and knot complements, ProQuest LLC, Ann Arbor, MI, 2003. Thesis (Ph.D.)–Harvard University. MR 2704683