Thin links and Conway spheres. (English) Zbl 1543.57012
A link is said to be a thin link if its \(\delta\)-graded homology is supported in at most one \(\delta\)-grading. It is clear that the notion of thin links depends on the homology theory in question and on the coefficient system. In the case of Heegaard Floer and Khovanov homology, this notion can be considered as a natural generalization of alternating links. The authors provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. These results are based on certain immersed curve invariants for Conway tangles, namely the Heegaard Floer invariant HFT and the Khovanov invariant \(\widetilde{\text{Kh}}\) that were developed by the authors in previous works.
Reviewer: Khaled Qazaqzeh (Irbid)
Keywords:
knot Floer homology; Khovanov homology; thin links; alternating links; tangle decompositions; Conway spheres; L-spaces; multicurvesReferences:
| [1] | Boyer, S. and Clay, A., Foliations, orders, representations, L-spaces and graph manifolds, Adv. Math. 310 (2017), 159-234. · Zbl 1381.57003 |
| [2] | Boyer, S., Gordon, C. M. A. and Watson, L., On L-spaces and left-orderable fundamental groups, Math. Ann. 356 (2013), 1213-1245. · Zbl 1279.57008 |
| [3] | Bar-Natan, D., On Khovanov’s categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002), 337-370. · Zbl 0998.57016 |
| [4] | Bar-Natan, D., Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443-1499. · Zbl 1084.57011 |
| [5] | Bar-Natan, D. and Burgos-Soto, H., Khovanov homology for alternating tangles, J. Knot Theory Ramifications23 (2014), 1450013. · Zbl 1291.57006 |
| [6] | Bowden, J., Approximating \(C^0\)-foliations by contact structures, Geom. Funct. Anal. 26 (2016), 1255-1296. · Zbl 1362.57037 |
| [7] | Conway, J. H., An enumeration of knots and links, and some of their algebraic properties, in Computational problems in abstract algebra (Pergamon, 1970), 329-358. · Zbl 0202.54703 |
| [8] | Dowlin, N., A spectral sequence from Khovanov homology to knot Floer homology, Preprint (2018), arXiv:1811.07848. |
| [9] | Dunfield, N. M., Floer homology, group orderability, and taut foliations of hyperbolic 3-manifolds, Geom. Topol. 24 (2020), 2075-2125. · Zbl 1469.57020 |
| [10] | Greene, J. and Morrison, S., JavaKh (2020), http://katlas.math.toronto.edu/wiki/Khovanov_Homology. |
| [11] | Greene, J. E., Homologically thin, non-quasi-alternating links, Math. Res. Lett. 17 (2010), 39-49. · Zbl 1232.57008 |
| [12] | Greene, J. E., Alternating links and definite surfaces, Duke Math. J. 166 (2017), 2133-2151, With an appendix by András Juhász and Marc Lackenby. · Zbl 1377.57009 |
| [13] | Greene, J. E. and Watson, L., Turaev torsion, definite 4-manifolds, and quasi-alternating knots, Bull. Lond. Math. Soc. 45 (2013), 962-972. · Zbl 1420.57021 |
| [14] | Hedden, M. and Ni, Y., Manifolds with small Heegaard Floer ranks, Geom. Topol. 14 (2010), 1479-1501. · Zbl 1206.57014 |
| [15] | Hedden, M. and Ording, P., The Ozsváth-Szabó and Rasmussen concordance invariants are not equal, Amer. J. Math. 130 (2008), 441-453. · Zbl 1139.57012 |
| [16] | Howie, J. A., A characterisation of alternating knot exteriors, Geom. Topol. 21 (2017), 2353-2371. · Zbl 1375.57011 |
| [17] | Hanselman, J., Rasmussen, J. A., Rasmussen, S. D. and Watson, L., L-spaces, taut foliations, and graph manifolds, Compos. Math. 156 (2020), 604-612. · Zbl 1432.57032 |
| [18] | Hanselman, J., Rasmussen, J. and Watson, L., Heegaard Floer homology for manifolds with torus boundary: properties and examples, Proc. Lond. Math. Soc. (3)125 (2022), 879-967. · Zbl 1530.57018 |
| [19] | Hanselman, J., Rasmussen, J. A. and Watson, L., Bordered Floer homology for manifolds with torus boundary via immersed curves, J. Amer. Math. Soc. 37 (2024), 391-498. · Zbl 1539.57014 |
| [20] | Hoste, J. and Thistlethwaite, M. B., knotscape, a program for studying knot theory and providing convenient access to tables of knots (2021), https://web.math.utk.edu/ morwen/knotscape.html. |
| [21] | Kirby, R., Problems in low-dimensional topology (1995), https://math.berkeley.edu/ kirby/. |
| [22] | Kronheimer, P. and Mrowka, T., Khovanov homology is an unknot-detector, Publ. Math. Inst. Hautes Études Sci.113 (2011), 97-208. · Zbl 1241.57017 |
| [23] | Kotelskiy, A., Bordered theory for pillowcase homology, Math. Res. Lett. 26 (2019), 1467-1516. · Zbl 1430.53098 |
| [24] | Kazez, W. H. and Roberts, R., \(C^0\) approximations of foliations, Geom. Topol. 21 (2017), 3601-3657. · Zbl 1381.57014 |
| [25] | Kotelskiy, A., Watson, L. and Zibrowius, C., Immersed curves in Khovanov homology, Preprint (2019), arXiv:1910.14584. |
| [26] | Kotelskiy, A., Watson, L. and Zibrowius, C., Database of Khovanov tangle invariants that are used in this paper and that were computed with kht++ [Zib21] (2021), https://cbz20.raspberryip.com/code/khtpp/examples/ThinLinksAndConwaySpheres.html. |
| [27] | Kotelskiy, A., Watson, L. and Zibrowius, C., Khovanov multicurves are linear, Preprint (2021), arXiv:2202.01460. |
| [28] | Kotelskiy, A., Watson, L. and Zibrowius, C., Khovanov homology and strong inversions, in Gauge theory and low-dimensional topology—progress and interaction, (Mathematical Science Publishers, Berkeley, CA, 2022), 223-244. · Zbl 1519.57011 |
| [29] | Lee, E. S., An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005), 554-586. · Zbl 1080.57015 |
| [30] | Lickorish, W. B. R., An introduction to knot theory, (Springer, New York, 1997). · Zbl 0886.57001 |
| [31] | Lidman, T., Heegaard Floer homology and triple cup products, Preprint (2010), arXiv:1011.4277. |
| [32] | Lidman, T., Moore, A. H. and Zibrowius, C., \(L\)-space knots have no essential Conway spheres, Geom. Topol. 26 (2022), 2065-2102. · Zbl 1510.57013 |
| [33] | Lipshitz, R., Ozsváth, P. S. and Thurston, D. P., Bimodules in bordered Heegaard Floer homology, Geom. Topol. 19 (2015), 525-724. · Zbl 1315.57036 |
| [34] | Lewark, L. and Zibrowius, C., Rasmussen invariants, Math. Res. Postc. 1 (2021). |
| [35] | Lewark, L. and Zibrowius, C., Rasmussen invariants of Whitehead doubles and other satellites, Preprint (2022), arXiv:2208.13612. |
| [36] | Manolescu, C. and Ozsváth, P. S., On the Khovanov and knot Floer homologies of quasi-alternating links, in Proceedings of Gökova geometry-topology conference 2007 (Gökova Geometry/Topology Conference (GGT), Gökova, 2008), 60-81. · Zbl 1195.57032 |
| [37] | Ozsváth, P. S. and Szabó, Z., Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003), 225-254. · Zbl 1130.57303 |
| [38] | Ozsváth, P. S. and Szabó, Z., On the Floer homology of plumbed three-manifolds, Geom. Topol. 7 (2003), 185-224. · Zbl 1130.57302 |
| [39] | Ozsváth, P. S. and Szabó, Z., Holomorphic disks and genus bounds, Geom. Topol. 8 (2004), 311-334. · Zbl 1056.57020 |
| [40] | Ozsváth, P. S. and Szabó, Z., Holomorphic disks and knot invariants, Adv. Math. 186 (2004), 58-116. · Zbl 1062.57019 |
| [41] | Ozsváth, P. S. and Szabó, Z., Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. (2)159 (2004), 1159-1245. · Zbl 1081.57013 |
| [42] | Ozsváth, P. S. and Szabó, Z., On Heegaard diagrams and holomorphic disks, in European congress of mathematics (European Mathematical Society, Zürich, 2005), 769-781. · Zbl 1076.57014 |
| [43] | Ozsváth, P. S. and Szabó, Z., On the Heegaard Floer homology of branched double-covers, Adv. Math. 194 (2005), 1-33. · Zbl 1076.57013 |
| [44] | Ozsváth, P. S. and Szabó, Z., Holomorphic disks, link invariants and the multi-variable Alexander polynomial, Algebr. Geom. Topol. 8 (2008), 615-692. · Zbl 1144.57011 |
| [45] | Positsel’skii, L. E., Nonhomogeneous quadratic duality and curvature, Funct. Anal. Appl. 27 (1993), 197-204. · Zbl 0826.16041 |
| [46] | Rasmussen, J. A. and Rasmussen, S. D., Floer simple manifolds and L-space intervals, Adv. Math. 322 (2017), 738-805. · Zbl 1379.57024 |
| [47] | Rasmussen, S. D., L-space intervals for graph manifolds and cables, Compos. Math. 153 (2017), 1008-1049. · Zbl 1420.57042 |
| [48] | Scaduto, C. and Stoffregen, M., Two-fold quasi-alternating links, Khovanov homology and instanton homology, Quantum Topol. 9 (2018), 167-205. · Zbl 1397.57031 |
| [49] | Shumakovitch, A. N., Torsion of Khovanov homology, Fund. Math. 225 (2014), 343-364. · Zbl 1297.57022 |
| [50] | Shumakovitch, A. N., Torsion in Khovanov homology of homologically thin knots, J. Knot Theory Ramifications30 (2021), Paper No. 2141015. · Zbl 1490.57012 |
| [51] | Szabó, Z., Knot Floer homology calculator (2020), https://web.math.princeton.edu/ szabo/HFKcalc.html. |
| [52] | Watson, L., A surgical perspective on quasi-alternating links, in Low-dimensional and symplectic topology, (American Mathematical Society, Providence, RI, 2011), 39-51. · Zbl 1243.57002 |
| [53] | Watson, L., Surgery obstructions from Khovanov homology, Selecta Math. (N.S.)18 (2012), 417-472. · Zbl 1280.57002 |
| [54] | Watson, L., Khovanov homology and the symmetry group of a knot, Adv. Math. 313 (2017), 915-946. · Zbl 1368.57008 |
| [55] | Xie, Y. and Zhang, B., Classification of links with Khovanov homology of minimal rank, Preprint (2018), arXiv:1909.10032. |
| [56] | Zarev, R., Bordered Floer homology for sutured manifolds, Preprint (2009), arXiv:0908.1106. |
| [57] | Zibrowius, C., On a polynomial Alexander invariant for tangles and its categorification, in Essay for the Smith-Knight and Rayleigh-Knight prize competition 2015 (Cambridge, 2015), arXiv:1601.04915. |
| [58] | Zibrowius, C., APT.m, a Mathematica package for computing the decategorified polynomial tangle invariants \(\nabla^s_t\) from [Zib19] (2018), https://cbz20.raspberryip.com/code/APT/APT.zip. |
| [59] | Zibrowius, C., PQM.m, a Mathematica package for computing peculiar modules of 4-ended tangles (2018), https://cbz20.raspberryip.com/code/PQM/PQMv1.1.zip. |
| [60] | Zibrowius, C., Kauffman states and Heegaard diagrams for tangles, Algebr. Geom. Topol. 19 (2019), 2233-2282. · Zbl 1443.57008 |
| [61] | Zibrowius, C., Peculiar modules for 4-ended tangles, J. Topol. 13 (2020), 77-158. · Zbl 1473.57023 |
| [62] | Zibrowius, C., kht++, a program for computing Khovanov invariants for links and tangles (2021), https://cbz20.raspberryip.com/code/khtpp/docs/. |
| [63] | Zibrowius, C., Heegaard Floer multicurves of double tangles, Proceedings of Frontiers in Geometry and Topology, a conference in honour of Tom Mrowka’s 60th birthday, to appear. Preprint (2023), arXiv:2212.08501. |
| [64] | Zibrowius, C., On symmetries of peculiar modules, or \(\delta \)-graded link Floer homology is mutation invariant, J. Eur. Math. Soc. (JEMS)25 (2023), 2949-3006. · Zbl 07714629 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
