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Khovanov homology and periodic links. (English) Zbl 1480.57012

From the paper: “Based on the results of the second author, we define an equivariant version of Lee and Bar-Natan homology for periodic links and show that there exists an equivariant spectral sequence from the equivariant Khovanov homology to equivariant Lee homology. As a result we obtain new obstructions for a link to be periodic. These obstructions generalize previous results of Przytycki and of the second author.”
This is a pioneering paper categorifying Kauffman bracket periodicity criteria. It would be very nice to generalize the authors’ results and categorify the Jaeger composition product [F. Jaeger, Enseign. Math. (2) 35, No. 3–4, 323–361 (1989; Zbl 0705.57004)]. This product allows very strong periodicity criteria as described in [J. H. Przytycki, Proc. Am. Math. Soc. 123, No. 5, 1607–1611 (1995; Zbl 0843.57009)]; see also [P. Traczyk, Invent. Math. 106, No. 1, 73–84 (1991; Zbl 0753.57008); Y. Yokota, Math. Ann. 291, No. 2, 281–292 (1991; Zbl 0724.57007)].

MSC:

57K18 Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.)
57K10 Knot theory

Software:

SnapPy; knotkit; GitHub
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Full Text: DOI arXiv

References:

[1] D. BAR-NATAN, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443-1499. http://dx.doi.org/10.2140/gt.2005.9.1443. MR2174270 · Zbl 1084.57011
[2] M. CULLER, N. DUNFIELD, M. GOERNER, and J. WEEKS, SnapPy, a computer program for studying the geometry and topology of 3-manifolds.
[3] C. W. CURTIS and I. REINER, Methods of Representation Theory. Vol. I: With Applications to Finite Groups and Orders, Reprint of the 1981 original, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. A Wiley-Interscience Publication. MR1038525
[4] J. A. HILLMAN, C. LIVINGSTON, and S. NAIK, Twisted Alexander polynomials of periodic knots, Algebr. Geom. Topol. 6 (2006), 145-169. http://dx.doi.org/10.2140/agt.2006.6. 145. MR2199457 · Zbl 1084.57011
[5] F. HOSOKAWA and S. KINOSHITA, On the homology group of branched cyclic covering spaces of links, Osaka Math. J. 12 (1960), 331-355. MR125579 · Zbl 0104.39904
[6] S. JABUKA and S. NAIK, Periodic knots and Heegaard Floer correction terms, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 8, 1651-1674. http://dx.doi.org/10.4171/JEMS/624. MR3519536 · Zbl 1352.57010
[7] V. F. R. JONES, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335-388. http://dx.doi.org/10.2307/1971403. MR908150 · Zbl 1097.57010
[8] M. KHOVANOV, Link homology and Frobenius extensions, Fund. Math. 190 (2006), 179-190. http://dx.doi.org/10.4064/fm190-0-6. MR2232858 · Zbl 1101.57004
[9] E. S. LEE, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005), no. 2, 554-586. http://dx.doi.org/10.1016/j.aim.2004.10.015. MR2173845 · Zbl 1352.57010
[10] J. P. MAYBERRY and K. MURASUGI, Torsion-groups of abelian coverings of links, Trans. Amer. Math. Soc. 271 (1982), no. 1, 143-173. http://dx.doi.org/10.2307/1998756. MR648083 · Zbl 0631.57005
[11] J. MCCLEARY, A User’s Guide to Spectral Sequences, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 58, Cambridge University Press, Cambridge, 2001. http://dx.doi.org/10. 1017/CBO9780511626289. MR1793722 · Zbl 1101.57004
[12] H. MURAKAMI, A recursive calculation of the Arf invariant of a link, J. Math. Soc. Japan 38 (1986), no. 2, 335-338. http://dx.doi.org/10.2969/jmsj/03820335. MR833206 · Zbl 1080.57015
[13] K. MURASUGI, On periodic knots, Comment. Math. Helv. 46 (1971), 162-174. http://dx. doi.org/10.1007/BF02566836. MR292060 · Zbl 0487.57001
[14] S. NAIK, New invariants of periodic knots, Math. Proc. Cambridge Philos. Soc. 122 (1997), no. 2, 281-290. http://dx.doi.org/10.1017/S0305004197001801. MR1458233 · Zbl 0891.57009
[15] P. OZSVÁTH and Z. SZABÓ, On the Heegaard Floer homology of branched double-covers, Adv. Math. 194 (2005), no. 1, 1-33. http://dx.doi.org/10.1016/j.aim.2004.05. 008. MR2141852 · Zbl 0605.57003
[16] W. POLITARCZYK, Equivariant Khovanov homology of periodic links (2015), preprint, available at http://arxiv.org/abs/arXiv:1504.03462. · Zbl 1477.57016
[17] , Equivariant Jones polynomials of periodic links, J. Knot Theory Ramifications 26 (2017), no. 3, 1741007, 21. http://dx.doi.org/10.1142/S0218216517410073. MR3627707
[18] J. H. PRZYTYCKI, On Murasugi’s and Traczyk’s criteria for periodic links, Math. Ann. 283 (1989), no. 3, 465-478. http://dx.doi.org/10.1007/BF01442739. MR985242 · Zbl 0642.57007
[19] J. RASMUSSEN, Khovanov homology and the slice genus, Invent. Math. 182 (2010), no. 2, 419-447. http://dx.doi.org/10.1007/s00222-010-0275-6. MR2729272 · Zbl 1211.57009
[20] M. SAKUMA, On the polynomials of periodic links, Math. Ann. 257 (1981), no. 4, 487-494. http://dx.doi.org/10.1007/BF01465869. MR639581 · Zbl 1376.57011
[21] , Periods of composite links, Math. Sem. Notes Kobe Univ. 9 (1981), no. 2, 445-452. MR650749
[22] C. SEED, Knotkit, with a module checking periodicity by W. POLITARCZYK (2016), available at https://github.com/wpolitarczyk/knotkit. · Zbl 1211.57009
[23] P. TRACZYK, 10 101 has no period 7: A criterion for periodic links, Proc. Amer. Math. Soc. 108 (1990), no. 3, 845-846. http://dx.doi.org/10.2307/2047810. MR1031676 · Zbl 0458.57002
[24] , Periodic knots and the skein polynomial, Invent. Math. 106 (1991), no. 1, 73-84. http://dx.doi.org/10.1007/BF01243905. MR1123374 · Zbl 0753.57008
[25] V. G. TURAEV, Reidemeister torsion in knot theory, Uspekhi Mat. Nauk 41 (1986), no. 1(247), 97-147, 240 (Russian).
[26] http://dx.doi.org/10.1070/RM1986v041n01ABEH003204. MR832411
[27] P. R. TURNER, Calculating Bar-Natan’s characteristic two Khovanov homology, J. Knot Theory Ramifications 15 (2006), no. 10, 1335-1356. http://dx.doi.org/10.1142/ S0218216506005111. MR2286127 · Zbl 0753.57008
[28] L. WATSON, Khovanov homology and the symmetry group of a knot, Adv. Math. 313 (2017), 915-946. http://dx.doi.org/10.1016/j.aim.2017.04.003. MR3649241 · Zbl 1368.57008
[29] J. R. WEEKS, Convex hulls and isometries of cusped hyperbolic 3-manifolds, Topology Appl. 52 (1993), no. 2, 127-149. · Zbl 0602.57005
[30] http://dx.doi.org/10.1016/0166-8641(93)90032-9. MR1241189
[31] C. A. WEIBEL, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathe-matics, vol. 38, Cambridge University Press, Cambridge, 1994. http://dx.doi.org/10.1017/ CBO9781139644136. MR1269324 · Zbl 1368.57008
[32] M. ZHANG, A rank inequality for the annular Khovanov homology of 2-periodic links, Algebr. Geom. Topol. 18 (2018), no. 2, 1147-1194. http://dx.doi.org/10.2140/agt.2018.18. 1147. MR3773751 · Zbl 1394.57009
[33] MACIEJ BORODZIK: Institute of Mathematics AND Institute of Mathematics
[34] Warsaw, Poland 02-097 Warsaw, Poland E-MAIL: mcboro@mimuw.edu.pl WOJCIECH POLITARCZYK: Dept. of Mathematics and Computer Science AND Institute of Mathematics · Zbl 1394.57009
[35] Warsaw, Poland E-MAIL: politarw@amu.edu.pl E-MAIL: wpolitarczyk@mimuw.edu.pl KEY WORDS AND PHRASES: Periodic links, Khovanov homology.
[36] MATHEMATICS SUBJECT CLASSIFICATION: 57M25. Received: June 4, 2018.
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