Turaev genus, knot signature, and the knot homology concordance invariants
HTML articles powered by AMS MathViewer
- by Oliver T. Dasbach and Adam M. Lowrance PDF
- Proc. Amer. Math. Soc. 139 (2011), 2631-2645 Request permission
Abstract:
We give bounds on knot signature, the Ozsváth-Szabó $\tau$ invariant, and the Rasmussen $s$ invariant in terms of the Turaev genus of the knot.References
- Tetsuya Abe, An estimation of the alternation number of a torus knot, J. Knot Theory Ramifications 18 (2009), no. 3, 363–379. MR 2514849, DOI 10.1142/S021821650900694X
- Abhijit Champanerkar and Ilya Kofman, Spanning trees and Khovanov homology, Proc. Amer. Math. Soc. 137 (2009), no. 6, 2157–2167. MR 2480298, DOI 10.1090/S0002-9939-09-09729-9
- Abhijit Champanerkar, Ilya Kofman, and Neal Stoltzfus, Graphs on surfaces and Khovanov homology, Algebr. Geom. Topol. 7 (2007), 1531–1540. MR 2366169, DOI 10.2140/agt.2007.7.1531
- Cornelia A. Van Cott, Ozsváth-Szabó and Rasmussen invariants of cable knots, Algebr. Geom. Topol. 10 (2010), no. 2, 825–836. MR 2629765, DOI 10.2140/agt.2010.10.825
- Oliver T. Dasbach, David Futer, Efstratia Kalfagianni, Xiao-Song Lin, and Neal W. Stoltzfus, Alternating sum formulae for the determinant and other link invariants, J. Knot Theory Ramifications 19 (2010), no. 6, 765–782, arXiv:math/0611025.
- Oliver T. Dasbach, David Futer, Efstratia Kalfagianni, Xiao-Song Lin, and Neal W. Stoltzfus, The Jones polynomial and graphs on surfaces, J. Combin. Theory Ser. B 98 (2008), no. 2, 384–399. MR 2389605, DOI 10.1016/j.jctb.2007.08.003
- Dieter Erle, Calculation of the signature of a 3-braid link, Kobe J. Math. 16 (1999), no. 2, 161–175. MR 1745024
- C. McA. Gordon, R. A. Litherland, and K. Murasugi, Signatures of covering links, Canadian J. Math. 33 (1981), no. 2, 381–394. MR 617628, DOI 10.4153/CJM-1981-032-3
- Joshua Greene, On closed $3$-braids with unknotting number one, arXiv:0902.1573., 2009.
- Matthew Hedden and Philip Ording, The Ozsváth-Szabó and Rasmussen concordance invariants are not equal, Amer. J. Math. 130 (2008), no. 2, 441–453. MR 2405163, DOI 10.1353/ajm.2008.0017
- Mikhail Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359–426. MR 1740682, DOI 10.1215/S0012-7094-00-10131-7
- Louis H. Kauffman and Laurence R. Taylor, Signature of links, Trans. Amer. Math. Soc. 216 (1976), 351–365. MR 388373, DOI 10.1090/S0002-9947-1976-0388373-0
- Eun Soo Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005), no. 2, 554–586. MR 2173845, DOI 10.1016/j.aim.2004.10.015
- Charles Livingston, Computations of the Ozsváth-Szabó knot concordance invariant, Geom. Topol. 8 (2004), 735–742. MR 2057779, DOI 10.2140/gt.2004.8.735
- Andrew Lobb, Computable bounds for Rasmussen’s concordance invariant, to appear in Compos. Math., arXiv:0908.2745, 2009.
- Adam M. Lowrance, On knot Floer width and Turaev genus, Algebr. Geom. Topol. 8 (2008), no. 2, 1141–1162. MR 2443110, DOI 10.2140/agt.2008.8.1141
- Adam Lowrance, The Khovanov width of twisted links and closed 3-braids, to appear in Comment. Math. Helv., arXiv:0901.2196, 2009.
- Kunio Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387–422. MR 171275, DOI 10.1090/S0002-9947-1965-0171275-5
- Kunio Murasugi, On closed $3$-braids, Memoirs of the American Mathematical Society, No. 151, American Mathematical Society, Providence, R.I., 1974. MR 0356023
- Kunio Murasugi, On invariants of graphs with applications to knot theory, Trans. Amer. Math. Soc. 314 (1989), no. 1, 1–49. MR 930077, DOI 10.1090/S0002-9947-1989-0930077-6
- Peter Ozsváth and Zoltán Szabó, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003), 225–254. MR 1988285, DOI 10.2140/gt.2003.7.225
- 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, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. (2) 159 (2004), no. 3, 1159–1245. MR 2113020, DOI 10.4007/annals.2004.159.1159
- Peter Ozsváth and Zoltán Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), no. 3, 1027–1158. MR 2113019, DOI 10.4007/annals.2004.159.1027
- Jacob Rasmussen, Floer homology and knot complements, Ph.D. thesis, Harvard University, 2003.
- Jacob Rasmussen, Khovanov homology and the slice genus, to appear in Invent. Math., arXiv:math.GT/0402131, 2004.
- Morwen B. Thistlethwaite, On the Kauffman polynomial of an adequate link, Invent. Math. 93 (1988), no. 2, 285–296. MR 948102, DOI 10.1007/BF01394334
- PawełTraczyk, A combinatorial formula for the signature of alternating diagrams, Fund. Math. 184 (2004), 311–316. MR 2128055, DOI 10.4064/fm184-0-17
- H. F. Trotter, Homology of group systems with applications to knot theory, Ann. of Math. (2) 76 (1962), 464–498. MR 143201, DOI 10.2307/1970369
- V. G. Turaev, A simple proof of the Murasugi and Kauffman theorems on alternating links, Enseign. Math. (2) 33 (1987), no. 3-4, 203–225. MR 925987
- S. Wehrli, A spanning tree model for Khovanov homology, J. Knot Theory Ramifications 17 (2008), no. 12, 1561–1574. MR 2477595, DOI 10.1142/S0218216508006762
Additional Information
- Oliver T. Dasbach
- Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803-4918
- MR Author ID: 612149
- Email: kasten@math.lsu.edu
- Adam M. Lowrance
- Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242-1419
- Email: alowrance@math.uiowa.edu
- Received by editor(s): March 9, 2010
- Received by editor(s) in revised form: July 6, 2010
- Published electronically: December 22, 2010
- Additional Notes: The first author was partially supported by NSF-DMS 0806539 and NSF-DMS FRG 0456275.
The second author was partially supported by NSF-DMS 0739382 (VIGRE) and NSF-DMS 0602242 (VIGRE) - Communicated by: Daniel Ruberman
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 2631-2645
- MSC (2010): Primary 57M25, 57M27
- DOI: https://doi.org/10.1090/S0002-9939-2010-10698-6
- MathSciNet review: 2784832