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Twisting quasi-alternating links
Authors:
Abhijit Champanerkar and Ilya Kofman
Journal:
Proc. Amer. Math. Soc. 137 (2009), 2451-2458
MSC (2000):
Primary 57M25
Posted:
March 10, 2009
MathSciNet review:
2495282
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References |
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Additional Information
Abstract: Quasi-alternating links are homologically thin for both Khovanov homology and knot Floer homology. We show that every quasi-alternating link gives rise to an infinite family of quasi-alternating links obtained by replacing a crossing with an alternating rational tangle. Consequently, we show that many pretzel links are quasi-alternating, and we determine the thickness of Khovanov homology for ``most'' pretzel links with arbitrarily many strands.
- 1.
J. Baldwin, Heegaard Floer homology and genus one, one boundary component open books (arXiv:0804.3624v2 [math.GT]).
- 2.
Gerhard
Burde and Heiner
Zieschang, Knots, 2nd ed., de Gruyter Studies in Mathematics,
vol. 5, Walter de Gruyter & Co., Berlin, 2003. MR 1959408
(2003m:57005)
- 3.
Abhijit
Champanerkar and Ilya
Kofman, On links with cyclotomic Jones polynomials, Algebr.
Geom. Topol. 6 (2006), 1655–1668 (electronic). MR 2253460
(2007f:57009), http://dx.doi.org/10.2140/agt.2006.6.1655
- 4.
-, Spanning trees and Khovanov homology, Proc. Amer. Math. Soc. 137 (2009), no. 6, 2157-2167.
- 5.
Abhijit
Champanerkar, Ilya
Kofman, and Neal
Stoltzfus, Graphs on surfaces and Khovanov homology, Algebr.
Geom. Topol. 7 (2007), 1531–1540. MR 2366169
(2008m:57005), http://dx.doi.org/10.2140/agt.2007.7.1531
- 6.
O. T. Dasbach, D. Futer, E. Kalfagianni, X. S. Lin, and N. W. Stoltzfus, Alternating sum formulae for the determinant and other link invariants (arXiv:math/0611025v2 [math.GT]).
- 7.
E. Eftekhary, Heegaard Floer homologies of pretzel knots (arXiv:math.GT/0311419).
- 8.
J. Greene, A spanning tree model for the Heegaard Floer homology of a branched double-cover (arXiv:0805.1381v1 [math.GT]).
- 9.
Eriko
Hironaka, The Lehmer polynomial and pretzel links, Canad.
Math. Bull. 44 (2001), no. 4, 440–451. MR 1863636
(2002g:57009), http://dx.doi.org/10.4153/CMB-2001-044-x
- 10.
Akio
Kawauchi, A survey of knot theory, Birkhäuser Verlag,
Basel, 1996. Translated and revised from the 1990 Japanese original by the
author. MR
1417494 (97k:57011)
- 11.
Mikhail
Khovanov, Patterns in knot cohomology. I, Experiment. Math.
12 (2003), no. 3, 365–374. MR 2034399
(2004m:57022)
- 12.
Paolo
Lisca and András
I. Stipsicz, Ozsváth-Szabó invariants and tight
contact 3-manifolds. III, J. Symplectic Geom. 5
(2007), no. 4, 357–384. MR 2413308
(2009k:57020)
- 13.
A. Lowrance, On knot Floer width and Turaev genus (arXiv:0709.0720v1).
- 14.
Ciprian
Manolescu, An unoriented skein exact triangle for knot Floer
homology, Math. Res. Lett. 14 (2007), no. 5,
839–852. MR 2350128
(2008m:57074)
- 15.
C. Manolescu and P. Ozsváth, On the Khovanov and knot Floer homologies of quasi-alternating links (arXiv:0708.3249v2 [math.GT]).
- 16.
V. Manturov, Minimal diagrams of classical and virtual links (arXiv:math.GT/0501393).
- 17.
Peter
Ozsváth and Zoltán
Szabó, Knot Floer homology, genus bounds, and mutation,
Topology Appl. 141 (2004), no. 1-3, 59–85. MR 2058681
(2005b:57028), http://dx.doi.org/10.1016/j.topol.2003.09.009
- 18.
Peter
Ozsváth and Zoltán
Szabó, On the Heegaard Floer homology of branched
double-covers, Adv. Math. 194 (2005), no. 1,
1–33. MR
2141852 (2006e:57041), http://dx.doi.org/10.1016/j.aim.2004.05.008
- 19.
A. Shumakovitch, KhoHo, available from http://www.geometrie.ch/KhoHo/ (2003).
- 20.
A. Shumakovitch, Private communication, March 2008.
- 21.
R. Suzuki, Khovanov homology and Rasmussen's s-invariants for pretzel knots (arXiv:math.QA/0610913).
- 22.
Morwen
B. Thistlethwaite, A spanning tree expansion of the Jones
polynomial, Topology 26 (1987), no. 3,
297–309. MR
899051 (88h:57007), http://dx.doi.org/10.1016/0040-9383(87)90003-6
- 23.
T. Widmer, Quasi-alternating Montesinos links (arXiv:math/0811.0270 [math.GT]).
- 1.
- J. Baldwin, Heegaard Floer homology and genus one, one boundary component open books (arXiv:0804.3624v2 [math.GT]).
- 2.
- G. Burde and H. Zieschang, Knots, second ed., de Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 2003. MR 1959408 (2003m:57005)
- 3.
- A. Champanerkar and I. Kofman, On links with cyclotomic Jones polynomials, Algebr. Geom. Topol. 6 (2006), 1655-1668. MR 2253460 (2007f:57009)
- 4.
- -, Spanning trees and Khovanov homology, Proc. Amer. Math. Soc. 137 (2009), no. 6, 2157-2167.
- 5.
- A. Champanerkar, I. Kofman, and N. Stoltzfus, Graphs on surfaces and Khovanov homology, Algebr. Geom. Topol. 7 (2007), 1531-1540. MR 2366169 (2008m:57005)
- 6.
- O. T. Dasbach, D. Futer, E. Kalfagianni, X. S. Lin, and N. W. Stoltzfus, Alternating sum formulae for the determinant and other link invariants (arXiv:math/0611025v2 [math.GT]).
- 7.
- E. Eftekhary, Heegaard Floer homologies of pretzel knots (arXiv:math.GT/0311419).
- 8.
- J. Greene, A spanning tree model for the Heegaard Floer homology of a branched double-cover (arXiv:0805.1381v1 [math.GT]).
- 9.
- E. Hironaka, The Lehmer polynomial and pretzel links, Canad. Math. Bull. 44 (2001), no. 4, 440-451. MR 1863636 (2002g:57009)
- 10.
- A. Kawauchi, A survey of knot theory, Birkhäuser Verlag, Basel, 1996; translated and revised from the 1990 Japanese original by the author. MR 1417494 (97k:57011)
- 11.
- M. Khovanov, Patterns in knot cohomology. I, Experiment. Math. 12 (2003), no. 3, 365-374. MR 2034399 (2004m:57022)
- 12.
- P. Lisca and A. Stipsicz, Ozsváth-Szabó invariants and tight contact three-manifolds. III, J. Symplectic Geom. 5 (2007), no. 4, 357-384. MR 2413308
- 13.
- A. Lowrance, On knot Floer width and Turaev genus (arXiv:0709.0720v1).
- 14.
- C. Manolescu, An unoriented skein exact triangle for knot Floer homology, Math. Res. Lett. 14 (2007), no. 5, 839-852. MR 2350128 (2008m:57074)
- 15.
- C. Manolescu and P. Ozsváth, On the Khovanov and knot Floer homologies of quasi-alternating links (arXiv:0708.3249v2 [math.GT]).
- 16.
- V. Manturov, Minimal diagrams of classical and virtual links (arXiv:math.GT/0501393).
- 17.
- P. Ozsváth and Z. Szabó, Knot Floer homology, genus bounds, and mutation, Topology Appl. 141 (2004), no. 1-3, 59-85. MR 2058681 (2005b:57028)
- 18.
- -, On the Heegaard Floer homology of branched double-covers, Adv. Math. 194 (2005), no. 1, 1-33. MR 2141852 (2006e:57041)
- 19.
- A. Shumakovitch, KhoHo, available from http://www.geometrie.ch/KhoHo/ (2003).
- 20.
- A. Shumakovitch, Private communication, March 2008.
- 21.
- R. Suzuki, Khovanov homology and Rasmussen's s-invariants for pretzel knots (arXiv:math.QA/0610913).
- 22.
- M. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987), 297-309. MR 899051 (88h:57007)
- 23.
- T. Widmer, Quasi-alternating Montesinos links (arXiv:math/0811.0270 [math.GT]).
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Additional Information
Abhijit Champanerkar
Affiliation:
Department of Mathematics, College of Staten Island, The City University of New York, Staten Island, New York 10314
Email:
abhijit@math.csi.cuny.edu
Ilya Kofman
Affiliation:
Department of Mathematics, College of Staten Island, The City University of New York, Staten Island, New York 10314
Email:
ikofman@math.csi.cuny.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-09-09876-1
PII:
S 0002-9939(09)09876-1
Keywords:
Khovanov homology,
knot Floer homology,
pretzel link
Received by editor(s):
April 22, 2008
Posted:
March 10, 2009
Additional Notes:
The first author was supported by NSF grant DMS-0844485.
The second author was supported by NSF grant DMS-0456227 and a PSC-CUNY grant.
Communicated by:
Daniel Ruberman
Article copyright:
© Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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