|
Virtual Yang-Baxter cocycle invariants
Author(s):
Jose
Ceniceros;
Sam
Nelson
Journal:
Trans. Amer. Math. Soc.
361
(2009),
5263-5283.
MSC (2000):
Primary 57M27, 18G60
Posted:
April 8, 2009
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virtual knots and links. In particular, they provide a method for detecting non-classicality of virtual knots and links.
References:
-
- 1.
- N. Andruskiewitsch and M. Graña. From racks to pointed Hopf algebras. Adv. Math. 178 (2003) 177-243. MR 1994219 (2004i:16046)
- 2.
- J. S. Carter, A. S. Crans, M. Elhamdadi and M. Saito. Cohomology of Categorical Self-Distributivity. arXiv:math/0607417
- 3.
- J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito. Quandle cohomology and state-sum invariants of knotted curves and surfaces. Trans. Amer. Math. Soc. 355 (2003) 3947-3989. MR 1990571 (2005b:57048)
- 4.
- J. S. Carter, M. Elhamdadi, and M. Saito. Twisted quandle homology theory and cocycle knot invariants. Algebr. Geom. Topol. 2 (2002) 95-135. MR 1885217 (2003a:57019)
- 5.
- J. S. Carter, M. Elhamdadi and M. Saito. Homology Theory for the Set-Theoretic Yang-Baxter Equation and Knot Invariants from Generalizations of Quandles. MR 2128041 (2005k:57009)
- 6.
- J. S. Carter and M. Saito. Set-Theoretic Yang-Baxter Solutions via Fox Calculus. J. Knot Theory Ramifications 15 (2006) 949-956. MR 2275090
- 7.
- C. Creel and S. Nelson, Symbolic computation with finite biquandles. J. Symbolic Comput. 42 (2007) 992-1000. MR 2361675
- 8.
- R. Fenn, M. Jordan-Santana and L. Kauffman. Biquandles and virtual links. Topology Appl. 145 (2004) 157-175. MR 2100870 (2005h:57015)
- 9.
- D. Hrencecin and L.H. Kauffman. Biquandles for virtual knots. J. Knot Theory Ramifications 16 (2007) 1361-1382. MR 2384830
- 10.
- N. Jackson. Extensions of racks and quandles. Homology, Homotopy Appl. 7 (2005) 151-167. MR 2155522 (2006f:18006)
- 11.
- N. Kamada and S. Kamada. Abstract link diagrams and virtual knots. J. Knot Theory Ramifications 9 (2000) 93-106. MR 1749502 (2001h:57007)
- 12.
- L. Kauffman. Virtual Knot Theory. European J. Combin. 20 (1999) 663-690. MR 1721925 (2000i:57011)
- 13.
- L. H. Kauffman and D. Radford. Bi-oriented quantum algebras, and a generalized Alexander polynomial for virtual links. Contemp. Math. 318 (2003) 113-140. MR 1973514 (2004c:57013)
- 14.
- L. H. Kauffman and V. O. Manturov. Virtual biquandles. Fundam. Math. 188 (2005) 103-146. MR 2191942 (2006k:57015)
- 15.
- S. Nelson and J. Rische. On Bilinear Biquandles. Colloq. Math. 112 (2008), 279-289.
- 16.
- S. Nelson and J. Vo. Matrices and Finite Biquandles. Homology, Homotopy Appl. 8 (2006) 51-73. MR 2246021 (2007j:57009)
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
57M27, 18G60
Retrieve articles in all Journals with MSC
(2000):
57M27, 18G60
Additional Information:
Jose
Ceniceros
Affiliation:
Department of Mathematics, Whittier College, 13406 Philadelphia, P.O. Box 634, Whittier, California 90608-0634
Email:
jcenicer@poets.whittier.edu
Sam
Nelson
Affiliation:
Department of Mathematics, Claremont McKenna College, 850 Columbia Avenue, Claremont, California 91711
Email:
knots@esotericka.org
DOI:
10.1090/S0002-9947-09-04751-5
PII:
S 0002-9947(09)04751-5
Received by editor(s):
September 6, 2007
Posted:
April 8, 2009
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org