Virtual Yang-Baxter cocycle invariants

Ceniceros, Jose; Nelson, Sam

doi:10.1090/S0002-9947-09-04751-5

Virtual Yang-Baxter cocycle invariants
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by Jose Ceniceros and Sam Nelson PDF

Trans. Amer. Math. Soc. 361 (2009), 5263-5283 Request permission

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

Nicolás Andruskiewitsch and Matías Graña, From racks to pointed Hopf algebras, Adv. Math. 178 (2003), no. 2, 177–243. MR 1994219, DOI 10.1016/S0001-8708(02)00071-3
J. S. Carter, A. S. Crans, M. Elhamdadi and M. Saito. Cohomology of Categorical Self-Distributivity. arXiv:math/0607417
J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, Laurel Langford, and Masahico Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc. 355 (2003), no. 10, 3947–3989. MR 1990571, DOI 10.1090/S0002-9947-03-03046-0
J. Scott Carter, Mohamed Elhamdadi, and Masahico Saito, Twisted quandle homology theory and cocycle knot invariants, Algebr. Geom. Topol. 2 (2002), 95–135. MR 1885217, DOI 10.2140/agt.2002.2.95
J. Scott Carter, Mohamed Elhamdadi, and Masahico Saito, Homology theory for the set-theoretic Yang-Baxter equation and knot invariants from generalizations of quandles, Fund. Math. 184 (2004), 31–54. MR 2128041, DOI 10.4064/fm184-0-3
J. Scott Carter and Masahico Saito, Set-theoretic Yang-Baxter solutions via Fox calculus, J. Knot Theory Ramifications 15 (2006), no. 8, 949–956. MR 2275090, DOI 10.1142/S0218216506004877
Conrad Creel and Sam Nelson, Symbolic computation with finite biquandles, J. Symbolic Comput. 42 (2007), no. 10, 992–1000. MR 2361675, DOI 10.1016/j.jsc.2007.08.006
Roger Fenn, Mercedes Jordan-Santana, and Louis Kauffman, Biquandles and virtual links, Topology Appl. 145 (2004), no. 1-3, 157–175. MR 2100870, DOI 10.1016/j.topol.2004.06.008
David Hrencecin and Louis H. Kauffman, Biquandles for virtual knots, J. Knot Theory Ramifications 16 (2007), no. 10, 1361–1382. MR 2384830, DOI 10.1142/S0218216507005841
Nicholas Jackson, Extensions of racks and quandles, Homology Homotopy Appl. 7 (2005), no. 1, 151–167. MR 2155522
Naoko Kamada and Seiichi Kamada, Abstract link diagrams and virtual knots, J. Knot Theory Ramifications 9 (2000), no. 1, 93–106. MR 1749502, DOI 10.1142/S0218216500000049
Louis H. Kauffman, Virtual knot theory, European J. Combin. 20 (1999), no. 7, 663–690. MR 1721925, DOI 10.1006/eujc.1999.0314
Louis H. Kauffman and David Radford, Bi-oriented quantum algebras, and a generalized Alexander polynomial for virtual links, Diagrammatic morphisms and applications (San Francisco, CA, 2000) Contemp. Math., vol. 318, Amer. Math. Soc., Providence, RI, 2003, pp. 113–140. MR 1973514, DOI 10.1090/conm/318/05548
Louis H. Kauffman and Vassily O. Manturov, Virtual biquandles, Fund. Math. 188 (2005), 103–146. MR 2191942, DOI 10.4064/fm188-0-6
S. Nelson and J. Rische. On Bilinear Biquandles. Colloq. Math. 112 (2008), 279–289.
Sam Nelson and John Vo, Matrices and finite biquandles, Homology Homotopy Appl. 8 (2006), no. 2, 51–73. MR 2246021