No Access

Twisted skein homology

Nguyen D. Duong

Department of Mathematics, The University of Alabama, Tuscaloosa, AL 35487-0350, USA

Search for more papers by this author

and

Lawrence P. Roberts

Department of Mathematics, The University of Alabama, Tuscaloosa, AL 35487-0350, USA

Search for more papers by this author

https://doi.org/10.1142/S0218216514500278Cited by:0 (Source: Crossref)

Abstract

We apply the techniques of totally twisted Khovanov homology to Asaeda, Przytycki, and Sikora's construction of Khovanov type homologies for links and tangles in I-bundles over (orientable) surfaces. As a result we describe a chain complex built out of resolutions with only noncontractible circles whose homology is an invariant of the tangle. We use these to understand the δ-graded homology for links with alternating diagrams in the surface.

Keywords:

AMSC: 57M27, 55N99

References

M. Asaeda , J. Przytycki and A. Sikora , Algebr. Geom. Topol. 4 , 1177 ( 2004 ) . Crossref, Google Scholar
M. Asaeda, J. Przytycki and A. Sikora, Primes and Knots, Contemporary Mathematics 416 (American Mathematical Society, Providence, RI, 2006) pp. 1–8. Crossref, Google Scholar
J. Baldwin and A. Levine, Adv. Math. 231(4), 1886 (2012). Crossref, Web of Science, Google Scholar
D. Bar-Natan , Algebr. Geom. Topol. 2 , 337 ( 2002 ) . Crossref, Google Scholar
D. Bar-Natan , Geom. Topol. 9 , 1443 ( 2005 ) . Crossref, Web of Science, Google Scholar
A. Champanerkar and I. Kofman, Proc. Amer. Math. Soc. 137(6), 2157 (2009). Crossref, Web of Science, Google Scholar
C. Gordon and R. Litherland, Invent. Math. 47(1), 53 (1978). Crossref, Web of Science, Google Scholar
T. Jaeger, J. Knot Theory Ramifications 22(6), 1350022 (2013). Link, Web of Science, Google Scholar
M. Khovanov, Duke Math. J. 101(3), 359 (2000). Crossref, Web of Science, Google Scholar
D. Kriz and I. Kriz , Adv. Math. 255 , 414 ( 2014 ) . Crossref, Web of Science, Google Scholar
E. S. Lee, Adv. Math. 197(2), 554 (2005). Crossref, Web of Science, Google Scholar
A. Manion, Algebr. Geom. Topol. 14(2), 753 (2014). Crossref, Web of Science, Google Scholar
P. Ozsváth and Z. Szabó , Geom. Topol. 7 , 225 ( 2003 ) . Crossref, Web of Science, Google Scholar
L. Roberts, Geom. Topol. 17(1), 413 (2013). Crossref, Web of Science, Google Scholar
L. Roberts, Totally twisted Khovanov homology, accepted for publication in Geom. Topol . Google Scholar
O. Viro , Fund. Math. 184 , 317 ( 2004 ) . Crossref, Web of Science, Google Scholar
S. Wehrli, J. Knot Theory Ramifications 17(12), 1561 (2008). Link, Web of Science, Google Scholar