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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Quandle cohomology and state-sum invariants of knotted curves and surfaces


Authors: J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, Laurel Langford and Masahico Saito
Journal: Trans. Amer. Math. Soc. 355 (2003), 3947-3989
MSC (2000): Primary 57M25, 57Q45; Secondary 55N99, 18G99
Published electronically: June 24, 2003
MathSciNet review: 1990571
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Abstract: The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed. Our proof is based on a state-sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles (also known as distributive groupoids).

A quandle is a set with a binary operation -- the axioms of which model the Reidemeister moves in classical knot theory. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in $3$-space and knotted surfaces in $4$-space.

Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants. Non-triviality of the invariants is proved for a variety of knots and links, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.


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Additional Information

J. Scott Carter
Affiliation: Department of Mathematics, University of South Alabama, Mobile, Alabama 36688
Email: carter@jaguar1.usouthal.edu

Daniel Jelsovsky
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620
Address at time of publication: Department of Mathematics, Florida Southern College, Lakeland, Florida 33801
Email: jelsovsk@math.usf.edu, djelsovsky@flsouthern.edu

Seiichi Kamada
Affiliation: Department of Mathematics, Osaka City University, Osaka 558-8585, Japan
Address at time of publication: Department of Mathematics, Hiroshima University, Hiroshima 739-8526, Japan
Email: kamada@sci.osaka-cu.ac.jp, kamada@math.sci.hiroshima-u.ac.jp

Laurel Langford
Affiliation: Department of Mathematics, University of Wisconsin at River Falls, River Falls, Wisconsin 54022
Email: laurel.langford@uwrf.edu

Masahico Saito
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620
Email: saito@math.usf.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03046-0
PII: S 0002-9947(03)03046-0
Keywords: Knots, links, knotted surfaces, quandle, rack, quandle cohomology, state-sum invariants, non-invertibility
Received by editor(s): August 21, 2001
Received by editor(s) in revised form: February 20, 2002
Published electronically: June 24, 2003
Dedicated: Dedicated to Professor Kunio Murasugi for his 70th birthday
Article copyright: © Copyright 2003 American Mathematical Society