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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Quandle cohomology and state-sum invariants of knotted curves and surfaces
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by J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, Laurel Langford and Masahico Saito PDF
Trans. Amer. Math. Soc. 355 (2003), 3947-3989 Request permission

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
  • MR Author ID: 682724
  • 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
  • MR Author ID: 288529
  • 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
  • MR Author ID: 196333
  • Email: saito@math.usf.edu
  • 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
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 355 (2003), 3947-3989
  • MSC (2000): Primary 57M25, 57Q45; Secondary 55N99, 18G99
  • DOI: https://doi.org/10.1090/S0002-9947-03-03046-0
  • MathSciNet review: 1990571