Quandle cohomology and state-sum invariants of knotted curves and surfaces
@article{Carter1999QuandleCA, title={Quandle cohomology and state-sum invariants of knotted curves and surfaces}, author={J. Scott Carter and Daniel Jelsovsky and Seiichi Kamada and Laurel Langford and Masahico Saito}, journal={Transactions of the American Mathematical Society}, year={1999}, volume={355}, pages={3947-3989} }
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…
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References
SHOWING 1-10 OF 59 REFERENCES
A Combinatorial Description of Knotted Surfaces and Their Isotopies
- Mathematics
- 1997
Abstract We discuss the diagrammatic theory of knot isotopies in dimension 4. We project a knotted surface to a three-dimensional space and arrange the surface to have generic singularities upon…
Braided surfaces and seifert ribbons for closed braids
- Mathematics
- 1983
Apositive band in the braid groupBn is a conjugate of one of the standard generators; a negative band is the inverse of a positive band. Using the geometry of the configuration space, a theory of…
RACKS AND LINKS IN CODIMENSION TWO
- Mathematics
- 1992
A rack, which is the algebraic distillation of two of the Reidemeister moves, is a set with a binary operation such that right multiplication is an automorphism. Any codimension two link has a…
Reidemeister-type moves for surfaces in four-dimensional space
- Mathematics
- 1998
We consider smooth knottings of compact (not necessarily orientable) n-dimensional manifolds in R (or S), for the cases n = 2 or n = 3. In a previous paper we have generalized the notion of the…
Knots And Physics
- Mathematics
- 1991
Physical Knots States and the Bracket Polynomial The Jones Polynominal and Its Generalizations Braids and Polynomials: Formal Feynman Diagrams, Bracket as Vacuum-Vacmum expectation and the Quantum…
Quantum Invariants of Knots and 3-Manifolds
- Physics
- 1994
This monograph, now in its second revised edition, provides a systematic treatment of topological quantum field theories in three dimensions, inspired by the discovery of the Jones polynomial of…