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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