%0 Journal Article %1 Carter1999a %A Carter, J. Scott %A Jelsovsky, Daniel %A Kamada, Seiichi %A Langford, Laurel %A Saito, Masahico %D 1999 %J Trans. Amer. Math. Soc. %K algebra homology knot-theory quandles topology %N 10 %P 2003 %T Quandle cohomology and state-sum invariants of knotted curves and surfaces %U http://arxiv.org/abs/math/9903135 %V 355 %X The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a state-sum invariant for classical knots and knotted surfaces is 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 the classical theory of knotted and linked curves in 3-space. 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 of classical knots and links and other linked surfaces. Non-triviality of the invariants are proved for variety of knots and links, including the trefoil and figure-eight knots, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.

@article{Carter1999a, abstract = {The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a state-sum invariant for classical knots and knotted surfaces is 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 the classical theory of knotted and linked curves in 3-space. 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 of classical knots and links and other linked surfaces. Non-triviality of the invariants are proved for variety of knots and links, including the trefoil and figure-eight knots, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles. }, added-at = {2009-05-25T19:36:48.000+0200}, author = {Carter, J. Scott and Jelsovsky, Daniel and Kamada, Seiichi and Langford, Laurel and Saito, Masahico}, biburl = {https://www.bibsonomy.org/bibtex/2bdf27ccfa665331aabe3e42b9666cd75/njj}, description = {Quandle Cohomology and State-sum Invariants of Knotted Curves and Surfaces}, interhash = {138e5164b0fccee42af23fe1f6797d89}, intrahash = {bdf27ccfa665331aabe3e42b9666cd75}, journal = {Trans. Amer. Math. Soc.}, keywords = {algebra homology knot-theory quandles topology}, number = 10, pages = 2003, timestamp = {2009-05-25T19:36:48.000+0200}, title = {Quandle cohomology and state-sum invariants of knotted curves and surfaces }, url = {http://arxiv.org/abs/math/9903135}, volume = 355, year = 1999 }