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\iteman{io-port 05998842}
\itemau{Nelson, Sam}
\itemti{The combinatorial revolution in knot theory.}
\itemso{Notices Am. Math. Soc. 58, No. 11, 1553-1561 (2011).}
\itemab
From the introduction: Doing knot theory in terms of knot diagrams, of course, is nothing new; the Reidemeister moves date back to the 1920s, and identifying knot invariants (functions used to distinguish different knot types) by checking invariance under the moves has been common ever since. A recent shift toward taking the combinatorial approach more seriously, however, has led to the discovery of new types of generalized knots and links that do not correspond to simple closed curves in $\bbfR^3$. Like the complex numbers arising from missing roots of real polynomials, the new generalized knot types appear as abstract solutions in knot equations that have no solutions among the classical geometric knots. Although they seem esoteric at first, these generalized knots turn out to have interpretations such as knotted circles or graphs in three-manifolds other than $\bbfR^3$, circuit diagrams, and operators in exotic algebras. Moreover, classical knot theory emerges as a special case of the new generalized knot theory. This diagram-based combinatorial approach to knot theory has revived interest in a related approach to algebraic knot invariants, applying techniques from universal algebra to turn the combinatorial structures into algebraic ones. The resulting algebraic objects, with names such as kei, quandles, racks, and biquandles, yield new invariants of both classical and generalized knots and provide new insights into old invariants. Much like groups arising from symmetries of geometric objects, these knot-inspired algebraic structures have connections to vector spaces, groups, Lie groups, Hopf algebras, and other mathematical structures. Consequentially, they have potential applications in disciplines from statistical mechanics to biochemistry to other areas of mathematics, with many promising open questions.
\itemrv{~}
\itemcc{}
\itemut{virtual knots; generalized knots; kei; quandle; knot invariant; enhancement}
\itemli{}
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