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Room P3.10, Mathematics Building
Knots are embeddings of $S_1$ into $3$-space and are classified modulo isotopy. Knot theorists have always been concerned with finding new ways of telling knots apart. In his PhD thesis, D. Joyce defined, for each knot, an algebraic gadget called knot quandle and proved it to be a classifying invariant for knots. In this way, the set of quandle homomorphisms (known as colorings) from the knot quandle to a given (labelling) quandle is also an invariant. We consider an apparently innocuous invariant, counting colorings, and illustrate its effectiveness on telling apart knots of up to and including ten crossings. In this talk we will go over these issues as well as introduce the CJKLS invariant, a knot invariant whose basic ingredients are quandle cohomology and colorings.