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Untying (Generalized) Colored Shoelaces Dr. Steven Wallace, Macon State University Mathematics Department


Abstract

A knot is a circle living inside a 3-dimensional manifold. Two knots are equivalent if one may be obtained from another by a sequence of moves which do not require the object to pass through itself or break apart as it sits inside the higher dimensional space. Since it is often difficult to tell two knots apart strictly by these "Reidemeister moves," we often assign to every knot an algebraic object which is invariant of these moves. Two such classical invariants are n-colorability, and unknotting number. The former is the existence of a labeling of a planar projection of the knot using the "colors" {0, 1,..., n-1} which satisfies an appropriate condition at each crossing. The latter is the minimum number of crossing changes, taken over all planar projections, required to change the knot into one equivalent to the standard unknotted circle. In this talk, we will give generalizations of these classical invariants.
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