Mathematics Colloquia and Seminars

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We will define quandles, show why these are natural algebraic objects to consider when studying oriented knots, and examine how they can be used to classify knots. We will specifically review the fundamental quandle (also called the knot quandle), quandle counting invariant, and quandle State-Sum Invariant. For the state-sum invariant, we will discuss emerging strategies for using this invariant to tabulate prime knots and distinguish between mirror images when possible. Then, we will give a brief introduction to Legendrian knots, define Legendrian racks (and quandles) via the front projection and show how to use these racks (and quandles) to distinguish Legendrian knots using a similar counting invariant and State-Sum Invariant.