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Polynomial Quandle Cocycles, Their knot invariants and Applications
by
Kheira Ameur
In this talk, a variety of n-cocycles for n ≥ 2 are constructed for some Alexander quandles, given by polynomial expressions. As an application, these cocycles are used to compute the invariants for (2, n)-torus knots, twist knots and their r-twist spins. The calculations in the case of (2, n)-torus knots resulted in formulas that involved the derivative of the Alexander polynomial. Non-triviality of some quandle homology groups is also proved using these cocycles. Another application is given for tangle embeddings. The quandle cocycle invariants are used as obstructions to embedding tangles in links. The formulas for the cocycle invariants of tangles are obtained using polynomial cocycles, and by comparing the invariant values, information is obtained on which tangles do not embed in which knots. Tangles and knots in the tables are examined, and concrete examples are listed.
Date received: November 13, 2006
Copyright © 2006 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # catp-13.