Mathematics Colloquia and Seminars

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Since the introduction of quantum invariants into knot theory, there has been a strong interest in relating them to the intrinsic geometry of a link complement. For example, the Volume Conjecture claims that the volume of a link is determined by its colored Jones polynomial. Another major task in knot theory is to relate the geometry of a link to its combinatorial picture, reflected in the link diagram. We will discuss how investigating the latter question might help to answer the former one. In particular, I will show how the upper bound for volume in terms of the twist number of a diagram (by M. Lackenby, I. Agol and D. Thurston) can be refined and expressed in terms of the coefficients of the colored Jones polynomial for alternating links. I will also show how to obtain the exact volume from a link diagram for some families of links (e.g. hyperbolic 2-bridge links). Different parts of this work are joint with O. Dasbach and M. Thistlethwaite.