Stephen Bigelow developed a new approach to the Jones polynomial of knots and more generally, links in the three sphere as an intersection number of a homo- logical cycle and a cocycle on a covering space of a configuration space. A link in this approach is represented by the plat closure of a braid in B2n.
The work in this thesis is to generalize several key parts of Bigelow’s construction.
A link will be obtained by taking the plat closure of a braid in the braid group of a punctured torus. A study of the braid group of the torus as well as punctured torus provides tools for the constructions of some functions, which turn out be a generalization of Bigelow’s function in the definition of the covering space. We explore some new relations between braid groups on a disk and their generalizations to braid groups of a torus and punctured torus. As a consequence, some useful abelian representation of configuration spaces are generalized to larger settings, involving a representation to some Heisengerg type group. The topological groupfeature of a torus is considered when studying its braid group, and this leads to an interesting subgroup K which turns out to be the normal closure of the braid group of the disk in the braid group of the torus. Several parts of Bigelow’s construction has been generalized to the preimage of K under the inclusion map of the punctured torus into the torus.
Some potential applications of the work is to develop a Jones-type invariant in any 3-Manifold admitting a Heegaard splitting of genus 1, where we can decompose the link as a plat closure of a punctured torus braid.
|Advisor:||Cappell, Sylvain E.|
|Commitee:||Miller, Edward, Pirutka, Alena, Yang, Deane, Young, Robert|
|School:||New York University|
|School Location:||United States -- New York|
|Source:||DAI-B 79/02(E), Dissertation Abstracts International|
|Keywords:||Braid group, Configuration space, Jones polynomial, Knots and links|
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