Georgia Institute of Technology College of Sciences

The slice-ribbon conjecture states that a knot in $S^3=partial D^4$ is the boundary of an embedded disc in $D^4$ if and only if it bounds a disc in $S^3$ which has only ribbon singularities. In this seminar we will prove the conjecture for a family of Montesinos knots. The proof is based on Donaldson's diagonalization theorem for definite four manifolds.

I will define character varieties of finitely generated groups and consider some applications of them in geometry/topology.

Abstract: We utilize the Ozsvath-Szabo contact invariant to detect the action of involutions on certain homology spheres that are surgeries on symmetric links, generalizing a previous result of Akbulut and Durusoy. Potentially this may be useful to detect different smooth structures on $4$-manifolds by cork twisting operation. This is a joint work with S. Akbulut.

The theorem of Birman and Hilden relates the mapping class group of a surface and its image under a covering map. I'll explore when we can extend the original theorem and possible applications for further work.

I will consider two constructions which lead to information about the topology of a 3-manifold from one of its triangulation. The first construction is a modification of the Turaev-Viro invariant based on re-normalized 6j-symbols. These re-normalized 6j-symbols satisfy tetrahedral symmetries. The second construction is a generalization of Kashaev's invariant defined in his foundational paper where he first stated the volume conjecture. This generalization is based on symmetrizing 6j-symbols using *charges* developed by W. Neumann, S. Baseilhac, and R. Benedetti.

I will give an example of transforming a knot into closed braid form using Yamada-Vogel algorithm. From this we can write down the corresponding element of the knot in the braid group. Finally, the definition of a colored Jones polynomial is given using a Yang-Baxter operator. This is a preparation for next week's talk by Anh.