Adam S. Sikora

Skein algebra of a group

We define for each group G the skein algebra of G. We show how it is related to the Kauffman bracket skein modules. We prove that skein algebras of abelian groups are isomorphic to symmetric subalgebras of corresponding group rings. Moreover, we show that, for any abelian group G, homomorphisms from the skein algebra of G to C correspond exactly to traces of SL(2, C)-representations of G. We also solve, for abelian groups, the conjecture of Bullock on SL(2, C) character varieties of groups – we show that skein algebras are isomorphic to the coordinate rings of corresponding to them character varieties.

Distributive Products and Their Homology

We develop a theory of sets with distributive products (called shelves and multi-shelves) and of their homology. We relate the shelf homology to the rack and quandle homology.

Confluence theory for graphs

We develop a theory of confluence of graphs. We describe an algorithm for proving that a given system of reduction rules for abstract graphs and graphs in surfaces is locally confluent. We apply this algorithm to show that each simple Lie algebra of rank at most 2, gives rise to a confluent system of reduction rules of graphs (via Kuperbergs spiders) in an arbitrary surface. As a further consequence of this result, we find canonical bases of SU_3-skein modules of cylinders over orientable surfaces.

KAUFFMAN-HARARY CONJECTURE HOLDS FOR MONTESINOS KNOTS

The Kauffman–Harary conjecture states that for any reduced alternating diagram K of a knot with a prime determinant p, every non-trivial Fox p-coloring of K assigns different colors to its arcs. We generalize this conjecture by stating it in terms of homology of the double cover of S3. In this way we extend the scope of the conjecture to all prime alternating links of arbitrary determinants. We first prove the Kauffman–Harary conjecture for pretzel knots and then we generalize our argument to show the generalized Kauffman–Harary conjecture for all Montesinos links. Finally, we discuss on the relation between the conjecture and Menascos work on incompressible surfaces in exteriors of alternating links.

_{}-character varieties as spaces of graphs

An SL_n-character of a group G is the trace of an SL_n-representation of G. We show that all algebraic relations between SL_n-characters of G can be visualized as relations between graphs (resembling Feynman diagrams) in any topological space X, with pi_1(X)=G. We also show that all such relations are implied by a single local relation between graphs. In this way, we provide a topological approach to the study of SL_n-representations of groups. The motivation for this paper was our work with J. Przytycki on invariants of links in 3-manifolds which are based on the Kauffman bracket skein relation. These invariants lead to a notion of a skein module of M which, by a theorem of Bullock, Przytycki, and the author, is a deformation of the SL_2-character variety of pi_1(M). This paper provides a generalization of this result to all SL_n-character varieties.

Cut numbers of 3-manifolds

We investigate the relations between the cut number, c(M), and the first Betti number, b 1 (M), of 3-manifolds M. We prove that the cut number of a generic 3-manifold M is at most 2. This is a rather unexpected result since specific examples of 3-manifolds with large b 1 (M) and c(M) < 2 are hard to construct. We also prove that for any complex semisimple Lie algebra g there exists a 3-manifold M with b 1 (M) = dims and c(M) < ranks. Such manifolds can be explicitly constructed.

Skein theory for SU(n)-quantum invariants

For any n ≥ 2 we define an isotopy invariant, h i n , for a certain set of n-valent ribbon graphs in R 3 , including all framed oriented links. We show that our bracket coincides with the Kauffman bracket for n = 2 and with the Kuperbergs bracket for n = 3. Furthermore, we prove that for any n, our bracket of a link L is equal, up to normalization, to the SUn- quantum invariant of L. We show a number of properties of our bracket extending those of the Kauffmans and Kuperbergs brackets, and we relate it to the bracket of Murakami-Ohtsuki-Yamada. Finally, on the basis of the skein relations satisfied by hi n , we define the SUn-skein module of any 3- manifold M and we prove that it determines the SLn-character variety of π1(M). AMS Classification 57M27; 17B37

SKEIN MODULES AT THE 4TH ROOTS OF UNITY

The Kauffman bracket skein modules,

VERIFICATION OF THE JONES UNKNOT CONJECTURE UP TO 22 CROSSINGS

{\mathcal S}(M,A)

Topological insights from the Chinese Rings

, have been calculated for A=±1 for all 3-manifolds M by relating them to the