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Adam S. Sikora

Publications and source records attributed to Adam S. Sikora.

34 records · Page 2Linked to original sources
arXiv2008

Quantizations of Character Varieties and Quantum Knot Invariants

Let G be a simple complex algebraic group and g its Lie algebra. We show that the g-Witten-Reshetikhin-Turaev quantum invariants determine a deformation-quantization, C_q[X_G(torus)], of the coordinate ring of the G-character variety of the torus. We prove that this deformation is in the direction of the Goldman's bracket. Furthermore, we show that every knot K defines an ideal I_K in C_q[X_G(torus)]. We conjecture that the homomorphism C_q[X_G(torus)] -> C[X_G(torus)], q -> 1, maps I_K to the ideal whose radical is the kernel of the map C[X_G(torus)] -> C[X_G(S^3 K)]. This conjecture is related to AJ-conjecture for sl(2,\C). The results of this paper are inspired by the theory of q-holonomic relations between quantum invariants of Garoufalidis and Le. Along the way, we disprove Conjecture 2 in Le's "The Colored Jones and the A-polynomial of Two-Bridge knots".

math.QA↗
arXiv2006

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 Kuperberg's 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.

math.QA↗
arXiv2006

A categorification of the skein module of tangles

We generalize our previous work on categorification of Kauffman bracket skein module of surfaces, by extending our homology to tangles in cylinders over surfaces, F x [0,1]. Our homology of 0-tangles and 1-tangles in D^3 coincides (up to normalization) with Khovanov link homology and the reduced Khovanov link homology. We prove the basic properties of our homology. In particular, the short exact sequence of homologies of skein related tangles and the Kunneth formula for the tensor product of tangles.

math.QA↗
arXiv2005

Skein theory for SU(n)-quantum invariants

For any n>1 we define an isotopy invariant, _n, for a certain set of n-valent ribbon graphs Gamma in R^3, including all framed oriented links. We show that our bracket coincides with the Kauffman bracket for n=2 and with the Kuperberg's bracket for n=3. Furthermore, we prove that for any n, our bracket of a link L is equal, up to normalization, to the SU_n-quantum invariant of L. We show a number of properties of our bracket extending those of the Kauffman's and Kuperberg's brackets, and we relate it to the bracket of Murakami-Ohtsuki-Yamada. Finally, on the basis of the skein relations satisfied by <.>_n, we define the SU_n-skein module of any 3-manifold M and we prove that it determines the SL_n-character variety of pi_1(M).

math.QA↗
arXiv2004

Categorification of the Kauffman bracket skein module of I-bundles over surfaces

Khovanov defined graded homology groups for links L in R^3 and showed that their polynomial Euler characteristic is the Jones polynomial of L. Khovanov's construction does not extend in a straightforward way to links in I-bundles M over surfaces F not D^2 (except for the homology with Z/2 coefficients only). Hence, the goal of this paper is to provide a nontrivial generalization of his method leading to homology invariants of links in M with arbitrary rings of coefficients. After proving the invariance of our homology groups under Reidemeister moves, we show that the polynomial Euler characteristics of our homology groups of L determine the coefficients of L in the standard basis of the skein module of M. Therefore, our homology groups provide a `categorification' of the Kauffman bracket skein module of M. Additionally, we prove a generalization of Viro's exact sequence for our homology groups. Finally, we show a duality theorem relating cohomology groups of any link L to the homology groups of the mirror image of L.

math.QA↗
arXiv2003

Topology on the spaces of orderings of groups

A natural topology on the space of left orderings of an arbitrary semi-group is introduced. It is proved that this space is compact and that for free abelian groups it is homeomorphic to the Cantor set. An application of this result is a new proof of the existence of universal Grobner bases.

math.GR↗
arXiv2003

Torus and Z/p actions on manifolds

Let G be either a finite cyclic group of prime order or S^1. We find new relations between cohomology of a manifold (or a Poincare duality space) M with a G-action on it and cohomology of the fixed point set, M^G. Our main tool is the notion of Poincare duality on the Leray spectral sequence of the map M_G -> BG. We apply our results to study group actions on 3-manifolds.

math.AT↗
arXiv2003

Cut numbers of 3-manifolds

The cut number of a manifold M, c(M), is the largest number of disjoint two-sided hypersurfaces in M which do not separate M. Equivalently, it is the largest rank of a free group being an epimorphic image of pi_1(M). We investigate the relations between the cut number and the first Betti number, b_1(M), of 3-manifolds M. We prove that the cut number of a ``generic'' 3-manifold is at most 2. This is a rather unexpected result since it is very hard to construct specific examples of 3-manifolds with with large b_1(M) and small c(M). On the other hand, we prove that for any complex semisimple Lie algebra g there exists a 3-manifold M with b_1(M)=dim g and c(M)<=rank g. Such manifolds can be explicitly constructed.

math.GT↗
arXiv2003

Analogies between group actions on 3-manifolds and number fields

Mazur, Kapranov, Reznikov, and others developed ``Arithmetic Topology,'' a theory describing some surprising analogies between 3-dimensional topology and number theory, which can be summarized by saying that knots are like prime numbers. We extend their work by proving several formulas concerning branched coverings of 3-manifolds and extensions of number fields and observe that these formulas are almost identical, via the dictionary of arithmetic topology. Until now there is no satisfactory explanation for the coincidences between our formulas. The proofs of topological results use equivariant cohomology and the Leray-Serre spectral sequence. The number theoretic proofs are based on an approach to class field theory via idele groups.

math.GT↗
arXiv2003

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 the conjecture by stating it in terms of homology of the double cover of S^3 branched along a link. 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 speculate on the relation between the conjecture and Menasco's work on incompressible surfaces in exteriors of alternating links.

math.GT↗
arXiv2001

Topological Insights from the Chinese Rings

L. Kauffman conjectured that a particular solution of the Chinese Rings puzzle is the simplest possible. We prove his conjecture by using low-dimensional topology and group theory. We notice also a surprising connection between the Chinese Rings and Habiro moves (related to Vassiliev invariants).

math.GT↗
arXiv2001

Skein modules at the 4th roots of unity

The Kauffman bracket skein modules, S(M,A), have been calculated for A=+1,-1, for all 3-manifolds M by relating them to the SL(2,C)-character varieties. We extend this description to the case when A is a 4-th root of 1 and M is either a surface x [0,1] or a rational homology sphere (or its submanifold).

math.GT↗
arXiv2001

Sl_n-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.

math.RT↗
arXiv1998

Polygon dissections and Euler, Fuss, Kirkman and Cayley numbers

We give a short proof for a formula for the number of divisions of a convex (sn+2)-gon along non-crossing diagonals into (sj+2)-gons, where 1<=j<=n-1. In other words, we consider dissections of an (sn+2)-gon into pieces which can be further subdivided into (s+2)-gons. This formula generalizes the formulas for classical numbers of polygon dissections: Euler-Catalan number, Fuss number and Kirkman-Cayley number. Our proof is elementary and does not use the method of generating functions.

math.CO↗
arXiv1998

On Skein Algebras And Sl_2(C)-Character Varieties

This paper gives insight into intriguing connections between two apparently unrelated theories: the theory of skein modules of 3-manifolds and the theory of representations of groups into special linear groups of 2 by 2 matrices. Let R be a ring with an invertible element A. For any 3-manifold M one can assign an R-module called the Kauffman bracket skein module of M. If A^2=1 then this module has a structure of an R-algebra. We investigate this structure and, in particular, we prove that if R is the field of complex numbers then this algebra is isomorphic to the (unreduced) coordinate ring of the SL_2-character variety of pi_1(M). Using that result we develop a theory of Sl_2-character varieties by use of topological methods. We also assign to any surface a relative Kauffman bracket skein algebra. We prove several results about this non-commutative algebra. Our work should be considered in the context of the book of Brumfiel and Hilden `SL(2) Representations of Finitely Presented Groups,' Cont. Math 187. In particular we give a topological interpretation to algebraic objects considered in that book.

q-alg↗