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Research Statement Research Statement Radmila Sazdanovi¶c My mathematical research interests include knots, links and their invariants, categori¯cation of combinatorial structures, and computational mathematics. My current research combines classical knot theory and homological algebra and explores relations between Khovanov homology of links and chromatic graph cohomology. I am also studying several diagrammatic categori¯cations of the ring of polynomials in one variable which lead to categori¯cation of orthogonal polynomials, including Chebyshev and Hermite polynomials. 1 Categori¯cations of knot and graph invariants Categori¯cation lifts numbers to vector spaces and vector spaces to categories. A prime example is turning Euler characteristic of a topological space into its homology groups. More exotic exam- ples include various link homology groups which lift polynomial invariants of knots. For instance, Khovanov homology lifts the Jones polynomial and Ozsv¶ath-Szab¶o-Rassmussen homology lifts the Alexander polynomial. Knots, via their diagrams, are closely related to planar graphs. Re- cently, several graph invariants have also been categori¯ed, such as the chromatic and the Tutte polynomial. To a graph ¡ and a ¯nite-dimensional commutative algebra A over a ¯eld k there is associated a complex C(G; A) whose Euler characteristic is the value of the chromatic polynomial at the dimension of A. We analyze the chromatic graph cohomology, de¯ned by L. Helme-Guizon and Y. Rong [HGR], for algebras of truncated polynomials, especially the degrees which corre- spond to those containing torsion in the Hochschild homology of this algebra [PPS, PS]. The torsion in Khovanov homology carries additional information about knots and their cobor- disms, not contained in the Jones polynomial. A. Shumakovitch proved a number of interesting results about torsion in Khovanov homology and used some of them to improve the upper bound on Thurston-Bennequin number. In particular, he proved that alternating links can have only 2-torsion and conjectured that any link which is not a connected or disjoint sum of Hopf links and trivial links has torsion in Khovanov homology [Sh]. Shumakovitch proved the conjecture for alternating links and M. Asaeda, J. Przytycki [AP] proved the existence of 2-torsion in Kho- vanov homology of a large class of adequate links. Using the correspondence between Khovanov homology of links and well developed theory of Hochschild homology via chromatic homology for graphs [Pr1], we have generalized previous results to semi-adequate knots whose corresponding graphs have no loops [PS, PPS], and obtained the following explicit formulas for the torsion: Theorem 1.1. [PS] Given a semi-adequate knot diagram D with n crossings let Gs(D) be a graph corresponding to the state s+ if D is +adequate and s¡ if D is ¡adequate. If a simple graph G obtained from Gs(D), is a loopless graph and D is +adequate, then ( Zp1(G); if G is bipartite; torH (D) = 2 n¡4;n+2jDs+ j¡8 p1(G)¡1 Z2 ; if G has an odd cycle. If G is a loopless graph and D is ¡adequate, then ( Zp1(G); if G is bipartite; torH (D) = 2 ¡n¡2;¡n¡2jDs¡ j+8 p1(G)¡1 Z2 ; if G has an odd cycle. where p1(G) denotes the cyclomatic number of a graph G. Research Statement Radmila Sazdanovi¶c As a corollary we have that Khovanov homology of knots corresponding to positive adequate 3-braids contains 2-torsion. Moreover, we analyze torsion in chromatic cohomology over algebras of truncated polynomials. For example, given a graph G, chromatic graph cohomology H1;2v¡3(G) can be approximated by A3 homology of a cell complexes whose 1-skeleton is the graph G and 2-skeleton consists of squares and triangles with additional identi¯cation of edges. Here is an example of our results: Theorem 1.2. [PPS] For the complete graph Kn,with n ¸ 4 vertices we have 1;2n¡3 n¡1 n(n¡1)(2n¡7) H (K ) = Z © Z © Z 6 : A3 n 2 3 This implies that if a graph G contains a triangle then H1;2v(G)¡3(G) contains Z torsion. More- A3 3 over, given any n 2 N we have constructed a graph with Zn torsion in chromatic graph cohomology. In the future, I plan to explore the following avenues of research: ² Functoriality. I would like to ¯nd the correct de¯nition of a cobordism between graphs that induces a homomorphism between their homology groups. There are long exact sequences of homology between triples of graphs, related by elementary transformations: edge con- tractions and deletions. These transformations are expected to be the building blocks for general graph cobordisms. The resulting structure should lift graph homology to a functor from the category of graph cobordisms to the category of bigraded abelian groups. ² Generalization to matroids. Each graph gives rise to a matroid of a very special kind. We hope that it is possible to rewrite the de¯nition of graph homology in the matroid language and then, ideally, generalize chromatic graph homology to more general matroids. ² Chromatic graph cohomology may have a geometric counterpart in the Eastwood-Huggett homology. Eastwood and Huggett [EH] de¯ned the family of singly-graded homology theo- ries for graphs, as the homology of a con¯guration space associated to a pair (G; M), where G is a graph and M a manifold. There should exist a spectral sequence from the bigraded graph homology associated to the cohomology algebra of M to the singly-graded Eastwood- Huggett homology of (G; M). We would like to understand this spectral sequence and study its convergence properties. ² Applications to graph theory. Hopefully, some applications should emerge from the functo- riality of graph homology, once it is properly understood. 2 Categori¯cation of combinatorial structures I am currently working, jointly with Mikhail Khovanov, on categori¯cation of various orthogonal polynomials in one variable, including the Chebyshev polynomials of the second kind and Hermite polynomials [KS]. We introduce several categories of modules over suitable algebras. In the Chebyshev case, the algebra is a degeneration of the Temperley-Lieb algebra. This degeneration is of a new kind, and its diagrams do not have isotopy invariance, unlike the original Temperley- Lieb algebra. The categories of modules have a monoidal structure, and their Grothendieck groups are rings. There is a natural identi¯cation of these Grothendieck rings with the ring of polynomials in one variable x. Under this isomorphism, projective modules correspond to monomials in x, while certain standard modules (the analogues of the Verma modules in the Lie algebra theory) correspond to the Chebyshev polynomials and, in the other example, to the Hermite polynomials, Fig. 1. Various basic structures of the theory of orthogonal polynomials, such as the kernels Kn(x; y) that approximate the identity operator, admit categorical lifting in our framework. In the future, I would like to push this categori¯cation to the limit and ¯nd a categorical lifting of more complicated parts of the orthogonal polynomials' theory. 2 Research Statement Radmila Sazdanovi¶c Figure 1: Hermite polynomials-projective, big standard and standard/Verma modules 3 Knots and links One of the principal questions in knot theory is describing and distinguishing knots and links. Various ways of representing knots (diagrams, braids, Dowker and Gauss codes, etc.) and numer- ous knot invariants have been developed but none of them provides a complete solution. In the book [JS1] we utilize Conway symbolic notation [Co] that gives interpretable and understandable information about knots and links. For example, all rational knots with an even number of cross- ings and with symmetrical (palindromic) Conway symbols are achiral and the equivalence of two knots is determined using continued fractions. Conway notation is eminently suitable for calculations of many algebraic knot invariants. The original motivation for [JS1, JS2] was to use Conway notation to explore the e®ect of 2n-moves [Pr] on various knot invariants. Applying a 2n-move on an integer tangle decreases or increases its Conway symbol by 2n. In¯nite families of links are obtained by iteratively applying 2n-moves on an arbitrary subset of integer tangles of a given link. Conceptually, the unknotting number of a knot is one of the simplest knot invariants, but it is extremely di±cult to compute. We de¯ne the upper bound of the unlinking number, called BJ{unlinking number [JS3], which is obtained only from minimal diagrams. De¯nition is based on Bernhard-Jablan Conjecture [Be, Ja] which says that unlinking number coincides with BJ- unlinking number. BJ-conjecture holds for all links for which the unlinking number has been computed. In particular, it holds for all knots up to 11 crossings [Liv] and 2-component links up to 9 crossings [Ko2]. Furthermore, T. Kanenobu, H. Murakami and P. Kohn proved that for unknotting number one rational links the unknotting crossing appears in the minimal diagram [KM, Ko1]. We illustrate the importance of the conjecture by the following example. Consider alternating pretzel knots P(a;b;c) where 0 < a · b · c and a; b; c are all odd numbers. We show that a+b uBJ (P(a;b;c)) = 2 . However, the unknotting numbers of these knots are still unknown, except for the smallest knots such as P(1;3;3) (with unknotting number 2 computed by W.B.R. Lickorish [Lic]) and P(3;3;3) (with unknotting number 3 computed recently by B. Owens [Ow]). In the similar manner, we derive formulas for BJ{unlinking number of several in¯nite families including rational knots with 2 or 3 parameters in Conway symbol. Notice that when BJ-unlinking number coincides with the lower bound based on the signature then we have obtained the unlinking number.
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