Research Statement

Radmila Sazdanovi´c

My mathematical research interests include , links and their invariants, categorification of combinatorial structures, and computational mathematics. My current research combines classical theory and homological algebra and explores relations between Khovanov of links and chromatic graph . I am also studying several diagrammatic categorifications of the ring of in one variable which lead to categorification of orthogonal polynomials, including Chebyshev and Hermite polynomials.

1 Categorifications of knot and graph invariants

Categorification lifts numbers to vector spaces and vector spaces to categories. A prime example is turning of a topological space into its homology groups. More exotic exam- ples include various homology groups which lift invariants of knots. For instance, lifts the and Ozsv´ath-Szab´o-Rassmussen homology lifts the . Knots, via their diagrams, are closely related to planar graphs. Re- cently, several graph invariants have also been categorified, such as the chromatic and the Tutte polynomial. To a graph Γ and a finite-dimensional commutative algebra A over a field 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, defined 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+2|Ds+ |−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−2|Ds− |+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 identification 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 ∈ 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: • Functoriality. I would like to find the correct definition 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. • Generalization to matroids. Each graph gives rise to a matroid of a very special kind. We hope that it is possible to rewrite the definition of graph homology in the matroid language and then, ideally, generalize chromatic graph homology to more general matroids. • Chromatic graph cohomology may have a geometric counterpart in the Eastwood-Huggett homology. Eastwood and Huggett [EH] defined the family of singly-graded homology theo- ries for graphs, as the homology of a configuration 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. • Applications to graph theory. Hopefully, some applications should emerge from the functo- riality of graph homology, once it is properly understood.

2 Categorification of combinatorial structures

I am currently working, jointly with , on categorification 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 identification 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 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 categorification to the limit and find 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 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 effect of 2n-moves [Pr] on various knot invariants. Applying a 2n-move on an integer decreases or increases its Conway symbol by 2n. Infinite families of links are obtained by iteratively applying 2n-moves on an arbitrary subset of integer tangles of a given link. Conceptually, the of a knot is one of the simplest knot invariants, but it is extremely difficult to compute. We define the upper bound of the unlinking number, called BJ–unlinking number [JS3], which is obtained only from minimal diagrams. Definition 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 infinite 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.

Proposition 3.1. Let R[a,b] denote a 2-bridge link with the Conway symbol a b. Then the following holds:

(a) If a, b are both odd then for a link R[a,b] = R[2m+1,2n+1] we have a+b uBJ (R[2m+1,2n+1]) = uM (R[2m+1,2n+1]) = u(R[2m+1,2n+1]) = 2 = m + n + 1.

3 Research Statement Radmila Sazdanovi´c

(b) If a is odd and b is even then for a knot R[a,b] = R[2m+1,2n] we have uBJ (R[2m+1,2n]) = uM (R[2m+1,2n]) = u(R[2m+1,2n+1]) = n.

(c) If a, b are both even then for a knot R[a,b] = R[2m,2n] we have 1 uBJ (R[2m,2n]) = uM (R[2m,2n]) = min(m, n).

Motivated by the example of knot 108 = 5 1 4 given by Y. Nakanishi [Na1] and S. Bleiler [Bl], we define BJ-unlinking gap as the difference between an unlinking number obtained from minimal diagrams and BJ-unlinking number. We construct infinite families of knots and links with non- zero BJ-unlinking gap and examples of links with BJ-unlinking gap arbitrarily large. Moreover, we have established correspondence between Conway symbols of knot and link families and minimum braids [Gt], which were used for studying graph trees, amphicheirality, unknotting numbers and periodic tables of knots and links. Among the set of all braid families representing the same family of knots or links, the braid family representative is the one that has the following properties: minimum number of braid crossings, minimum reduced braid, minimum source braid. In fact, braid family representative is a source of a link family expressed as a minimum braid word [JS2]. In the case of non-algebraic (polyhedral) knots and links, braid family representatives have an important advantage over Conway symbols: no artificial conventions, such as knowing the corresponding graphs and orientation of tangles in the basic polyhedra, are required. For example, family of antiprismatic basic polyhedra 2n∗, n ≥ 3 is given by a braid family representative (Ab)n. In addition to unknotting number problem, the book I coauthored, ”LinKot- Knot Theory by Computer” contains an abundance of results, conjectures, as well as new methods for handling open problems such as invertibility and amphicheirality of knots, n-moves and (2,2)-moves [DIP], undetectability, non-algebraic tangles, polyhedral knots. It would be interesting to continue exploring dependencies between Conway symbols and various knot invariants and use them for efficient classification and enumeration of knots and links which will reflect their properties (unlike the commonly used classification based on the number of crossings).

4 Computational mathematics

Besides the theoretical approach to problem solving I enjoy using computers for a number of different reasons- combining these two has proved to be very beneficial in my research. Computa- tional results obtained using my software for determining the torsion of Khovanov type chromatic graph cohomology over algebra of truncated polynomials have facilitated deeper understanding of the topic, provided useful examples and motivated new results. Knot theory software LinKnot, which I developed with Slavik Jablan [JS4], is a powerful research tool for experimenting with large families of knots, providing computational results for numerous knot theory problems. For example, we obtained efficient lower and upper bounds for unknotting numbers of knots and links with 11 [JS6] and 12 crossings. The lower bound was computed using signature and the upper bound is obtained by an algorithm based on Bernhard-Jablan Conjecture [JS1] whose output is the unknotting sequence built on minimal diagrams. Together with Pierre Dehornoy we have created tables of Lorentz knots up to 49 crossings including all that are obtained from 3-braids. Using CoHo by A. Shumakovitch we have computed odd Khovanov homology of Lorentz braids; results imply that all positive minimal braids are odd Khovanov homology thick. LinKnot is also available online as webMathematica package (with the support from Wolfram Research, Ministry of Science and Technology, Serbia and ICT). The program is accompanied with the web version of the book ’LinKnot-Knot Theory by Computer’ [JS5] and a detailed user guide which makes it easy to use, both as teaching and research tool.

1 Unknotting number u(R[2m,2n]) is unknown for most values of m, n > 1.

4 Research Statement Radmila Sazdanovi´c

Prior to Linknot I have developed Tess [SS], the only computer program which creates all possible tessellations, both uniform and non-uniform, corresponding to a given vertex symbol and produces drawing in Euclidean plane, spherical geometry or any of three models of the hyperbolic plane. I plan to continue using computer programming in my research (classification of knots and tangles and their structures) as well as for creating computer graphics whose beauty comes both from aesthetically pleasing appearance and mathematical content.

References

[AP] M. M. Asaeda, J. H. Przytycki, Khovanov homology: torsion and thickness, Advances in Topological Quantum Field Theory, 135–166, Kluwer Acad. Publ., Dordrecht, 2004; arxiv:math.GT/0402402.

[Bl] S. A. Bleiler, A note on unknotting number, Math. Proc. Camb. Phil. Soc., 96 (1984) 469–471.

[Be] J. A. Bernhard, Unknotting numbers and their minimal knot diagrams, J. Knot Theory Ramifications, 3, 1 (1994) 1–5.

[Co] J. Conway, An enumeration of knots and links and some of their related properties, in Com- putational Problems in Abstract Algebra, Proc. Conf. Oxford 1967 (Ed. J. Leech), 329–358, Pergamon Press, New York (1970).

[DIP] M. K. Dabkowski, M. Ishiwata, J.H. Przytycki, 5-move equivalence classes of links and their algebraic invariants,Journal of Knot Theory and Its Ramifications 16 (10) (2007) 1413-1450, arXiv:0712.0985.

[EH] M. Estwood, S. Hugget, Euler characteristics and chromatic polynomials, European Journal of Combinatorics 28 (6) 2007, 1553–1560

[Ga] D. Garity, Unknotting Numbers are not Realized in Minimal Projections for a Class of Ra- tional Knots, Proceedings of the ”II Italian-Spanish Congress on General Topology and its Applications” (Italian) (Trieste, 1999). Rend. Istit. Mat. Univ. Trieste, 32 (2001), suppl. 2, 59–72 (2002).

[Gt] T. Gittings, Minimum braids: a complete invariant of knots and links, arxiv:math.GT/0401051.

[HGR] L. Helme-Guizon, Y. Rong, A Categorification for the Chromatic Polynomial, Algebraic and Geometric Topology (AGT), 5, 2005, 1365-1388, arXiv:math.CO/0412264.

[Ja] S. Jablan, Unknotting number and ∞-unknotting number of a knot, Filomat, 12, 1, (1998) 113–120.

[JS1] S. Jablan, R., Sazdanovi´c,’LinKnot- Knot Theory by Computer’, World Scientific edition ’Knots and Everything’ 21(2007) pp.500 ISBN 978-981-277-223-7

[JS2] S. Jablan, R., Sazdanovi´c,Braid Family Representatives, Journal of Knot Theory and Its Ramifications 17 (7) (2008) 817-833, arxiv: math.GT/0504479.

[JS3] S. Jablan, R., Sazdanovi´c,Unlinking number and unlinking gap, Journal of Knot Theory and Its Ramifications 16 (10) (2007) 1331-1355, arxiv: math/0503270.

[JS4] S. Jablan, R., Sazdanovi´c,LinKnot- Mathematica package (http://www.mi.sanu.ac.yu/vismath/linknot/index.html

5 Research Statement Radmila Sazdanovi´c

[JS5] S. Jablan, R., Sazdanovi´c,’LinKnot Knot Theory by Computer’ Online book with interac- tive webMathematica version of LinKnot software. (http://math.ict.edu.yu:8080/webMathematica/LinkSL/cont.htm)

[JS6] S. Jablan, R., Sazdanovi´c,Unknotting numbers of 11 crossing knots, Knot Tables by C. Livingston http://www.indiana.edu/˜knotinfo/descriptions/unknotting number.html

[JRS] S. Jablan, Lj. Radovi´c,R., Sazdanovi´c,Basic Polyhedra in Knot Theory, Kragujevac Jour- nal of Mathematics, 28(2005), 155-164 (http://elib.mi.sanu.ac.yu/files/journals/kjm/28/12.pdf)

[Kh1] M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101, 2000, no. 3, 359–426, arxiv: math.QA/9908171

[KS] M. Khovanov, R. Sazadanovi´c,Categorifications of Hermite and other orthogonal polyno- mials, in preparation.

[Ko1] P. Kohn, Two Bridge Links with Unlinking Number One, Proceedings of the American Mathematical Society, 98, 4 (1991) 1135–1147.

[Ko2] P. Kohn, Unlinking two component links, Osaka J. Math., 30 (1993) 741–752.

[KM] T. Kanenobu, H. Murakami Two-bridge knots with Unknotting Number One, Proceedings of the American Mathematical Society, 98, 3 (1986) 499–502.

[Lic] W.B.R. Lickorish, The unknotting number of a classical knot, in Combinatorial methods in topology and algebraic geometry (Rocherster, N.Y., 1982), Vol. 44 of Cont. Math., 117–121.

[Liv] C. Livingston, Knot Tables, http://www.indiana.edu/∼knotinfo/ accessed on June 6, 2007.

[Na1] Y. Nakanishi, Unknotting numbers and knot diagrams with the minimum crossings, Math. Sem. Notes Kobe Univ., 11 (1983) 257–258.

[Ow] B. Owens, Unknotting information from Heegaard , Accepted for publication in Advances in Mathematics; arxiv: math.GT/0506485

[PPS] M. Pabiniak, J. Przytycki, R., Sazdanovi´c,On the first group of the chromatic coho- mology of graphs, Geometriae Dedicata,Vol 140, No. 1 (2009)19-48, ISSN 0046-5755, arxiv: math.GT/0607326.

[Pr] J. Przytycki, tk moves on links, Contemporary Math., Vol. 78, Braids - Proceed- ings of the Santa Cruz conference on Artin’s braid groups (July 1986), 1988, 615-656, arxiv:math.GT/0606633

[Pr1] J. Przytycki, When the theories meet: Khovanov homology as Hochschild homology of links, arxiv:math.GT/0509334

[PS] J. Przytycki, R., Sazdanovi´c,Torsion in Khovanov homology of semi-adequate links, in preparation

[Sh] A. Shumakovitch, Torsion of the Khovanov homology, arXiv:math/0405474

[SS] R. Sazdanovi´c,M., Sremcevi´c,’Tessellations of the Euclidean, Elliptic and Hyperbolic Plane’, MathSource, Wolfram research, 2002 http://library.wolfram.com/infocenter/MathSource/4540/

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