A major focus of my current research is to develop new techniques for studying deformations of tangle replacement moves and skein modules of three-dimensional manifolds. Development of such techniques is not only important for knot theory, but has significance in the problem of determining the full three-dimensional structure of recombination systems. Various mathematical approaches could be used, including techniques used in knot theory, in order to study DNA recombination systems. In particular, Sumners and Ernst, in their work, proposed a theoretical model for studying DNA recombination. Using the methods of 3-D topology techniques (cyclic surgery method), Sumners and Ernst derived first results concerning the three-dimensional structure of such systems. Their pioneering work opened series of research investigations concerning applications of topology, in particular knot theory, to problems concerning DNA. The applications of topology to biology were, for example, discussed recently during U.S.-Mexico Workshop, Knots in Biological Sciences. The workshop was sponsored by CIMAT, UT Dallas, and UT Southwestern Medical Center and held at UTD on April 29, 2005.
The recombination model, proposed by Sumners and Ernst, could be understood in terms of tangles and the problem of determining the full three-dimensional structure of recombination systems could be viewed as an instance of a tangle embedding problem. This problem leads to the necessity of developing new techniques for studying tangle replacement moves on links, which is the major focus of my current research. My main contributions to this area include defining and developing a new family of link invariants - n-Burnside groups of links. Various applications of the new invariants allow us to solve some of the open problems in this area. In particular, in my published work the new invariants were used to solve the following problems: Montesinos-Nakanishi 3-move conjecture (1981), Harikae-Nakanishi (2,2)-move conjecture (1992) and to answer Kawauchi's question concerning Nakanishi's 4-move conjecture (1985). The new techniques and some of their extensions to non-associative invariants could be successfully used to derive criteria for the tangle embedding problem and, in this way, have important applications in study of DNA recombination mechanisms.
Another area of my mathematical research investigations concerns properties of known algebraic invariants of 3-manifolds called fundamental groups. In my research work, I investigated the property of the spaces of orders on 3-manifold groups. This topic of mathematical research has recently gained a considerable attention due to its applications to other areas of 3-dimensional topology. My main contributions to this area include results about existence of left orders on important classes of 3-manifold groups. The results have applications to the important problem of the existence of foliations.
In my current research I study computational properties of the spaces of orders. We showed, in particular, that many classes of 3-manifolds groups admit infinitely many orders that are arbitrarily computationally complex. That is, there are examples of 3-manifold groups that admit an order of arbitrary Turing degree. The spaces of orders on such groups also admit an embedding of the Cantor set, which establishes new connections between topology and recursion theory.
The topic that I find very interesting for my future research is the applications of knots invariants to protein structures similarity problem. Assessing similarity between two protein structures is among one of the most challenging and important problems in computational biology. Since the number of known structures is constantly growing, the need for faster and more accurate methods persists. The problem has been approached using a variety of methods from almost all areas of sciences. In particular, some recent results in this area have been obtained by using knot invariants. This suggests that ideas coming from knot theory have potential for another important range of applications.