arXiv subjects
Publications and source records attributed to Dongseok Kim.
We show that nontrivial classical pretzel knots L(p,q,r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n-pretzel links using a new computation tree. As applications, we compute the genera of n-pretzel links using these polynomials and find the basket number of pretzel links by showing that the genus and the canonical genus of a pretzel link are the same.
In this paper, we find computational formulae for generalized characteristic polynomials of graph bundles. We show that the number of spanning trees in a graph is the partial derivative (at (0,1)) of the generalized characteristic polynomial of the graph. Since the reciprocal of the Bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph, consequently, the Bartholdi zeta function of a graph bundle can be computed by using our computational formulae.
We characterize the equivalence and the weak equivalence of Cayley graphs for a finite group $\C{A}$. Using these characterizations, we find enumeration formulae of the equivalence classes and weak equivalence classes of Cayley graphs. As an application, we find the number of weak equivalence classes of circulant graphs.
Temperley-Lieb algebras have been generalized to sl(3) web spaces. Since a cubic bipartite planar graph with suitable directions on edges is a web, the quantum sl(3) invariants naturally extend to all cubic bipartite planar graphs. First we completely classify them as a connected sum of primes webs. We also provide a method to find all prime webs and exhibit all prime webs up to 20 vertices. Using quantum sl(3) invariants, we provide a criterion which determine the symmetry of graphs.
Several natural partial orders on integral partitions, such as the embeddability, the stable embeddability, the bulk embeddability and the supermajorization, raise in the quantum computation, bin-packing and matrix analysis. We find the implications between these partial orders. For integral partitions whose entries are all powers of a fixed number $p$, we show that the embeddability is completely determined by the supermajorization order and we find an algorithm to determine the stable embeddability.
We develop graphical calculation methods. Jones-Wenzl projectors for U_q(sl(2,C)) are very powerful tools to find not only invariants of links but also invariants of 3-manifolds. We find single clasp expansions of generalized Jones-Wenzl projectors for simple Lie algebras of rank 2. Trihedron coefficients of the representation theory for U_q(sl(2,C)) has significant meaning and it is called 3j symbols. Using single clasp expansions for U_q(sl(3,C)), we find some trihedron coefficients of the representation theory of U_q(sl(3,C)). We study representation theory for U_q(sl(4,C)). We conjecture a complete set of relations for U_q(sl(4,C)).