Seungsang Oh

Reducing dehn fillings and small surfaces

In this paper we investigate the distances between Dehn fillings on a hyperbolic 3-manifold that yield 3-manifolds containing essential small surfaces including non-orientable surfaces. In particular, we study the situations where one filling creates an essential sphere or projective plane, and the other creates an essential sphere, projective plane, annulus, M?bius band, torus or Klein bottle, for all eleven pairs of such non-hyperbolic manifolds.

Small knot mosaics and partition matrices

Lomonaco and Kauffman introduced knot mosaic system to give a definition of quantum knot system. This definition is intended to represent an actual physical quantum system. A knot (m, n)-mosaic is an matrix of mosaic tiles which are T0 through T10 depicted, representing a knot or a link by adjoining properly that is called suitably connected. An interesting question in studying mosaic theory is how many knot (m, n)-mosaics are there. denotes the total number of all knot (m, n)-mosaics. This counting is very important because the total number of knot mosaics is indeed the dimension of the Hilbert space of these quantum knot mosaics. In this paper, we find a table of the precise values of for . Mainly we use a partition matrix argument which turns out to be remarkably efficient to count small knot mosaics.

Mosaic number of knots

Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot n-mosaic is an n × n matrix of 11 kinds of specific mosaic tiles representing a knot or a link. The mosaic number m(K) of a knot K is the smallest integer n for which K is representable as a knot n-mosaic. In this paper, we establish an upper bound on the mosaic number of a knot or a link K in terms of the crossing number c(K). Let K be a nontrivial knot or a non-split link except the Hopf link. Then m(K) ≤ c(K) + 1. Moreover if K is prime and non-alternating except link, then m(K) ≤ c(K) - 1.

AN ELEMENTARY SET FOR θn-CURVE PROJECTIONS

A finite set of nontrivial θn-curves is shown to be minimal among those which produce all projections of nontrivial θn-curves.

Quantum knots and the number of knot mosaics

Lomonaco and Kauffman developed a knot mosaic system to introduce a precise and workable definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot

Knots with small lattice stick numbers

(m,n)

STRONGLY ALMOST TRIVIAL θ-CURVES

(m,n)-mosaic is an