진교택 교수

The minimal number of straight line segments required to form a given knot or link in ℝ 3 is determined for a family of torus knots and links.

Every knot can be embedded in the union of finitely many half planes with a common boundary line in such a way that the portion of the knot in each half plane is a properly embedded arc. The minimal number of such half planes is called the arc index of the knot. We have identified all prime knots with arc index up to 11. We also proved that the crossing number is an upperbound of arc index for non-alternating knots. As a result the arc index is determined for prime knots up to twelve crossings.

For a 1-dimensional semi-boundary link, Cochran constructed a sequence of Sato-Levine invariants of successively derived links. This is a linear recurrence sequence and conversely any linear recurrence sequence can be constructed in this way. An upper bound for the growth of this sequence is obtained.

Abstract We present examples showing that certain results on the invariance of link polynomials under generalized mutation are the best possible. They show, moreover, that this generalized mutation cannot be effected by a sequence of ordinary mutations. One of the examples also shows that the reduced Jones polynomial can be a more sensitive invariant than the Jones polynomial itself.

We study the situation where we have two exceptional Dehn fillings on a given hyperbolic 3-manifold. We consider two cases that one filling creates a projective plane, and the other creates an essential torus or a Klein bottle, and give the best possible upper bound on the distance between two fillings for each case.

We show that the integer-valued knot invariants appearing as the nontrivial coefficients of the HOMFLY polynomial, the Kauffman polynomial and the Q-polynomial are not of finite type.

It is known that the arc index of alternating knots is the minimal crossing number plus two and the arc index of prime nonalternating knots is less than or equal to the minimal crossing number. We study some cases when the arc index is strictly less than the minimal crossing number. We also give minimal grid diagrams of some prime nonalternating knots with 13 crossings and 14 crossings whose arc index is the minimal crossing number minus one.

The arc index of a knot is the minimal number of arcs in all arc presentations of the knot. An arc presentation of a knot can be shown in the form of a grid diagram which is a closed plane curve consisting of finitely many horizontal line segments and the same number of vertical line segments. The arc index of an alternating knot is its minimal crossing number plus two. In this paper, we give a list of minimal grid diagrams of the 11 crossing prime alternating knots obtained from arc presentatio

In this paper, we give a list of minimal grid diagrams of the 12 crossing prime alternating knots. This is a continuation of the work in [G. T. Jin and H. J. Lee, Minimal grid diagrams of 11 crossing prime alternating knots, J. Knot Theory Ramifications 29(11) (2020) Article ID: 2050076, 14 pp.].

An upper bound of the superbridge index of the connected sum of two knots is given in terms of the braid index of the summands. Using this upper bound and minimal polygonal presentations, we give an upper bound in terms of the superbridge index and the bridge index of the summands when they are torus knots. In contrast to the fact that the difference between the sum of bridge indices of two knots and the bridge index of their connected sum is always one, the corresponding difference for the supe

Abstract Mutants of boundary links may not be boundary links, not even homology boundary links. Hence mutants of homology boundary links may not be homology boundary links.

Let S be a connected open subset of 2-sphere S 2 which is identified with the extended plane R 2 ∪ {∞}. We assume that S contains the n segments {1, 2, …, n} × [-1, 1]. An n-lace ℓ (in S) is the union ℓ 1 ∪ … ∪ ℓ n of disjoint simple arcs in S such that ∂ ℓ i = {(i, 1), (π (i),-1)}, i = 1, …, n, for some permutation π of {1, 2, …, n}. In this paper we will do present mapping class group description of n-laces, and 1-lace in 1-punctured plane, and 2-laces in S 2 . We will also describe the isotro