강연 초록

Title  : On classification of rational 3-tangles and normal form

Abstract: An important issue in classifying the rational 3-tangle is how to know whether or not the given tangle is the trivial rational 3-tangle called infinity-tangle. The author provided a certain algorithm to detect the infinity-tangle. In this paper, we give a much simpler method to detect the infinity-tangle by using the bridge arc replacement.  We also introduce the normal form of rational 3-tangles based on the bridge arc replacement. We hope that this method can help prove many application problems such as a classification of 3-bridge knots.

Title :  Self-distributive structures in Knot Theory: a homological perspective

Abstract : The Yang-Baxter equation was first introduced as statistical mechanical model and is now used to study theoretical physics, especially quantum field theory. Self-distributive algebraic structures form a significant class of solutions to the set-theoretic Yang-Baxter equation. J. D. H. Smith obtained solutions to the Yang-Baxter equation using quantum idempotence, distributivity, and quantum quasigroup techniques. W. Rump showed a one-to-one correspondence between rumples and involutive nondegenerate solutions of the Yang-Baxter equation. Although homology theories of associative structures have been extensively investigated since the work of Eilenberg and Hochschild, it has not been long since the study of homology theories of non-associative distributive structures became active.

In this talk, we introduce homology theories of self-distributive structures and their relevance to knot theory.

Title : Satellite knots obtained from torus knots by twisting two strands

Abstract :  Consider the knots obtained from torus knots by twisting two adjacent strands. Among these knots, we determine which are satellite knots.

Title :  An upper bound on the simple hexagonal lattice stick number of a knot

Abstract : The simple hexagonal lattice(sh-lattice) is a lattice space based on four types of sticks: x=<1, 0, 0>,  y=<1/2, √3/2, 0>, z=<1/2, √3/2, 0>, and w=<0, 0, 1>. In this talk, I introduce the relationships and distinctions between lattice stick knots and sh-lattice stick knots. Furthermore, I suggest an upper bound on the sh-lattice stick number of a given knot expressed in terms of its crossing number.

Title :  Optimization of ropelength for knots and links

Abstract : The ropelength of a knotted string with volume is defined as the ratio of the length of its central curve to the radius of its sectional disc. In a physical context, achieving minimal ropelength corresponds to a state of minimal potential energy, and geometrically, it signifies a tightly-packed conformation. The quest to establish a connection between the topological complexity of knotted strings and their minimal ropelength has persisted into recent years.

In this talk, I introduce the upper bound on the minimal ropelength of some knots and links.

Title :  Restricted arc-presentations of knots and links

Abstract : A knot is a simple closed curve in R³ (or S³).

An arc presentation of a knot is an embedding into the open book decomposition of R³ such that each half plane contains a properly embedded single simple arc.

In this talk, I limit the number of pages in the arc presentation to three and instead have multiple arcs on each page. This type of arc presentation is called a three-page presentation. Also I introduce the results of three-page presentation for torus knots and 2-bridge knots. These results induce the exact value of three-page index of trefoil knot.

Title :   SELF-HOMOTOPY INVARIANTS ON THE REDUCED SUSPENSION SPACES

Abstract: For a based CW-complex X, the self-closeness number X, Nε(X), determines the minimum number at which a certain monoid set and the self-homotopy equivalence group ε(X) become the same. The self-length, ℒε(X), is related to the self-closeness number, representing the number of strict inclusions in the descending chain of certain monoid sets. Nε(X) and ℒε(X) are called self-homotopy invariants. In this talk, we give some properties and theorems related to the self-homotopy invariants of X and ΣX, where ΣX is the reduced suspension of X. Furthermore, we have calculated some examples.