A SHORT INTRODUCTION TO KNOT THEORY AND CRYPTOGRAPHY RESEARCH GROUP

Jae Choon Cha(jccha@postech.ac.kr)

In the language of experts, the mathematical theory of knots and links is the study of embeddings of one manifold in another. For example, An easy-to-imagine case is embeddings of circles into the Euclidean three space. Tying a string and joining the ends, one may obtain an embedding of a circle in the three dimensional space. By mathematicians, it is called a classical knot. This is generalized to higher dimensional knots and links.

The theory of knots and links is a hot subject which is being rapidly developed, based on the remarkable progress of topology in the 20th century. Recently, surprising new relationships of knot theory with other fields than topology have been recovered and actively studied. It includes fields outside mathematics like quantum mechanics, statistical physics, chemistry, and the study of structures of DNA, as well as other branches of mathematics like representation theory, combinatorics, and cryptology.

The research fields of our group cover all the subjects of the theory of knots and links, including:

- Algorithms in braid groups
- Applications of knot theory in cryptology
- One-bridge torus knots
- Laces in the plane
- Dehn fillings and 3-manifolds
- Classical numeric invariants of knots and links
- Polynomial invariants and finite type invariants
- The algebraic topology of knots and links
- Concordance of knots and links
- Signature invariants, invariants of periodic knots

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Copyright 2000, Knot Theory and Cryptography Research Group.
Last modified: Apr. 14, 2008