Proceedings of Symposia in Applied Mathematics Volume 66, 2009 A Brief Introduction to Knot Theory from the Physical Point of View Colin Adams Abstract. In recent years, there has been an upsurge in interest in physical knot theory, which is the investigation of questions related to how a knot physically sits in Euclidean 3-space. This is motivated by applications that are concerned with exactly these questions. This article is an introduction to knot theory but with particular emphasis on those aspects of the theory that are relevant to physical knot theory. In particular, the last section focuses on superinvariants, which as of yet have not been fully explored. 1. Introduction Knots have existed in some form or another for thousands of years. Initially, interest in knots came out of a desire to apply them in very practical situations. Early humans needed means to tie skins to the body for warmth. Sailors needed to have a variety of means to tie ropes to hoist sails. Surgeons needed to tie off sutures. Eventually, in the late nineteenth century, the mathematical beauty of the field became apparent and pure mathematicians began to consider the mathematical theory of knots. There followed a period of mathematical exploration. But in recent years, we have seen a explosion of interest in the applications of knot theory. In this article, we will discuss the basic background necessary to get started in knot theory. We will focus on those aspects relevant to the applications and to the subsequent articles in this volume. In particular, unlike most introductions to knot theory, we will be interested in those geometric quantities related to the physical realizations of knots in 3-space. In no sense is this an inclusive introduction, but rather, focuses on the author’s particular interests. Much of the following material appears in greater detail in the books [1], [8], [9], [10], [14], [20],[25], [26], [27], and others, with the exception of the last section, which has not appeared before. Take a piece of string. Tie a knot in it and then glue the two loose ends of the string together. The result is what we call a knot. We consider two knots to be equivalent if, when made out of string, one can be rearranged to look like the other without cutting the knotted loop of string open. 2000 Mathematics Subject Classification. 57M25. c 2009 American Mathematical Society 1 http://dx.doi.org/10.1090/psapm/066/2508726
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