A Survey of Knots and Links Matt Lawson Appalachian State University 8 December 2001

Abstract A summary of knots and knot theory, starting with a definition of what a knot is (what makes a knot a knot) and leading into to the motivation for a knot invariant. An important part of this is a discussion of the Redermeister moves, which preserve knot invariants (such as coloring). The topological significance of the knot will also be covered. The latter part of the talk will delve into knot algebras and possible applications of knot theory to fields in physics and biology. Lastly a catalogue of knots will be displayed, and, if time permits, the program KnotPlot by Rob Scharein will be demonstrated.

What Is a Knot? An embedding of S1 into R3 Mathematical knot vs. knot on a string Knot (and link) vs. knot diagram vs. knot projection

Some Examples of Knots Note the similarity between the trefoil and unknot#2. Suggests uncrossing no.

Examples of Links

Knot Theory-Knots and Links Problem 1: How do you prove the trefoil is knotted? Crossings: the heart of knots

The Redermeister Moves how to untangle a knot Type I – the untwist Type II – ±2 crossings Type III -- sliding

Illustrated Redermeister Moves Don’t forget about R0 and R4