The use of knots goes back to pre-history, but the mathematical study of knots goes back only to the 19th century. Some of the early investigators were Gauss, Listing, Kirkman, Tait, and Little. Due to connections with applied fields like physics and biology there is increased research today. For examples look at the Titles in Series on Knots and Everything edited by Louis H. Kauffman. There are books which explain the topic for a more popular audience. One favorite is The Knot Book by Colin Adams. This is listed in a bibilography of knot theory. Now I prefer Knots: Mathematics With A Twist by Alexei Sossinsky (translation published 2002) which is more elementary and interesting. The story behind Making a Mathematical Exhbition by Ronald Brown and Tim Porter describes some of the issues involved in presenting knot theory. There are web sites which you can visit to find out more. You can start with the Mega-Math section on knots. Then switch to the KnotPlot site for great color graphics. A less ambitious site is Geometry and the Imagination section on knot notation. Knots on the Web by Peter Suber has a long well-annotated list of links on knot theory as well as many other aspects of knots. Mouse Bousfield has a nice Knot Theory site. Morwen Thislethwaite has images of some nice symmetric knots and a "Knotscape" program. Mathematics and Knots is an exhibition in Wales. Robert J. Jenkins Jr. has written a paper A Dynamic Approach to Calculating the HOMFLY Polynomial for Directed Knots and Links.