Knots and Their Polynomials

For a comprehensive guide to the knot resources of the Web, visit Peter Suber's Knots on the Web.

Given a piece of string and asked to tie an overhand knot, the chances are you will come up with one of these two:

Left and right-handed overhand knots

Practically speaking, these knots are different. They make the difference between a square knot and a granny, according to which one you use when. In the one on the left the string twists to the left as you follow it through the knot; in the one on the right, it twists to the right; they are often called ``left-handed'' and ``right-handed'' knots.

In these pages we will look at the mathematical theory of knots and perform an elementary but somewhat intricate calculation that is part of the mathematical proof that these two knots are really different.

• Knots as topological objects

   

• The Jones polynomial

   

• Calculation of the Jones polynomial of the right trefoil

   

• What happens if we change the orientation?

   

• What about the left trefoil?

   

• Food for thought