The Knot Group

The Fundamental (first homotopy) Group

Let M a manifold and L the set of continuous functions from the circle to M (also called the set of loops). Two such functions are considered equivalent if they can be continuously deformed to each other. The set of equivalence classes under loop multiplication is called the fundamental (or first homotopy) group of M.

Let M be the complement of a knot K in the three-sphere. The fundamental group of M is called the knot group of K. Obviously this group does not depend on any particular projection of a knot and is thus a knot invariant. Two knots possessing distinct groups are definitely inequivalent. Recently the converse has also been proved: if two knots possess identical groups, they are either ambient isotopic, or mirror symmetric, or inversely oriented, or each is the mirror symmetry of the inverse orientation of the other.

In the 1910's Wirtinger gave a presentation of the knot groups that is still used today. To each strand of a knot he assigned a distinct generator, and to each crossing the generators assigned to the three strands that meet are related by CB=BA, where B is the generator for the overcrossing strand, while A and C the generators for the undercrossing strands.

Let P and P' be two distinct projections of ambient isotopic knots. Their Wirtinger presentations will be distinct, but the actual groups are identical. Directly comparing distinct presentations is just as hard as performing the Reidemeister moves. One is thus forced to use alternative methods, such as the color tests. A color test is an attempt to assign to each generator of the knot group, an element of a particular group. If the attempt is successful, this implies that the particular group is a module of the knot group. If a particular group G is a module of the group of a knot K, but not of a knot K', then K and K' are inequivalent. In particular, the three-color test discussed before corresponds to the 3-permutation group, while the three "colors" to the permutations (2,1,3), (3,2,1), (1,3,2). This group is a module of the group of the trefoil, but not of the group of the trivial knot. In fact one may directly calculate that the trefoil group is the 3-braid group, while the trivial knot group is Z.

Please click to see more about the construction of the color tests.

Charilaos Aneziris,

Copyright 1995


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