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A simple way to describe a knot projection. The advantage of this notation is that it enables a knot diagram to be drawn quickly. For an oriented alternating knot with n ...

The Alexander invariant H_*(X^~) of a knot K is the homology of the infinite cyclic cover of the complement of K, considered as a module over Lambda, the ring of integral ...

If a map f:G->G^' from a group G to a group G^' satisfies f(ab)=f(b)f(a) for all a,b in G, then f is said to be an antihomomorphism.

Consider a knot as being formed from two tangles. The following three operations are called mutations. 1. Cut the knot open along four points on each of the four strings ...

A change in a knot projection such that a pair of oppositely oriented strands are passed through another pair of oppositely oriented strands.

A group L is a component of H if L is a quasisimple group which is a subnormal subgroup of H.

The Kauffman X-polynomial, also called the normalized bracket polynomial, is a 1-variable knot polynomial denoted X (Adams 1994, p. 153), L (Kauffman 1991, p. 33), or F ...

A prime factorization algorithm.

The unknot, also called the trivial knot (Rolfsen 1976, p. 51), is a closed loop that is not knotted. In the 1930s Reidemeister first proved that knots exist which are ...

The smallest number of times u(K) a knot K must be passed through itself to untie it. Lower bounds can be computed using relatively straightforward techniques, but it is in ...