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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 ...

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.

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 ...

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 ...

The knot move obtained by fixing disk 1 in the figure above and flipping disks 2 and 3.

A knot having the property that no surgery could possibly yield a counterexample to the Poincaré conjecture is said to satisfy Property P (Adams 1994, p. 262).

A relationship between knot polynomials for links in different orientations (denoted below as L_+, L_0, and L_-). J. H. Conway was the first to realize that the Alexander ...

An orientable surface with one boundary component such that the boundary component of the surface is a given knot K. In 1934, Seifert proved that such a surface can be ...