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A sphere with four punctures occurring where a knot passes through the surface.

A planar diagram depicting a link (or knot) as a sequence of segments with gaps representing undercrossings and solid lines overcrossings. In such a diagram, only two ...

An Alexander matrix is a presentation matrix for the Alexander invariant H_1(X^~) of a knot K. If V is a Seifert matrix for a tame knot K in S^3, then V^(T)-tV and V-tV^(T) ...

A knot that secures a rope to a post, ring, another rope, etc., but does not keep its shape by itself.

A closed curve associated with a knot which is displaced along the normal by a small amount. For a knot K parameterized as x^mu(s) for 0<=s<=L along the length of the knot by ...

Let the stick number s(K) of a knot K be the least number of straight sticks needed to make a knot K. The smallest stick number of any knot is s(T)=6, where T is the trefoil ...

Two distinct knots cannot have the same exterior. Or, equivalently, a knot is completely determined by its knot exterior (Cipra 1988; Adams 1994, p. 261). The question was ...

The span of an unoriented link diagram (also called the link spread) is the difference between the highest and lowest degrees of its bracket polynomial. The span is a ...

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 ...