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The least number of crossings that occur in any projection of a link. In general, it is difficult to find the crossing number of a given link. Knots and links are generally ...

Any link can be represented by a closed braid.

A type I Markov move.

Any tangle obtained by additions and multiplications of rational tangles (Adams 1994).

A category of link encompassing both alternating knots and torus knots.

A type I move (conjugation) takes AB->BA for A, B in B_n where B_n is a braid group. A type II move (stabilization) takes A->Ab_n or A->Ab_n^(-1) for A in B_n, and b_n, Ab_n, ...

The Reidemeister move of type II.

The Reidemeister move of type III.

The contact triangle of a triangle DeltaABC, also called the intouch triangle, is the triangle DeltaC_AC_BC_C formed by the points of tangency of the incircle of DeltaABC ...

The self-intersection of a one-sided surface. The word "cross-cap" is sometimes also written without the hyphen as the single word "crosscap." The cross-cap can be thought of ...