Linking Number - Generalizations

Generalizations

• Just as closed curves can be linked in three dimensions, any two closed manifolds of dimensions m and n may be linked in a Euclidean space of dimension . Any such link has an associated Gauss map, whose degree is a generalization of the linking number.
• Any framed knot has a self-linking number obtained by computing the linking number of the knot C with a new curve obtained by slightly moving the points of C along the framing vectors. The self-linking number obtained by moving vertically (along the blackboard framing) is known as Kauffman's self-linking number.
• The linking number is defined for two linked circles; given three or more circles, one can define the Milnor invariants, which are a numerical invariant generalizing linking number.
• In algebraic topology, the cup product is a far-reaching algebraic generalization of the linking number, with the Massey products being the algebraic analogs for the Milnor invariants.
• A linkless embedding of an undirected graph is an embedding into three-dimensional space such that every two cycles have zero linking number. The graphs that have a linkless embedding have a forbidden minor characterization as the graphs with no Petersen family minor.