Alternating knot

It is conjectured that as the crossing number increases, the percentage of knots that are alternating goes to 0 exponentially quickly.

[1] In November 2015, Joshua Evan Greene published a preprint that established a characterization of alternating links in terms of definite spanning surfaces, i.e. a definition of alternating links (of which alternating knots are a special case) without using the concept of a link diagram.

An alternating knot diagram is in one-to-one correspondence with a planar graph.

Each crossing is associated with an edge and half of the connected components of the complement of the diagram are associated with vertices in a checker board manner.

Marc Lackenby has shown that the volume has upper and lower linear bounds as functions of the number of twist regions of a reduced, alternating diagram.