Crossing Number


Any regular diagram of a knot has a finite number of crossing points, this number is called the crossing number of the diagram.  Given all the regular diagrams of a knot, the crossing number of the regular diagram with the fewest crossings is the crossing number of the knot.

Knots are often listed on tables by their crossing numbers.  The crossing number of the unknot is zero.  No nontrivial knots have crossing number 1 or 2, and the trefoil is the only knot with a crossing number of 3.  There is one 4-crossing knot, two 5-crossing knots, three 6-crossing knots, seven 7-crossing knots, 21 8-crossing knots, 49 9-crossing knots, 165 10-crossing knots, and 552 11-crossing knots.  So, on a knot table 31 represents the trefoil and 71 , 72, . . . , 77 represent the knots with 7 crossings.  Knots with 11 crossings are divided into two groups based on whether they are alternating or nonalternating.  There are 367 alternating 11-crossing knots and 185 nonalternating 11-crossing knots.