Unknotting Number

The unknotting number of a knot is the smallest number of times the knot must be passed through itself to untie it. Lower bounds can be computed using relatively straightforward techniques, but it is in general difficult to determine exact values. Many unknotting numbers can be determined from a knot's knot signature. A knot with unknotting number 1 is a prime knot (Scharlemann 1985). It is not always true that the unknotting number is achieved in a projection with the minimal number of crossings.

The unknotting number is not additive for two combined knots (Brittenham and Hermiller 2025). In particular, for the combined torus knot and its mirror image,

The following table is from Kirby (1997, pp. 88-89), with the values for 10-139 and 10-152 taken from Kawamura (1998). In the following table, Kirby's (1997, p. 88) value has been corrected to reflect the fact that is only currently known to be 1 or 2 (Kawauchi 1996, p. 271). The value has been computed by Stoimenow (2002). The unknotting numbers for 10-154 and 10-161 can be found using the slice-Bennequin inequality (Stoimenow 1998).

Knots for which the unknotting number is not known are 10-11, 10-47, 10-51, 10-54, 10-61, 10-76, 10-77, 10-79, 10-100 (Cha and Livingston 2008).

	0		2		2		3		2		3		2		2		1
	1		1		1		1		2		2		2		2		2
	1		2		1		2		2		2		2		3		2
	2		3		1		1		2		2		2		1		1
	1		1		2		1		2		2		?		2		2
	1		1		1		?		2		1		3		1		3
	1		4		2		2		1		2		1		1		2
	1		1		2		2		2		1		3		1		2
	3		3		1		2		1		2		1		3		1
	1		2		1		2		2		2		2		2		2
	2		2		3		2		3		?		2		1
	2		3		2		1		?		?		1		1
	2		2		2		1		2		2		2		2
	1		2		3		2		3		?		2		4
	1		3		1		2		2		3		2		2
	1		3		2		2		?		2		2		1
	2		2		2		2		2		1		2		3
	2		1		1		1		3		2		1		1
	2		3		2		2		?		1		1		2
	2		1		1		2		2		2		2		2
	2		2		1		1		2		2		2		1
	1		3		2		1		2		2		2		1
	2		2		2		2		2		1		1		2
	1		2		2		2		1		2		1		2
	2		1		3		1		1		2		3		2
	1		2		1		1		?		1		2		2
	2		1		3		1		2		2		2		4
	1		1		2		1		2		2		2		2
	1		2		2		2		2		2		4		3
	2		1		2		2		2		1		2		2