The smallest number of times a 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 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).