Unknotting Number

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 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. The unknotting numbers for 10-154 and 10-161 can be found using Menasco's Theorem (Stoimenow 1998).

03-001	1	08-009	1	09-010	2 or 3	09-032	1 or 2
04-001	1	08-010	1 or 2	09-011	2	09-033	1
05-001	2	08-011	1	09-012	1	09-034	1
05-002	1	08-012	2	09-013	2 or 3	09-035	2 or 3
06-001	1	08-013	1	09-014	1	09-036	2
06-002	1	08-014	1	09-015	2	09-037	2
06-003	1	08-015	2	09-016	3	09-038	2 or 3
07-001	3	08-016	2	09-017	2	09-039	1
07-002	1	08-017	1	09-018	2	09-040	2
07-003	2	08-018	2	09-019	1	09-041	2
07-004	2	08-019	3	09-020	2	09-042	1
07-005	2	08-020	1	09-021	1	09-043	2
07-006	1	08-021	1	09-022	1	09-044	1
07-007	1	09-001	4	09-023	2	09-045	1
08-001	1	09-002	1	09-024	1	09-046	2
08-002	2	09-003	3	09-025	2	09-047	2
08-003	2	09-004	2	09-026	1	09-048	2
08-004	2	09-005	2	09-027	1	09-049	2 or 3
08-005	2	09-006	3	09-028	1	10-139	4
08-006	2	09-007	2	09-029	1	10-152	4
08-007	1	09-008	2	09-030	1	10-154	3
08-008	2	09-009	3	09-031	2	10-161	3

See also Bennequin's Conjecture, Menasco's Theorem, Milnor's Conjecture, Signature (Knot)

References

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 57-64, 1994.

Cipra, B. ``From Knot to Unknot.'' What's Happening in the Mathematical Sciences, Vol. 2. Providence, RI: Amer. Math. Soc., pp. 8-13, 1994.

Kawamura, T. ``The Unknotting Numbers of $10_{139}$ and $10_{152}$ are 4.'' To appear in Osaka J. Math. http://ms421sun.ms.u-tokyo.ac.jp/~kawamura/worke.html.

Kirby, R. (Ed.) ``Problems in Low-Dimensional Topology.'' AMS/IP Stud. Adv. Math., 2.2, Geometric Topology (Athens, GA, 1993). Providence, RI: Amer. Math. Soc., pp. 35-473, 1997.

Scharlemann, M. ``Unknotting Number One Knots are Prime.'' Invent. Math. 82, 37-55, 1985.

Stoimenow, A. ``Positive Knots, Closed Braids and the Jones Polynomial.'' Rev. May, 1997. http://www.informatik.hu-berlin.de/~stoimeno/pos.ps.gz.

Weisstein, E. W. ``Knots and Links.'' Mathematica notebook Knots.m.