“Unknotting Number”的概念、定义、习题解题方法及翻译-数学辞典

The smallest number of times a Knot must be passed through itself to untie it. Lower bounds can be computed usingrelatively straightforward techniques, but it is in general difficult to determine exact values. Many unknotting numberscan be determined from a knot's Signature. A Knot withunknotting 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 10-139and 10-152taken fromKawamura. The unknotting numbers for 10-154and 10-161can 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

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 and 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.