Tait’s conjectures and odd crossing number amphicheiral knots

Stoimenow, A.

doi:10.1090/S0273-0979-08-01196-8

Tait’s conjectures and odd crossing number amphicheiral knots
HTML articles powered by AMS MathViewer

by A. Stoimenow PDF

Bull. Amer. Math. Soc. 45 (2008), 285-291 Request permission

Abstract:

We give a brief historical overview of the Tait conjectures, made 120 years ago in the course of his pioneering work in tabulating the simplest knots, and solved a century later using the Jones polynomial. We announce the solution, again based on a substantial study of the Jones polynomial, of one (possibly his last remaining) problem of Tait, with the construction of amphicheiral knots of almost all odd crossing numbers. An application to the non-triviality problem for the Jones polynomial is also outlined.

References

J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), no. 2, 275–306. MR 1501429, DOI 10.1090/S0002-9947-1928-1501429-1
Joan S. Birman, New points of view in knot theory, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 253–287. MR 1191478, DOI 10.1090/S0273-0979-1993-00389-6
J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) Pergamon, Oxford, 1970, pp. 329–358. MR 0258014

A volume-ish theorem for the Jones polynomial of alternating knots

math.GT/0403448

Oliver T. Dasbach and Xiao-Song Lin, On the head and the tail of the colored Jones polynomial, Compos. Math. 142 (2006), no. 5, 1332–1342. MR 2264669, DOI 10.1112/S0010437X06002296
Shalom Eliahou, Louis H. Kauffman, and Morwen B. Thistlethwaite, Infinite families of links with trivial Jones polynomial, Topology 42 (2003), no. 1, 155–169. MR 1928648, DOI 10.1016/S0040-9383(02)00012-5
Joel Hass and Jeffrey C. Lagarias, The number of Reidemeister moves needed for unknotting, J. Amer. Math. Soc. 14 (2001), no. 2, 399–428. MR 1815217, DOI 10.1090/S0894-0347-01-00358-7
Jim Hoste, The enumeration and classification of knots and links, Handbook of knot theory, Elsevier B. V., Amsterdam, 2005, pp. 209–232. MR 2179263, DOI 10.1016/B978-044451452-3/50006-X
Jim Hoste, Morwen Thistlethwaite, and Jeff Weeks, The first 1,701,936 knots, Math. Intelligencer 20 (1998), no. 4, 33–48. MR 1646740, DOI 10.1007/BF03025227
Vaughan F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103–111. MR 766964, DOI 10.1090/S0273-0979-1985-15304-2
Louis H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), no. 2, 417–471. MR 958895, DOI 10.1090/S0002-9947-1990-0958895-7
Louis H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395–407. MR 899057, DOI 10.1016/0040-9383(87)90009-7
Louis H. Kauffman, Formal knot theory, Mathematical Notes, vol. 30, Princeton University Press, Princeton, NJ, 1983. MR 712133
W. B. R. Lickorish and M. B. Thistlethwaite, Some links with nontrivial polynomials and their crossing-numbers, Comment. Math. Helv. 63 (1988), no. 4, 527–539. MR 966948, DOI 10.1007/BF02566777
William W. Menasco and Morwen B. Thistlethwaite, The Tait flyping conjecture, Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 2, 403–412. MR 1098346, DOI 10.1090/S0273-0979-1991-16083-0
Kunio Murasugi, Jones polynomials and classical conjectures in knot theory, Topology 26 (1987), no. 2, 187–194. MR 895570, DOI 10.1016/0040-9383(87)90058-9
Kunio Murasugi, Jones polynomials and classical conjectures in knot theory. II, Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 2, 317–318. MR 898151, DOI 10.1017/S0305004100067335
Kenneth A. Perko Jr., On the classification of knots, Proc. Amer. Math. Soc. 45 (1974), 262–266. MR 353294, DOI 10.1090/S0002-9939-1974-0353294-X
Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
D. S. Silver, Knot Theory’s Odd Origins, American Scientist 94(2) (2006), 158–165.
A. Stoimenow, On polynomials and surfaces of variously positive links, J. Eur. Math. Soc. (JEMS) 7 (2005), no. 4, 477–509. MR 2159224, DOI 10.4171/JEMS/36
Morwen B. Thistlethwaite, On the Kauffman polynomial of an adequate link, Invent. Math. 93 (1988), no. 2, 285–296. MR 948102, DOI 10.1007/BF01394334
Morwen B. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987), no. 3, 297–309. MR 899051, DOI 10.1016/0040-9383(87)90003-6
Morwen Thistlethwaite, On the structure and scarcity of alternating links and tangles, J. Knot Theory Ramifications 7 (1998), no. 7, 981–1004. MR 1654669, DOI 10.1142/S021821659800053X