An invariant of regular isotopy

Kauffman, Louis H.

doi:10.1090/S0002-9947-1990-0958895-7

An invariant of regular isotopy
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Trans. Amer. Math. Soc. 318 (1990), 417-471 Request permission

Abstract:

This paper studies a two-variable Laurent polynomial invariant of regular isotopy for classical unoriented knots and links. This invariant is denoted ${L_K}$ for a link $K$, and it satisfies the axioms: 1. Regularly isotopic links receive the same polynomial. 2. ${L_{[{\text {unk}}]}} = 1$. 3. ${L_{[{\text {unk}}]}} = aL,\qquad {L_{[{\text {unk}}]}} = {a^{ - 1}}L$. 4. ${L_{[{\text {unk}}]}} + {L_{[{\text {unk]}}}} = z({L_{[{\text {unk]}}}} + {L_{[{\text {unk]}}}})$. Small diagrams indicate otherwise identical parts of larger diagrams. Regular isotopy is the equivalence relation generated by the Reidemeister moves of type II and type III. Invariants of ambient isotopy are obtained from $L$ by writhe-normalization.

References

Yasuhiro Akutsu and Miki Wadati, Exactly solvable models and new link polynomials. I. $N$-state vertex models, J. Phys. Soc. Japan 56 (1987), no. 9, 3039–3051. MR 914721, DOI 10.1143/JPSJ.56.3039
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
R. Ball and M. L. Mehta, Sequence of invariants for knots and links, J. Physique 42 (1981), no. 9, 1193–1199 (English, with French summary). MR 630350, DOI 10.1051/jphys:019810042090119300
Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR 0375281

Braids, link polynomials and a new algebra

Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR 808776
Robert D. Brandt, W. B. R. Lickorish, and Kenneth C. Millett, A polynomial invariant for unoriented knots and links, Invent. Math. 84 (1986), no. 3, 563–573. MR 837528, DOI 10.1007/BF01388747

On algebras which are connected with the semisimple continuous groups

38

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
Richard H. Crowell and Ralph H. Fox, Introduction to knot theory, Ginn and Company, Boston, Mass., 1963. Based upon lectures given at Haverford College under the Philips Lecture Program. MR 0146828
P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 2, 239–246. MR 776477, DOI 10.1090/S0273-0979-1985-15361-3
W. Graeub, Die semilinearen Abbildungen, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1950 (1950), 205–272 (German). MR 0042709

A new polynomial invariant for knots and links—preliminary report

6

V. F. R. Jones, A new knot polynomial and von Neumann algebras, Notices Amer. Math. Soc. 33 (1986), no. 2, 219–225. MR 830613

A polynomial invariant for links via von Neumann algebras

12

V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388. MR 908150, DOI 10.2307/1971403
V. F. R. Jones, On knot invariants related to some statistical mechanical models, Pacific J. Math. 137 (1989), no. 2, 311–334. MR 990215

A note on the Kauffman polynomial

Louis H. Kauffman, The Conway polynomial, Topology 20 (1981), no. 1, 101–108. MR 592573, DOI 10.1016/0040-9383(81)90017-3

Formal knot theory

On knots

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

New invariants in the theory of knots

Louis H. Kauffman, New invariants in the theory of knots, Amer. Math. Monthly 95 (1988), no. 3, 195–242. MR 935433, DOI 10.2307/2323625
Louis H. Kauffman, Statistical mechanics and the Jones polynomial, Braids (Santa Cruz, CA, 1986) Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 263–297. MR 975085, DOI 10.1090/conm/078/975085

An invariant of regular isotopy

Knots and physics

Link polynomials and a graphical calculus

L. H. Kauffman, Knots, abstract tensors and the Yang-Baxter equation, Knots, topology and quantum field theories (Florence, 1989) World Sci. Publ., River Edge, NJ, 1989, pp. 179–334. MR 1146944
Mark E. Kidwell, On the degree of the Brandt-Lickorish-Millett-Ho polynomial of a link, Proc. Amer. Math. Soc. 100 (1987), no. 4, 755–762. MR 894450, DOI 10.1090/S0002-9939-1987-0894450-0

The enumeration, description and construction of knots with fewer than

crossings

32

W. B. R. Lickorish and Kenneth C. Millett, A polynomial invariant of oriented links, Topology 26 (1987), no. 1, 107–141. MR 880512, DOI 10.1016/0040-9383(87)90025-5
W. B. R. Lickorish, A relationship between link polynomials, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 1, 109–112. MR 838656, DOI 10.1017/S0305004100065890

Smith’s prize essay

Non-alternate

knots

35

H. R. Morton, Seifert circles and knot polynomials, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 107–109. MR 809504, DOI 10.1017/S0305004100063982

Calculating the

variable polynomial for knots presented as closed braids

Hitoshi Murakami, A formula for the two-variable link polynomial, Topology 26 (1987), no. 4, 409–412. MR 919727, DOI 10.1016/0040-9383(87)90039-5
Jun Murakami, The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (1987), no. 4, 745–758. MR 927059
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

Jones polynomials and classical conjectures in knot theory

Conway formulas for Jones-Conway and Kauffman polynomials

Knotentheorie

Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288

On knots

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 B. Thistlethwaite, Kauffman’s polynomial and alternating links, Topology 27 (1988), no. 3, 311–318. MR 963633, DOI 10.1016/0040-9383(88)90012-2
V. G. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), no. 3, 527–553. MR 939474, DOI 10.1007/BF01393746
Bruce Trace, On the Reidemeister moves of a classical knot, Proc. Amer. Math. Soc. 89 (1983), no. 4, 722–724. MR 719004, DOI 10.1090/S0002-9939-1983-0719004-4
Hassler Whitney, On regular closed curves in the plane, Compositio Math. 4 (1937), 276–284. MR 1556973
Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399. MR 990772
David N. Yetter, Markov algebras, Braids (Santa Cruz, CA, 1986) Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 705–730. MR 975104, DOI 10.1090/conm/078/975104