An invariant of regular isotopy

Kauffman, Louis H.

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

An invariant of regular isotopy

Author: Louis H. Kauffman
Journal: Trans. Amer. Math. Soc. 318 (1990), 417-471
MSC: Primary 57M25
DOI: https://doi.org/10.1090/S0002-9947-1990-0958895-7
MathSciNet review: 958895
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 , 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 by writhe-normalization.

1. 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, https://doi.org/10.1143/JPSJ.56.3039
[1] J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), no. 2, 275–306. MR 1501429, https://doi.org/10.1090/S0002-9947-1928-1501429-1
[2] 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
[3] Joan S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 82. MR 0375281
[4] J. S. Birman and H. Wenzel, Braids, link polynomials and a new algebra, preprint, 1986.
[5] Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR 808776
[6] 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, https://doi.org/10.1007/BF01388747
[7] R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. of Math. 38 (1937).
[8] 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
[9] Richard H. Crowell and Ralph H. Fox, Introduction to knot theory, Based upon lectures given at Haverford College under the Philips Lecture Program, Ginn and Co., Boston, Mass., 1963. MR 0146828
[10] 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, https://doi.org/10.1090/S0273-0979-1985-15361-3
[11] W. Graeub, Die semilinearen Abbildungen, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1950 (1950), 205–272 (German). MR 0042709
[12] C. F. Ho, A new polynomial invariant for knots and links--preliminary report, Abstracts Amer. Math. Soc. 6 (1985), 300.
[13] V. F. R. Jones, A new knot polynomial and von Neumann algebras, Notices Amer. Math. Soc. 33 (1986), no. 2, 219–225. MR 830613
[14] -, A polynomial invariant for links via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985), 103-112.
[15] 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, https://doi.org/10.2307/1971403
[15] V. F. R. Jones, On knot invariants related to some statistical mechanical models, Pacific J. Math. 137 (1989), no. 2, 311–334. MR 990215
[16] T. Kanenobu and M. Sakuma, A note on the Kauffman polynomial, preprint, 1986.
[17] Louis H. Kauffman, The Conway polynomial, Topology 20 (1981), no. 1, 101–108. MR 592573, https://doi.org/10.1016/0040-9383(81)90017-3
[18] -, Formal knot theory, Princeton Univ. Press Math. Notes, no. 30, Princeton Univ. Press, 1983.
[19] -, On knots, Ann. of Math. Stud., no. 115, Princeton Univ. Press, Princeton, N. J., 1987.
[20] Louis H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395–407. MR 899057, https://doi.org/10.1016/0040-9383(87)90009-7
[21] -, New invariants in the theory of knots (lectures given in Rome, June 1986), (to appear).
[22] Louis H. Kauffman, New invariants in the theory of knots, Amer. Math. Monthly 95 (1988), no. 3, 195–242. MR 935433, https://doi.org/10.2307/2323625
[23] 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, https://doi.org/10.1090/conm/078/975085
[24] -, An invariant of regular isotopy, Announcement, 1985.
[25] -, Knots and physics (in preparation).
[26] L. H. Kauffman and P. Vogel, Link polynomials and a graphical calculus, Announcement, 1987.
[26] 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
[27] 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, https://doi.org/10.1090/S0002-9939-1987-0894450-0
[28] T. P. Kirkman, The enumeration, description and construction of knots with fewer than crossings, Trans. Roy. Soc. Edinburgh 32 (1865), 281-309.
[29] W. B. R. Lickorish and Kenneth C. Millett, A polynomial invariant of oriented links, Topology 26 (1987), no. 1, 107–141. MR 880512, https://doi.org/10.1016/0040-9383(87)90025-5
[31] W. B. R. Lickorish, A relationship between link polynomials, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 1, 109–112. MR 838656, https://doi.org/10.1017/S0305004100065890
[32] A. S. Lipson, Smith's prize essay, Univ. of Cambridge, 1987.
[33] C. N. Little, Non-alternate knots, Trans. Roy. Soc. Edinburgh 35 (1889), 663-664.
[34] H. R. Morton, Seifert circles and knot polynomials, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 107–109. MR 809504, https://doi.org/10.1017/S0305004100063982
[35] H. R. Morton and H. B. Short, Calculating the -variable polynomial for knots presented as closed braids, preprint, 1986.
[36] Hitoshi Murakami, A formula for the two-variable link polynomial, Topology 26 (1987), no. 4, 409–412. MR 919727, https://doi.org/10.1016/0040-9383(87)90039-5
[37] Jun Murakami, The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (1987), no. 4, 745–758. MR 927059
[38] Kunio Murasugi, Jones polynomials and classical conjectures in knot theory, Topology 26 (1987), no. 2, 187–194. MR 895570, https://doi.org/10.1016/0040-9383(87)90058-9
[39] -, Jones polynomials and classical conjectures in knot theory. II, preprint, 1986.
[40] J. Przytycki, Conway formulas for Jones-Conway and Kauffman polynomials, preprint, 1986.
[41] K. Reidemeister, Knotentheorie, Chelsea, New York, 1948.
[42] Dale Rolfsen, Knots and links, Publish or Perish, Inc., Berkeley, Calif., 1976. Mathematics Lecture Series, No. 7. MR 0515288
[43] P. G. Tait, On knots. I, II, III, Scientific Papers, Vol. I, Cambridge Univ. Press, London, 1898, pp. 273-347.
[44] Morwen B. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987), no. 3, 297–309. MR 899051, https://doi.org/10.1016/0040-9383(87)90003-6
[45] Morwen B. Thistlethwaite, Kauffman’s polynomial and alternating links, Topology 27 (1988), no. 3, 311–318. MR 963633, https://doi.org/10.1016/0040-9383(88)90012-2
[45] V. G. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), no. 3, 527–553. MR 939474, https://doi.org/10.1007/BF01393746
[46] Bruce Trace, On the Reidemeister moves of a classical knot, Proc. Amer. Math. Soc. 89 (1983), no. 4, 722–724. MR 719004, https://doi.org/10.1090/S0002-9939-1983-0719004-4
[47] Hassler Whitney, On regular closed curves in the plane, Compositio Math. 4 (1937), 276–284. MR 1556973
[48] Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399. MR 990772
[49] 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, https://doi.org/10.1090/conm/078/975104