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An invariant of regular isotopy. (English) Zbl 0763.57004
Two links in the 3-sphere are said to be regular isotopic if one is transformed into the other by a finite sequence of Reidemeister moves of type II and III (excluding of type I). One of the objectives of this paper is to demonstrate that polynomial invariants defined by skein relations are, in fact, obtained as certain normalization of regular isotopy invariants. The Jones polynomial, Kauffman polynomial and Dubrovnik polynomial are examples of this type . In the later section, the author gives a complete proof of the well-definedness and uniqueness of the L-polynomial as a regular isotopy invariant. Another objective is to give ingenious diagrammatic interpretation of several algebras, e.g. a braid monoid, connection monoid, Brauer monoid. These monoids and their interpretations have been used extensively in recent knot theory. This paper is rather a research oriented expository paper than a pure research paper.

MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57M15 Relations of low-dimensional topology with graph theory 82B23 Exactly solvable models; Bethe ansatz 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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References:
 [1] Yasuhiro Akutsu and Miki Wadati, Exactly solvable models and new link polynomials. I. \?-state vertex models, J. Phys. Soc. Japan 56 (1987), no. 9, 3039 – 3051. · Zbl 0719.57003 [2] J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), no. 2, 275 – 306. · JFM 54.0603.03 [3] 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). [4] 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. · Zbl 0297.57001 [5] J. S. Birman and H. Wenzel, Braids, link polynomials and a new algebra, preprint, 1986. [6] Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. · Zbl 0568.57001 [7] 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. · Zbl 0595.57009 [8] R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. of Math. 38 (1937). · Zbl 0017.39105 [9] 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. [10] 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. · Zbl 0362.55001 [11] 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. · Zbl 0572.57002 [12] W. Graeub, Die semilinearen Abbildungen, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1950 (1950), 205 – 272 (German). · Zbl 0041.52302 [13] C. F. Ho, A new polynomial invariant for knots and links–preliminary report, Abstracts Amer. Math. Soc. 6 (1985), 300. [14] V. F. R. Jones, A new knot polynomial and von Neumann algebras, Notices Amer. Math. Soc. 33 (1986), no. 2, 219 – 225. [15] -, A polynomial invariant for links via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985), 103-112. [16] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335 – 388. · Zbl 0631.57005 [17] V. F. R. Jones, On knot invariants related to some statistical mechanical models, Pacific J. Math. 137 (1989), no. 2, 311 – 334. · Zbl 0695.46029 [18] T. Kanenobu and M. Sakuma, A note on the Kauffman polynomial, preprint, 1986. [19] Louis H. Kauffman, The Conway polynomial, Topology 20 (1981), no. 1, 101 – 108. · Zbl 0456.57004 [20] -, Formal knot theory, Princeton Univ. Press Math. Notes, no. 30, Princeton Univ. Press, 1983. · Zbl 0537.57002 [21] -, On knots, Ann. of Math. Stud., no. 115, Princeton Univ. Press, Princeton, N. J., 1987. · Zbl 0627.57002 [22] Louis H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395 – 407. · Zbl 0622.57004 [23] -, New invariants in the theory of knots (lectures given in Rome, June 1986), (to appear). [24] Louis H. Kauffman, New invariants in the theory of knots, Amer. Math. Monthly 95 (1988), no. 3, 195 – 242. · Zbl 0657.57001 [25] 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. [26] -, An invariant of regular isotopy, Announcement, 1985. [27] -, Knots and physics (in preparation). · Zbl 0782.57004 [28] L. H. Kauffman and P. Vogel, Link polynomials and a graphical calculus, Announcement, 1987. · Zbl 0795.57001 [29] 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. [30] 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. · Zbl 0638.57004 [31] T. P. Kirkman, The enumeration, description and construction of knots with fewer than $$10$$ crossings, Trans. Roy. Soc. Edinburgh 32 (1865), 281-309. · JFM 17.0521.02 [32] W. B. R. Lickorish and Kenneth C. Millett, A polynomial invariant of oriented links, Topology 26 (1987), no. 1, 107 – 141. · Zbl 0608.57009 [33] W. B. R. Lickorish, A relationship between link polynomials, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 1, 109 – 112. · Zbl 0603.57005 [34] A. S. Lipson, Smith’s prize essay, Univ. of Cambridge, 1987. [35] C. N. Little, Non-alternate $$+ -$$ knots, Trans. Roy. Soc. Edinburgh 35 (1889), 663-664. · JFM 21.0545.01 [36] H. R. Morton, Seifert circles and knot polynomials, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 107 – 109. · Zbl 0588.57008 [37] H. R. Morton and H. B. Short, Calculating the $$2$$-variable polynomial for knots presented as closed braids, preprint, 1986. · Zbl 0738.57003 [38] Hitoshi Murakami, A formula for the two-variable link polynomial, Topology 26 (1987), no. 4, 409 – 412. · Zbl 0633.57003 [39] Jun Murakami, The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (1987), no. 4, 745 – 758. · Zbl 0666.57006 [40] Kunio Murasugi, Jones polynomials and classical conjectures in knot theory, Topology 26 (1987), no. 2, 187 – 194. · Zbl 0628.57004 [41] -, Jones polynomials and classical conjectures in knot theory. II, preprint, 1986. [42] J. Przytycki, Conway formulas for Jones-Conway and Kauffman polynomials, preprint, 1986. · Zbl 0766.57005 [43] K. Reidemeister, Knotentheorie, Chelsea, New York, 1948. · JFM 58.1202.04 [44] Dale Rolfsen, Knots and links, Publish or Perish, Inc., Berkeley, Calif., 1976. Mathematics Lecture Series, No. 7. · Zbl 0339.55004 [45] P. G. Tait, On knots. I, II, III, Scientific Papers, Vol. I, Cambridge Univ. Press, London, 1898, pp. 273-347. [46] Morwen B. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987), no. 3, 297 – 309. · Zbl 0622.57003 [47] Morwen B. Thistlethwaite, Kauffman’s polynomial and alternating links, Topology 27 (1988), no. 3, 311 – 318. · Zbl 0667.57002 [48] V. G. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), no. 3, 527 – 553. · Zbl 0648.57003 [49] Bruce Trace, On the Reidemeister moves of a classical knot, Proc. Amer. Math. Soc. 89 (1983), no. 4, 722 – 724. · Zbl 0554.57003 [50] Hassler Whitney, On regular closed curves in the plane, Compositio Math. 4 (1937), 276 – 284. · Zbl 0016.13804 [51] Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351 – 399. · Zbl 0667.57005 [52] David N. Yetter, Markov algebras, Braids (Santa Cruz, CA, 1986) Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 705 – 730.
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