zbMATH — the first resource for mathematics

A polynomial invariant for knots via von Neumann algebras. (English) Zbl 0564.57006
The author introduces a new polynomial invariant $$V_ L(t)$$ for tame oriented links via certain representations of the braid group. That the invariant depends only on the closed braid is a direct consequence of Markov’s theorem and a certain trace formula, which was discovered because of the uniqueness of the trace on certain von Neumann algebras. There is an alternate way to calculate $$V_ L(t)$$ without converting L into a closed braid, using only a Conway type relation; $$V_{unknot}=1$$ and 1/t $$V_{L-}-t V_{L+}=(\sqrt{t}-1/\sqrt{t})V_ L$$. This is also interesting from a view point of formal knot theory. The author gives many results using this invariant. For an example, $$V_{L\sim}(t)=V_ L(1/t)$$ where $$L\sim$$ means the mirror image of L, $$V_{L_ 1\#L_ 2}=V_{L_ 1}\cdot V_{L_ 2}$$ where # means a connected sum of links, $$V_ L(-1)=\Delta_ L(-1)$$ where $$\Delta_ L$$ means the Alexander polynomial, $$V_ L(1)=(-2)^{p-1}$$ where p is the number of components of L, $$V_ K(e^{2\pi i/3})=1$$ and d/dt $$V_ K(1)=0$$ if K is a knot. If K is a knot and $$| \Delta_ K(i)| >3$$, then k cannot be represented as a closed 3 braid. If K is a knot and $$\Delta (e^{2\pi i/5})>6.5$$, then K cannot be represented as a closed 4 braid.
Reviewer: Y.Nakanishi

MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 46L99 Selfadjoint operator algebras ($$C^*$$-algebras, von Neumann ($$W^*$$-) algebras, etc.)
Full Text:
References:
 [1] J. W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. 9 (1923), 93-95. [2] Точно решаемые модели в статистической механике, ”Мир”, Мосцощ, 1985 (Руссиан). Транслатед фром тхе Енглиш бы Е. П. Вол$$^{\приме}$$ский анд Л. И. Дайхин; Транслатион едитед анд щитх а префаце бы А. М. Бродский. [3] D. Bennequin, Entrelacements et structures de contact, These, Paris, 1982. [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] W. Burau, Uber Zopfgruppen und gleichsinning verdrillte Verkettunger, Abh. Math. Sem. Hanischen Univ. 11 (1936), 171-178. [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. [7] H. S. M. Coxeter, Regular complex polytopes, Cambridge University Press, London-New York, 1974. · Zbl 0296.50009 [8] F. A. Garside, The braid group and other groups, Quart. J. Math. Oxford Ser. (2) 20 (1969), 235 – 254. · Zbl 0194.03303 · doi:10.1093/qmath/20.1.235 · doi.org [9] V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1 – 25. · Zbl 0508.46040 · doi:10.1007/BF01389127 · doi.org [10] V. F. R. Jones, Braid groups, Hecke algebras and type II1 factors, Japan-U.S. Conf. Proc. 1983. · Zbl 0659.46054 [11] Louis H. Kauffman, Formal knot theory, Mathematical Notes, vol. 30, Princeton University Press, Princeton, NJ, 1983. · Zbl 0537.57002 [12] Shin’ichi Kinoshita and Hidetaka Terasaka, On unions of knots, Osaka Math. J. 9 (1957), 131 – 153. · Zbl 0080.17001 [13] J. Lannes, Sur l’invariant de Kervaire pour les noeuds classiques, École Polytechnique, Palaiseau, 1984 (preprint). · Zbl 0571.57005 [14] J. Levine, Polynomial invariants of knots of codimension two, Ann. of Math. (2) 84 (1966), 537 – 554. · Zbl 0196.55905 · doi:10.2307/1970459 · doi.org [15] A. A. Markov, Uber die freie Aquivalenz geschlossener Zopfe, Mat. Sb. 1 (1935), 73-78. [16] Kunio Murasugi, On closed 3-braids, American Mathematical Society, Providence, R.I., 1974. Memoirs of the American Mathmatical Society, No. 151. · Zbl 0327.55001 [17] Kenneth A. Perko Jr., On the classification of knots, Proc. Amer. Math. Soc. 45 (1974), 262 – 266. · Zbl 0256.55004 [18] Mihai Pimsner and Sorin Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 57 – 106. · Zbl 0646.46057 [19] Robert T. Powers, Representations of uniformly hyperfinite algebras and their associated von Neumann rings, Ann. of Math. (2) 86 (1967), 138 – 171. · Zbl 0157.20605 · doi:10.2307/1970364 · doi.org [20] Dale Rolfsen, Knots and links, Publish or Perish, Inc., Berkeley, Calif., 1976. Mathematics Lecture Series, No. 7. · Zbl 0339.55004 [21] Lee Rudolph, Nontrivial positive braids have positive signature, Topology 21 (1982), no. 3, 325 – 327. · Zbl 0495.57003 · doi:10.1016/0040-9383(82)90014-3 · doi.org [22] S. Svensson, Handbook of Seaman’s ropework, Dodd, Mead, New York, 1971. [23] H. N. V. Temperley and E. H. Lieb, Relations between the ”percolation” and ”colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ”percolation” problem, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1549, 251 – 280. · Zbl 0211.56703 · doi:10.1098/rspa.1971.0067 · doi.org [24] Hans Wenzl, On sequences of projections, C. R. Math. Rep. Acad. Sci. Canada 9 (1987), no. 1, 5 – 9. · Zbl 0622.47019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.