A polynomial invariant for knots via von Neumann algebras. (English) Zbl 0564.57006

Jones, Vaughan F. R.

Bull. Am. Math. Soc., New Ser. 12, 103-111 (1985).

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.)

Keywords:

polynomial invariant for tame oriented links; trace on von Neumann algebras; representations of the braid group; connected sum of links; Alexander polynomial

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
[9]	V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1 – 25. · Zbl 0508.46040
[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
[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
[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
[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
[24]	Hans Wenzl, On sequences of projections, C. R. Math. Rep. Acad. Sci. Canada 9 (1987), no. 1, 5 – 9. · Zbl 0622.47019