The Jones polynomial, Knots, diagrams, and categories

Kauffman, Louis

doi:10.1090/bull/1792

The Jones polynomial, Knots, diagrams, and categories
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by Louis H. Kauffman;

Bull. Amer. Math. Soc. 60 (2023), 507-537

DOI: https://doi.org/10.1090/bull/1792

Published electronically: July 5, 2023

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Abstract:

This essay is a remembrance of Vaughan Jones and a diagrammatic exposition of the remarkable breakthroughs in knot theory and low-dimensional topology that were catalyzed by his work.

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
Yasuhiro Akutsu, Tetsuo Deguchi, and Miki Wadati, Exactly solvable models and new link polynomials. II. Link polynomials for closed $3$-braids, J. Phys. Soc. Japan 56 (1987), no. 10, 3464–3479. MR 923086, DOI 10.1143/JPSJ.56.3464
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
J. W. Alexander and G. B. Briggs, On types of knotted curves, Ann. of Math. (2) 28 (1926/27), no. 1-4, 562–586. MR 1502807, DOI 10.2307/1968399
Daniel Altschuler and Laurent Freidel, Vassiliev knot invariants and Chern-Simons perturbation theory to all orders, Comm. Math. Phys. 187 (1997), no. 2, 261–287. MR 1463829, DOI 10.1007/s002200050136
Michael Atiyah, The geometry and physics of knots, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1990. MR 1078014, DOI 10.1017/CBO9780511623868
John A. Baldwin and Adam Simon Levine, A combinatorial spanning tree model for knot Floer homology, Adv. Math. 231 (2012), no. 3-4, 1886–1939. MR 2964628, DOI 10.1016/j.aim.2012.06.006
Dror - dror Bar-Natan, Perturbative aspects of the Chern-Simons topological quantum field theory, ProQuest LLC, Ann Arbor, MI, 1991. Thesis (Ph.D.)–Princeton University. MR 2686517
Dror Bar-Natan, Perturbative Chern-Simons theory, J. Knot Theory Ramifications 4 (1995), no. 4, 503–547. MR 1361082, DOI 10.1142/S0218216595000247
Dror Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), no. 2, 423–472. MR 1318886, DOI 10.1016/0040-9383(95)93237-2
Dror Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002), 337–370. MR 1917056, DOI 10.2140/agt.2002.2.337
Dror Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443–1499. MR 2174270, DOI 10.2140/gt.2005.9.1443
Rodney J. Baxter, Exactly solved models in statistical mechanics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1982. MR 690578
Joan S. Birman and Xiao-Song Lin, Knot polynomials and Vassiliev’s invariants, Invent. Math. 111 (1993), no. 2, 225–270. MR 1198809, DOI 10.1007/BF01231287
S. Chmutov, S. Duzhin, and J. Mostovoy, Introduction to Vassiliev knot invariants, Cambridge University Press, Cambridge, 2012. MR 2962302, DOI 10.1017/CBO9781139107846
Vincent Coll, Anthony Giaquinto, and Colton Magnant, Meanders and Frobenius seaweed Lie algebras, J. Gen. Lie Theory Appl. 5 (2011), Art. ID G110103, 7. MR 2846732, DOI 10.4303/jglta/G110103
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-New York-Toronto, Ont., 1970, pp. 329–358. MR 258014
V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
Heather A. Dye, Aaron Kaestner, and Louis H. Kauffman, Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots, J. Knot Theory Ramifications 26 (2017), no. 3, 1741001, 57. MR 3627701, DOI 10.1142/S0218216517410012
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
Mikhail Goussarov, Michael Polyak, and Oleg Viro, Finite-type invariants of classical and virtual knots, Topology 39 (2000), no. 5, 1045–1068. MR 1763963, DOI 10.1016/S0040-9383(99)00054-3
C. F. Ho, A new polynomial invariant for knots and links—preliminary report, Abstracts Amer. Math. Soc. 6 (1985), 300.
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 323842, DOI 10.1007/978-1-4612-6398-2
Vaughan F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103–111. MR 766964, DOI 10.1090/S0273-0979-1985-15304-2
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, DOI 10.2140/pjm.1989.137.311
Louis H. Kauffman, Formal knot theory, Mathematical Notes, vol. 30, Princeton University Press, Princeton, NJ, 1983. MR 712133
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
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, On knots, Annals of Mathematics Studies, vol. 115, Princeton University Press, Princeton, NJ, 1987. MR 907872
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
Louis H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), no. 2, 417–471. MR 958895, DOI 10.1090/S0002-9947-1990-0958895-7
Louis H. Kauffman, Knots and physics, 4th ed., Series on Knots and Everything, vol. 53, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. MR 3013186, DOI 10.1142/8338
Louis H. Kauffman and Sóstenes L. Lins, Temperley-Lieb recoupling theory and invariants of $3$-manifolds, Annals of Mathematics Studies, vol. 134, Princeton University Press, Princeton, NJ, 1994. MR 1280463, DOI 10.1515/9781400882533
Louis H. Kauffman, Knot diagrammatics, Handbook of knot theory, Elsevier B. V., Amsterdam, 2005, pp. 233–318. MR 2179264, DOI 10.1016/B978-044451452-3/50007-1
Louis H. Kauffman, Chern-Simons theory, Vassiliev invariants, loop quantum gravity and functional integration without integration, Internat. J. Modern Phys. A 30 (2015), no. 35, 1530067, 27. MR 3437102, DOI 10.1142/S0217751X15300677
Louis H. Kauffman, An introduction to Khovanov homology, Knot theory and its applications, Contemp. Math., vol. 670, Amer. Math. Soc., Providence, RI, 2016, pp. 105–139. MR 3568526, DOI 10.1090/conm/670/13447
Mikhail Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359–426. MR 1740682, DOI 10.1215/S0012-7094-00-10131-7
Robion Kirby and Paul Melvin, The $3$-manifold invariants of Witten and Reshetikhin-Turaev for $\textrm {sl}(2,\textbf {C})$, Invent. Math. 105 (1991), no. 3, 473–545. MR 1117149, DOI 10.1007/BF01232277
P. B. Kronheimer and T. S. Mrowka, Khovanov homology is an unknot-detector, Publ. Math. Inst. Hautes Études Sci. 113 (2011), 97–208. MR 2805599, DOI 10.1007/s10240-010-0030-y
W. B. R. Lickorish, The skein method for three-manifold invariants, J. Knot Theory Ramifications 2 (1993), no. 2, 171–194. MR 1227009, DOI 10.1142/S0218216593000118
Dan Margalit, The mathematics of Joan Birman, Notices Amer. Math. Soc. 66 (2019), no. 3, 341–353. MR 3889349
Peter Ozsváth and Zoltán Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004), no. 1, 58–116. MR 2065507, DOI 10.1016/j.aim.2003.05.001
Peter Ozsváth and Zoltán Szabó, Kauffman states, bordered algebras, and a bigraded knot invariant, Adv. Math. 328 (2018), 1088–1198. MR 3771149, DOI 10.1016/j.aim.2018.02.017
Roger Penrose, Applications of negative dimensional tensors, Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) Academic Press, London-New York, 1971, pp. 221–244. MR 281657
Józef H. Przytycki and PawełTraczyk, Conway algebras and skein equivalence of links, Proc. Amer. Math. Soc. 100 (1987), no. 4, 744–748. MR 894448, DOI 10.1090/S0002-9939-1987-0894448-2
Kurt Reidemeister, Elementare Begründung der Knotentheorie, Abh. Math. Sem. Univ. Hamburg 5 (1927), no. 1, 24–32 (German). MR 3069462, DOI 10.1007/BF02952507
K. Reidemeister, Knotentheorie. Julius Springer (1932).
N.Y. Reshetikhin, Quantized universal enveloping algebras, the Yang–Baxter equation and invariants of links, I and II, LOMI reprints E-4, E-17, Steklov Institute, Leningrad, USSR (1987).
N. Reshetikhin and V. G. Turaev, Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547–597. MR 1091619, DOI 10.1007/BF01239527
Ted Stanford, Finite-type invariants of knots, links, and graphs, Topology 35 (1996), no. 4, 1027–1050. MR 1404922, DOI 10.1016/0040-9383(95)00056-9
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
V. G. Turaev and O. Ya. Viro, State sum invariants of $3$-manifolds and quantum $6j$-symbols, Topology 31 (1992), no. 4, 865–902. MR 1191386, DOI 10.1016/0040-9383(92)90015-A
V. A. Vassiliev, Cohomology of knot spaces, Theory of singularities and its applications, Adv. Soviet Math., vol. 1, Amer. Math. Soc., Providence, RI, 1990, pp. 23–69. MR 1089670
Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399. MR 990772, DOI 10.1007/BF01217730
Matthias Wolfrum, Enumeration of positive meanders, Patterns of dynamics, Springer Proc. Math. Stat., vol. 205, Springer, Cham, 2017, pp. 203–212. MR 3775413, DOI 10.1007/978-3-319-64173-7_{1}3