Positive knots, closed braids and the Jones polynomial
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 2, p. 237-285

Using the recent Gauß diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of the polynomials of Alexander/Conway, Jones, HOMFLY, Brandt-Lickorish-Millett-Ho and Kauffman there are only finitely many positive knots with the same polynomial and no positive knot with trivial polynomial. We also discuss an extension of the Bennequin inequality, showing that the unknotting number of a positive knot is not less than its genus, which recovers some recent unknotting number results of A'Campo, Kawamura and Tanaka, and give applications to the Jones polynomial of a positive knot.

Classification:  57M25,  05C99
@article{ASNSP_2003_5_2_2_237_0,
     author = {Stoimenow, Alexander},
     title = {Positive knots, closed braids and the Jones polynomial},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {2},
     year = {2003},
     pages = {237-285},
     zbl = {1170.57300},
     mrnumber = {2004964},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2003_5_2_2_237_0}
}
Stoimenow, Alexander. Positive knots, closed braids and the Jones polynomial. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 2, pp. 237-285. http://www.numdam.org/item/ASNSP_2003_5_2_2_237_0/

[A] N. A'Campo, Generic immersions of curves, knots, monodromy and gordian number, Inst. Hautes Études Sci. Publ. Math. 88 (1998), 151-169. | Numdam | MR 1733329 | Zbl 0960.57007

[Ad] C. C. Adams, “The knot book”, W. H. Freeman & Co., New York, 1994. | MR 1266837 | Zbl 0840.57001

[AM] S. Akbulut - J. D. Mccarthy, “Casson's invariant for oriented 3-spheres”, Mathematical notes 36, Princeton, 1990. | MR 1030042 | Zbl 0695.57011

[Al] J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), 275-306. | JFM 54.0603.03 | MR 1501429

[BN] D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423-472. | MR 1318886 | Zbl 0898.57001

[BN2] D. Bar-Natan, Bibliography of Vassiliev invariants, available from the web site http:// www.math.toronto.edu/drorbn/VasBib/VasBib.html.

[BS] D. Bar-Natan - A. Stoimenow, The Fundamental Theorem of Vassiliev invariants, In “Geometry and Physics", Lecture Notes in Pure & Appl.Math. 184, M.Dekker, New York, 1996, 101-134. | MR 1423158 | Zbl 0878.57004

[Be] D. Bennequin, Entrelacements et équations de Pfaff, Soc. Math. de France, Astérisque 107-108 (1983), 87-161. | MR 753131 | Zbl 0573.58022

[Bi] J. S. Birman, “Braids, links and mapping class groups”, Ann. of Math. Studies 82, Princeton, 1976. | Zbl 0305.57013

[Bi2] J. S. Birman, New Points of View in Knot Theory, Bull. Amer. Math. Soc. 28 (1993), 253-287. | MR 1191478 | Zbl 0785.57001

[BL] J. S. Birman - X. S. Lin, Knot polynomials and Vassiliev's invariants, Invent. Math. 111 (1993), 225-270. | MR 1198809 | Zbl 0812.57011

[BM] J. S. Birman - W. W. Menasco, Studying knots via braids V: The unlink, Trans. Amer. Math. Soc. 329 (1992), 585-606. | MR 1030509 | Zbl 0758.57005

[BW] J. S. Birman - R. F. Williams, Knotted periodic orbits in dynamical systems - I, Lorenz's equations, Topology 22 (1983), 47-82. | MR 682059 | Zbl 0507.58038

[BoW] M. Boileau - C. Weber, Le problème de J. Milnor sur le nombre gordien des nœuds algébriques, Enseign. Math. 30 (1984), 173-222. | MR 767901 | Zbl 0556.57002

[BLM] R. D. Brandt - W. B. R. Lickorish - K. Millett, A polynomial invariant for unoriented knots and links, Invent. Math. 84 (1986), 563-573. | MR 837528 | Zbl 0595.57009

[Bu] J. v. Buskirk, Positive links have positive Conway polynomial, Springer Lecture Notes in Math. 1144 (1983), 146-159. | Zbl 0586.57004

[CG] T. D. Cochran - R. E. Gompf, Applications of Donaldson's theorems to classical knot concordance, homology 3-spheres and Property P, Topology 27 (1988), 495-512. | MR 976591 | Zbl 0669.57003

[Co] J. H. Conway, On enumeration of knots and links, In “Computational Problems in abstract algebra", J. Leech (ed.), 329-358. Pergamon Press, 1969. | MR 258014 | Zbl 0202.54703

[Cr] P. R. Cromwell, Homogeneous links, J. London Math. Soc. (series 2) 39 (1989), 535-552. | MR 1002465 | Zbl 0685.57004

[Cr2] P. R. Cromwell, Positive braids are visually prime, Proc. London Math. Soc. 67 (1993), 384-424. | MR 1226607 | Zbl 0818.57004

[CM] P. R. Cromwell - H. R. Morton, Positivity of knot polynomials on positive links, J. Knot Theory Ramif. 1 (1992), 203-206. | MR 1164116 | Zbl 0757.57006

[DT] C. H. Dowker - M. B. Thistlethwaite, Classification of knot projections, Topol. Appl. 16 (1983), 19-31. | MR 702617 | Zbl 0516.57002

[Fi] T. Fiedler, On the degree of the Jones polynomial, Topology 30 (1991), 1-8. | MR 1081930 | Zbl 0724.57004

[Fi2] T. Fiedler, A small state sum for knots, Topology 32 (1993), 281-294. | MR 1217069 | Zbl 0787.57007

[Fi3] T. Fiedler, “Gauss sum invariants for knots and links”, Kluwer Academic Publishers, Mathematics and Its Applications Vol. 532, 2001. | MR 1948012 | Zbl 1009.57001

[Fi4] T. Fiedler, Die Casson-Invariante eines positiven Knotens ist nicht kleiner als sein Geschlecht, talk given at the knot theory workshop in Siegen, Germany, 1993.

[FS] T. Fiedler - A. Stoimenow, New knot and link invariants, Proceedings of the International Conference on Knot Theory “Knots in Hellas, 98", Series on Knots and Everything 24, World Scientific, 2000. | MR 1865701 | Zbl 0976.57014

[FW] J. Franks - R. F. Williams, Braids and the Jones-Conway polynomial, Trans. Amer. Math. Soc. 303 (1987), 97-108. | MR 896009 | Zbl 0647.57002

[H] P. Freyd - J. Hoste - W. B. R. Lickorish - K. Millett - A. Ocneanu - D. Yetter, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985), 239-246. | MR 776477 | Zbl 0572.57002

[Ga] D. Gabai, Genera of the alternating links, Duke Math. J. 53 (1986), 677-681. | MR 860665 | Zbl 0631.57004

[Ho] C. F. Ho, A polynomial invariant for knots and links - preliminary report, Abstracts Amer. Math. Soc. 6 (1985), 300.

[HT] J. Hoste - M. Thistlethwaite, KnotScape, a knot polynomial calculation and table access program, available at http://www.math.utk.edu/~morwen.

[J] V. F. R. Jones, A polynomial invariant of knots and links via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985), 103-111. | MR 766964 | Zbl 0564.57006

[J2] V. F. R. Jones, Hecke algebra representations of of braid groups and link polynomials, Ann. of Math. 126 (1987), 335-388. | MR 908150 | Zbl 0631.57005

[K] T. Kanenobu, Kauffman polynomials for 2-bridge knots and links, Yokohama Math. J. 38 (1991), 145-154. | MR 1105072 | Zbl 0744.57006

[K2] T. Kanenobu, Examples of polynomial invariants for knots and links, Math. Ann. 275 (1986), 555-572. | MR 859330 | Zbl 0584.57005

[K3] T. Kanenobu, An evaluation of the first derivative of the Q polynomial of a link, Kobe J. Math. 5 (1988), 179-184. | MR 990819 | Zbl 0675.57004

[KM] T. Kanenobu - H. Murakami, 2-bridge knots of unknotting number one, Proc. Amer. Math. Soc. 98(3) (1986), 499-502. | MR 857949 | Zbl 0613.57002

[Ka] L. H. Kauffman, “Knots and physics” (second edition), World Scientific, Singapore, 1993. | MR 1306280 | Zbl 0868.57001

[Ka2] L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), 417-471. | MR 958895 | Zbl 0763.57004

[Ka3] L. H. Kauffman, New invariants in the theory of knots, Amer. Math. Mon. 3 (1988), 195-242. | MR 935433 | Zbl 0657.57001

[Km] T. Kawamura, The unknotting numbers of 10 139 and 10 152 are 4, Osaka J. Math. 35, (3) (1998), 539-546. | MR 1648364 | Zbl 0909.57003

[Km2] T. Kawamura, Relations among the lowest degree of the Jones polynomial and geometric invariants for a closed positive braid, Comment. Math. Helv. 77 (1), (2002), 125-132. | MR 1898395 | Zbl 0991.57006

[Kw] A. Kawauchi, “A survey of Knot Theory”, Birkhäuser, Basel-Boston-Berlin, 1991. | Zbl 0861.57001

[KMr] P. B. Kronheimer - T. Mrowka, On the genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994), 797-808. | MR 1306022 | Zbl 0851.57023

[Li] W. B. R. Lickorish, The unknotting number of a classical knot, In “Contemporary Mathematics" 44 (1985), 117-119. | MR 813107 | Zbl 0607.57002

[L] X.-S. Lin, Finite type link invariants of 3-manifolds, Topology 33, (1) (1994), 45-71. | MR 1259514 | Zbl 0816.57013

[Me] W. W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1986), 37-44. | MR 721450 | Zbl 0525.57003

[Me2] W. W. Menasco, The Bennequin-Milnor Unknotting Conjectures, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), 831-836. | MR 1273914 | Zbl 0817.57008

[MT] W. W. Menasco - M. B. Thistlethwaite, The Tait flyping conjecture, Bull. Amer. Math. Soc. 25 (1991), 403-412. | MR 1098346 | Zbl 0745.57002

[Mi] J. Milnor, "Singular points of complex hypersurfaces", Annals of Math. Studies 61 (1968). | MR 239612 | Zbl 0184.48405

[Mo] H. R. Morton, An irreducible 4-string braid with unknotted closure, Math. Proc. Camb. Phil. Soc. 93 (1983), 259-261. | MR 691995 | Zbl 0522.57006

[Mo2] H. R. Morton, Seifert circles and knot polynomials, Proc. Cambridge Philos. Soc. 99 (1986), 107-109. | MR 809504 | Zbl 0588.57008

[Mu] K. Murasugi, Jones polynomial and classical conjectures in knot theory, Topology 26 (1987), 187-194. | MR 895570 | Zbl 0628.57004

[MP] K. Murasugi - J. Przytycki, The skein polynomial of a planar star product of two links, Math. Proc. Cambridge Philos. Soc. 106 (1989), 273-276. | MR 1002540 | Zbl 0734.57010

[N] T. Nakamura, Positive alternating links are positively alternating, J. Knot Theory Ramif. 9, (1) (2000), 107-112. | MR 1749503 | Zbl 0999.57005

[N2] T. Nakamura, Four-genus and unknotting number of positive knots and links, Osaka J. Math. 37, (2) (2000), 441-451. | MR 1772843 | Zbl 0968.57008

[Oh] Y. Ohyama, On the minimal crossing number and the braid index of links, Canad. J. Math. 45, (1) (1993), 117-131. | MR 1200324 | Zbl 0780.57006

[PV] M. Polyak - O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Notes 11 (1994), 445-454. | MR 1316972 | Zbl 0851.57010

[PV2] M. Polyak - O. Viro, On the Casson knot invariant, J. Of Knot Theory and Its Ram. 10 (2001) (Special volume of the International Conference on Knot Theory “Knots in Hellas, 98"), 711-738. | MR 1839698 | Zbl 0997.57021

[Ro] D. Rolfsen, "Knots and links", Publish or Perish, 1976. | MR 515288 | Zbl 0339.55004

[Ru] L. Rudolph, Braided surfaces and Seifert ribbons for closed braids, Comment. Math. Helv. 58 (1983), 1-37. | MR 699004 | Zbl 0522.57017

[Ru2] L. Rudolph, Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. 29 (1993), 51-59. | MR 1193540 | Zbl 0789.57004

[Ru3] L. Rudolph, Positive links are strongly quasipositive, “Geometry and Topology Monographs" 2 (1999), Proceedings of the Kirbyfest, 555-562. See also http://www. maths.warwick.ac.uk/gt/GTMon2/ paper25.abs.html. | MR 1734423 | Zbl 0962.57004

[St] A. Stoimenow, Gauss sum invariants, Vassiliev invariants and braiding sequences, J. Knot Theory Ramif. 9 (2000), 221-269. | MR 1749498 | Zbl 0998.57032

[St2] A. Stoimenow, Polynomials of knots with up to 10 crossings, available on http://www. math.toronto. edu/stoimeno/.

[St3] A. Stoimenow, Genera of knots and Vassiliev invariants, J. Of Knot Theory and Its Ram. 8 (2) (1999), 253-259. | MR 1687529 | Zbl 0937.57010

[St4] A. Stoimenow, A Survey on Vassiliev Invariants for knots, “Mathematics and Education in Mathematics", Proceedings of the XXVII. Spring Conference of the Union of Bulgarian Mathematicians, 1998, 37-47.

[St5] A. Stoimenow, On some restrictions to the values of the Jones polynomial, preprint. | MR 2136821 | Zbl 1076.57015

[St6] A. Stoimenow, Polynomial values, the linking form and unknotting numbers, preprint. | MR 2106240 | Zbl 1068.57009

[St7] A. Stoimenow, Knots of genus one, Proc. Amer. Math. Soc. 129, (7) (2001), 2141-2156. | MR 1825928 | Zbl 0971.57012

[Ta] T. Tanaka, Unknotting numbers of quasipositive knots, Topology and its Applications 88, (3) (1998), 239-246. | MR 1632085 | Zbl 0928.57007

[Th] M. B. Thistlethwaite, A spanning tree expansion for the Jones polynomial, Topology 26 (1987), 297-309. | MR 899051 | Zbl 0622.57003

[Tr] P. Traczyk, Non-trivial negative links have positive signature, Manuscripta Math. 61 (1988), 279-284. | MR 949818 | Zbl 0665.57008

[Tr2] P. Traczyk, A criterion for signed unknotting number, Contemporary Mathematics 233 (1999), 215-220. | MR 1701685 | Zbl 0934.57003

[Va] V. A. Vassiliev, Cohomology of knot spaces, “Theory of Singularities and its Applications" (Providence) V. I. Arnold (ed.), Amer. Math. Soc., Providence, 1990. | MR 1089670 | Zbl 0727.57008

[Vo] P. Vogel, Algebraic structures on modules of diagrams, to appear in Invent. Math.

[Vo2] P. Vogel, Representation of links by braids: A new algorithm, Comment. Math. Helv. 65 (1990), 104-113. | MR 1036132 | Zbl 0703.57004

[We] H. Wendt, Die Gordische Auflösung von Knoten, Math. Z. 42 (1937), 680-696. | MR 1545700 | Zbl 0016.42005

[Wi] S. Willerton, On the first two Vassiliev invariants, preprint math.GT/0104061. | MR 1959269 | Zbl 1116.57300

[Yo] Y. Yokota, Polynomial invariants of positive links, Topology 31 (1992), 805-811. | MR 1191382 | Zbl 0772.57017

[Zu] L. Zulli, The rank of the trip matrix of a positive knot diagram, J. Knot Theory Ramif. 6 (1997), 299-301. | MR 1452443 | Zbl 0880.57004