Positive knots, closed braids and the Jones polynomial
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)
- Volume: 2, Issue: 2, page 237-285
- ISSN: 0391-173X
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topStoimenow, Alexander. "Positive knots, closed braids and the Jones polynomial." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.2 (2003): 237-285. <http://eudml.org/doc/84502>.
@article{Stoimenow2003,
abstract = {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.},
author = {Stoimenow, Alexander},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {2},
pages = {237-285},
publisher = {Scuola normale superiore},
title = {Positive knots, closed braids and the Jones polynomial},
url = {http://eudml.org/doc/84502},
volume = {2},
year = {2003},
}
TY - JOUR
AU - Stoimenow, Alexander
TI - Positive knots, closed braids and the Jones polynomial
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2003
PB - Scuola normale superiore
VL - 2
IS - 2
SP - 237
EP - 285
AB - 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.
LA - eng
UR - http://eudml.org/doc/84502
ER -
References
top- [A] N. A’Campo, Generic immersions of curves, knots, monodromy and gordian number, Inst. Hautes Études Sci. Publ. Math. 88 (1998), 151–169. Zbl0960.57007MR1733329
- [Ad] C. C. Adams, “The knot book”, W. H. Freeman & Co., New York, 1994. Zbl0840.57001MR1266837
- [AM] S. Akbulut – J. D. McCarthy, “Casson’s invariant for oriented 3-spheres”, Mathematical notes 36, Princeton, 1990. Zbl0695.57011MR1030042
- [Al] J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928), 275–306. Zbl54.0603.03MR1501429JFM54.0603.03
- [BN] D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423-472. Zbl0898.57001MR1318886
- [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. Zbl0878.57004MR1423158
- [Be] D. Bennequin, Entrelacements et équations de Pfaff, Soc. Math. de France, Astérisque 107-108 (1983), 87-161. Zbl0573.58022MR753131
- [Bi] J. S. Birman, “Braids, links and mapping class groups”, Ann. of Math. Studies 82, Princeton, 1976. Zbl0305.57013
- [Bi2] J. S. Birman, New Points of View in Knot Theory, Bull. Amer. Math. Soc. 28 (1993), 253-287. Zbl0785.57001MR1191478
- [BL] J. S. Birman – X. S. Lin, Knot polynomials and Vassiliev’s invariants, Invent. Math. 111 (1993), 225-270. Zbl0812.57011MR1198809
- [BM] J. S. Birman – W. W. Menasco, Studying knots via braids V: The unlink, Trans. Amer. Math. Soc. 329 (1992), 585-606. Zbl0758.57005MR1030509
- [BW] J. S. Birman – R. F. Williams, Knotted periodic orbits in dynamical systems - I, Lorenz’s equations, Topology 22 (1983), 47-82. Zbl0507.58038MR682059
- [BoW] M. Boileau – C. Weber, Le problème de J. Milnor sur le nombre gordien des nœuds algébriques, Enseign. Math. 30 (1984), 173-222. Zbl0556.57002MR767901
- [BLM] R. D. Brandt – W. B. R. Lickorish – K. Millett, A polynomial invariant for unoriented knots and links, Invent. Math. 84 (1986), 563-573. Zbl0595.57009MR837528
- [Bu] J. v. Buskirk, Positive links have positive Conway polynomial, Springer Lecture Notes in Math. 1144 (1983), 146-159. Zbl0586.57004
- [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. Zbl0669.57003MR976591
- [Co] J. H. Conway, On enumeration of knots and links, In “Computational Problems in abstract algebra", J. Leech (ed.), 329-358. Pergamon Press, 1969. Zbl0202.54703MR258014
- [Cr] P. R. Cromwell, Homogeneous links, J. London Math. Soc. (series 2) 39 (1989), 535-552. Zbl0685.57004MR1002465
- [Cr2] P. R. Cromwell, Positive braids are visually prime, Proc. London Math. Soc. 67 (1993), 384-424. Zbl0818.57004MR1226607
- [CM] P. R. Cromwell – H. R. Morton, Positivity of knot polynomials on positive links, J. Knot Theory Ramif. 1 (1992), 203-206. Zbl0757.57006MR1164116
- [DT] C. H. Dowker – M. B. Thistlethwaite, Classification of knot projections, Topol. Appl. 16 (1983), 19-31. Zbl0516.57002MR702617
- [Fi] T. Fiedler, On the degree of the Jones polynomial, Topology 30 (1991), 1-8. Zbl0724.57004MR1081930
- [Fi2] T. Fiedler, A small state sum for knots, Topology 32 (1993), 281-294. Zbl0787.57007MR1217069
- [Fi3] T. Fiedler, “Gauss sum invariants for knots and links”, Kluwer Academic Publishers, Mathematics and Its Applications Vol. 532, 2001. Zbl1009.57001MR1948012
- [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. Zbl0976.57014MR1865701
- [FW] J. Franks – R. F. Williams, Braids and the Jones-Conway polynomial, Trans. Amer. Math. Soc. 303 (1987), 97-108. Zbl0647.57002MR896009
- [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. Zbl0572.57002MR776477
- [Ga] D. Gabai, Genera of the alternating links, Duke Math. J. 53 (1986), 677-681. Zbl0631.57004MR860665
- [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. Zbl0564.57006MR766964
- [J2] V. F. R. Jones, Hecke algebra representations of of braid groups and link polynomials, Ann. of Math. 126 (1987), 335-388. Zbl0631.57005MR908150
- [K] T. Kanenobu, Kauffman polynomials for 2-bridge knots and links, Yokohama Math. J. 38 (1991), 145-154. Zbl0744.57006MR1105072
- [K2] T. Kanenobu, Examples of polynomial invariants for knots and links, Math. Ann. 275 (1986), 555-572. Zbl0584.57005MR859330
- [K3] T. Kanenobu, An evaluation of the first derivative of the Q polynomial of a link, Kobe J. Math. 5 (1988), 179-184. Zbl0675.57004MR990819
- [KM] T. Kanenobu – H. Murakami, 2-bridge knots of unknotting number one, Proc. Amer. Math. Soc. 98(3) (1986), 499-502. Zbl0613.57002MR857949
- [Ka] L. H. Kauffman, “Knots and physics” (second edition), World Scientific, Singapore, 1993. Zbl0868.57001MR1306280
- [Ka2] L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), 417-471. Zbl0763.57004MR958895
- [Ka3] L. H. Kauffman, New invariants in the theory of knots, Amer. Math. Mon. 3 (1988), 195-242. Zbl0657.57001MR935433
- [Km] T. Kawamura, The unknotting numbers of and are 4, Osaka J. Math. 35, (3) (1998), 539-546. Zbl0909.57003MR1648364
- [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. Zbl0991.57006MR1898395
- [Kw] A. Kawauchi, “A survey of Knot Theory”, Birkhäuser, Basel-Boston-Berlin, 1991. Zbl0861.57001
- [KMr] P. B. Kronheimer – T. Mrowka, On the genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994), 797-808. Zbl0851.57023MR1306022
- [Li] W. B. R. Lickorish, The unknotting number of a classical knot, In “Contemporary Mathematics" 44 (1985), 117-119. Zbl0607.57002MR813107
- [L] X.-S. Lin, Finite type link invariants of 3-manifolds, Topology 33, (1) (1994), 45-71. Zbl0816.57013MR1259514
- [Me] W. W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1986), 37-44. Zbl0525.57003MR721450
- [Me2] W. W. Menasco, The Bennequin-Milnor Unknotting Conjectures, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), 831-836. Zbl0817.57008MR1273914
- [MT] W. W. Menasco – M. B. Thistlethwaite, The Tait flyping conjecture, Bull. Amer. Math. Soc. 25 (1991), 403-412. Zbl0745.57002MR1098346
- [Mi] J. Milnor, "Singular points of complex hypersurfaces", Annals of Math. Studies 61 (1968). Zbl0184.48405MR239612
- [Mo] H. R. Morton, An irreducible 4-string braid with unknotted closure, Math. Proc. Camb. Phil. Soc. 93 (1983), 259-261. Zbl0522.57006MR691995
- [Mo2] H. R. Morton, Seifert circles and knot polynomials, Proc. Cambridge Philos. Soc. 99 (1986), 107-109. Zbl0588.57008MR809504
- [Mu] K. Murasugi, Jones polynomial and classical conjectures in knot theory, Topology 26 (1987), 187-194. Zbl0628.57004MR895570
- [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. Zbl0734.57010MR1002540
- [N] T. Nakamura, Positive alternating links are positively alternating, J. Knot Theory Ramif. 9, (1) (2000), 107-112. Zbl0999.57005MR1749503
- [N2] T. Nakamura, Four-genus and unknotting number of positive knots and links, Osaka J. Math. 37, (2) (2000), 441-451. Zbl0968.57008MR1772843
- [Oh] Y. Ohyama, On the minimal crossing number and the braid index of links, Canad. J. Math. 45, (1) (1993), 117-131. Zbl0780.57006MR1200324
- [PV] M. Polyak – O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Notes 11 (1994), 445-454. Zbl0851.57010MR1316972
- [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. Zbl0997.57021MR1839698
- [Ro] D. Rolfsen, "Knots and links", Publish or Perish, 1976. Zbl0339.55004MR515288
- [Ru] L. Rudolph, Braided surfaces and Seifert ribbons for closed braids, Comment. Math. Helv. 58 (1983), 1-37. Zbl0522.57017MR699004
- [Ru2] L. Rudolph, Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. 29 (1993), 51-59. Zbl0789.57004MR1193540
- [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. Zbl0962.57004MR1734423
- [St] A. Stoimenow, Gauss sum invariants, Vassiliev invariants and braiding sequences, J. Knot Theory Ramif. 9 (2000), 221-269. Zbl0998.57032MR1749498
- [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. Zbl0937.57010MR1687529
- [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. Zbl1076.57015MR2136821
- [St6] A. Stoimenow, Polynomial values, the linking form and unknotting numbers, preprint. Zbl1068.57009MR2106240
- [St7] A. Stoimenow, Knots of genus one, Proc. Amer. Math. Soc. 129, (7) (2001), 2141-2156. Zbl0971.57012MR1825928
- [Ta] T. Tanaka, Unknotting numbers of quasipositive knots, Topology and its Applications 88, (3) (1998), 239-246. Zbl0928.57007MR1632085
- [Th] M. B. Thistlethwaite, A spanning tree expansion for the Jones polynomial, Topology 26 (1987), 297-309. Zbl0622.57003MR899051
- [Tr] P. Traczyk, Non-trivial negative links have positive signature, Manuscripta Math. 61 (1988), 279-284. Zbl0665.57008MR949818
- [Tr2] P. Traczyk, A criterion for signed unknotting number, Contemporary Mathematics 233 (1999), 215-220. Zbl0934.57003MR1701685
- [Va] V. A. Vassiliev, Cohomology of knot spaces, “Theory of Singularities and its Applications" (Providence) V. I. Arnold (ed.), Amer. Math. Soc., Providence, 1990. Zbl0727.57008MR1089670
- [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. Zbl0703.57004MR1036132
- [We] H. Wendt, Die Gordische Auflösung von Knoten, Math. Z. 42 (1937), 680-696. Zbl0016.42005MR1545700
- [Wi] S. Willerton, On the first two Vassiliev invariants, preprint math.GT/0104061. Zbl1116.57300MR1959269
- [Yo] Y. Yokota, Polynomial invariants of positive links, Topology 31 (1992), 805-811. Zbl0772.57017MR1191382
- [Zu] L. Zulli, The rank of the trip matrix of a positive knot diagram, J. Knot Theory Ramif. 6 (1997), 299-301. Zbl0880.57004MR1452443
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