We apply the concept of braiding sequences to the Conway and skein polynomial, and some geometric invariants of positive links. Using degree and coefficient growth properties of the Conway polynomial, estimates of braid index and Legendrian invariants are given. We enumerate alternating (and some other classes of) links of given genus asymptotically up to constants by braid index.
Nous appliquons le concept de séquences de tressage aux polynômes de skein et de Conway, mais aussi à quelques invariants géométriques des entrelacs positifs. On donne des estimations pour l’indice des tresses et pour des invariants legendriens, en utilisant le degré et des propriétés de croissance des coefficients du polynôme de Conway. Nous énumérons asymptotiquement à une constante près les entrelacs alternants (et quelques autres) de genre donné par leur indice de tresses.
Accepted:
Published online:
Keywords: positive knot, alternating knot, braid index, genus, link polynomial, Legendrian knot, Bennequin inequality, enumeration
Mot clés : noeud positif, noeud alternant, indice de tresses, genre, invariant polynomial d’entrelacs, noeud legendrien, inégalité de Bennequin, énumération des noeuds
Stoimenow, Alexander 1
@article{AIF_2020__70_4_1431_0, author = {Stoimenow, Alexander}, title = {Application of braiding sequences {IV:} link polynomials and geometric invariants}, journal = {Annales de l'Institut Fourier}, pages = {1431--1475}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {70}, number = {4}, year = {2020}, doi = {10.5802/aif.3371}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3371/} }
TY - JOUR AU - Stoimenow, Alexander TI - Application of braiding sequences IV: link polynomials and geometric invariants JO - Annales de l'Institut Fourier PY - 2020 SP - 1431 EP - 1475 VL - 70 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3371/ DO - 10.5802/aif.3371 LA - en ID - AIF_2020__70_4_1431_0 ER -
%0 Journal Article %A Stoimenow, Alexander %T Application of braiding sequences IV: link polynomials and geometric invariants %J Annales de l'Institut Fourier %D 2020 %P 1431-1475 %V 70 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3371/ %R 10.5802/aif.3371 %G en %F AIF_2020__70_4_1431_0
Stoimenow, Alexander. Application of braiding sequences IV: link polynomials and geometric invariants. Annales de l'Institut Fourier, Volume 70 (2020) no. 4, pp. 1431-1475. doi : 10.5802/aif.3371. https://aif.centre-mersenne.org/articles/10.5802/aif.3371/
[1] Generic immersions of curves, knots, monodromy and Gordian number, Publ. Math., Inst. Hautes Étud. Sci., Volume 88 (1998), pp. 151-169 | DOI | MR | Zbl
[2] Topological invariants of knots and links, Trans. Am. Math. Soc., Volume 30 (1928) no. 2, pp. 275-306 | DOI | MR
[3] On the Vassiliev knot invariants, Topology, Volume 34 (1995) no. 2, pp. 423-472 | DOI | MR | Zbl
[4] The Fundamental Theorem of Vassiliev invariants, Geometry and Physics (Aarhus, 1995) (Lecture Notes in Pure and Applied Mathematics), Volume 184 (1997), pp. 101-134 | MR | Zbl
[5] Entrelacements et équations de Pfaff, Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982) (Astérisque), Volume 107-108 (1983), pp. 87-161 | Numdam | MR | Zbl
[6] Studying knots via braids VI: A non-finiteness theorem, Pac. J. Math., Volume 156 (1992) no. 2, pp. 265-285 | DOI | Zbl
[7] Knotted periodic orbits in dynamical systems - I, Lorenz’s equations, Topology, Volume 22 (1983) no. 1, pp. 47-82 | DOI | MR | Zbl
[8] Le problème de J. Milnor, sur le nombre gordien des nœuds algébriques, Enseign. Math., Volume 30 (1984) no. 3-4, pp. 173-222 | Zbl
[9] Positive links have positive Conway polynomials, Knot Theory and Manifolds (Lecture Notes in Mathematics), Volume 1144 (1983), pp. 146-159 | DOI | Zbl
[10] Regular Legendrian knots and the HOMFLY polynomial of immersed plane curves, Math. Ann., Volume 317 (2000) no. 3, pp. 389-413 | DOI | MR | Zbl
[11] An enumeration of knots and links, and some of their algebraic properties, Computational Problems in abstract algebra. Proc. Conf., (Oxford, 1967) (1970), pp. 329-358
[12] Homogeneous links, J. Lond. Math. Soc., Volume 39 (1989) no. 3, pp. 535-552 | DOI | MR
[13] Knots and Links, Cambridge University Press, 2004, xviii+328 pages | MR | Zbl
[14] Positivity of knot polynomials on positive links, J. Knot Theory Ramifications, Volume 1 (1992) no. 2, pp. 203-206 | DOI | MR | Zbl
[15] Gauss sum invariants for knots and links, Mathematics and its Applications, 532, Kluwer Academic Publishers, 2002, xvi+412 pages | Zbl
[16] New knot and link invariants, Knots in Hellas, 98. Proceedings of the International Conference on Knot Theory an its ramifications, European Cultural Centre of Delphi, Greece, August 7–15, 1998 (Series on Knots and Everything), Volume 24 (2000), pp. 59-79 | MR | Zbl
[17] Braids and the Jones-Conway polynomial, Trans. Am. Math. Soc., Volume 303 (1987), pp. 97-108 | DOI | Zbl
[18] A new polynomial invariant of knots and links, Bull. Am. Math. Soc., Volume 12 (1985), pp. 239-246 | DOI | MR
[19] Invariants of Legendrian and transverse knots in the standard contact space, Topology, Volume 36 (1997) no. 5, pp. 1025-1053 | DOI | MR | Zbl
[20] Notions of positivity and the Ozsváth-Szabó concordance invariant, J. Knot Theory Ramifications, Volume 19 (2010) no. 5, pp. 617-629 | DOI | MR | Zbl
[21] On –polynomials of links, Osaka J. Math., Volume 10 (1958), pp. 273-282 | MR | Zbl
[22] The first coefficient of the Conway polynomial, Proc. Am. Math. Soc., Volume 95 (1985) no. 2, pp. 299-302 | DOI | MR | Zbl
[23] A polynomial invariant of knots and links via von Neumann algebras, Bull. Am. Math. Soc., Volume 12 (1985), pp. 103-111 | DOI | Zbl
[24] On the Thurston–Bennequin invariant of Legendrian knots and nonexactness of Bennequin’s inequality, Invent. Math., Volume 133 (1998) no. 2, pp. 227-242 | DOI | MR | Zbl
[25] Kauffman polynomials and Vassiliev link invariants, Knots 96. Proceedings of the international conference and worshop on knot theory, Tokyo, Japan, July, 22–26, 1996 (1997), pp. 411-431 | Zbl
[26] Vassiliev knot invariants of order 6, J. Knot Theory Ramifications, Volume 10 (2001) no. 5, pp. 645-665 (Proceedings of the International Conference “Knots in Hellas 98”, Delphi, Vol. 3) | DOI | MR | Zbl
[27] An invariant of regular isotopy, Trans. Am. Math. Soc., Volume 318 (1990) no. 2, pp. 417-471 | DOI | MR | Zbl
[28] On the degree of the Brandt–Lickorish–Millett–Ho polynomial of a link, Proc. Am. Math. Soc., Volume 100 (1987), pp. 755-762 | DOI | MR | Zbl
[29] Knots and Feynman diagrams, Cambridge Lecture Notes in Physics, 13, Cambridge University Press, 2000 | MR | Zbl
[30] Polynomials for links, Bull. Lond. Math. Soc., Volume 20 (1988) no. 6, pp. 558-588 | DOI | MR
[31] A polynomial invariant for oriented links, Topology, Volume 26 (1987) no. 1, pp. 107-141 | DOI | MR | Zbl
[32] Positive knots and knots with braid index three have property P, J. Knot Theory Ramifications, Volume 12 (2003) no. 4, pp. 427-444 | DOI | Zbl
[33] Singular points of complex hypersurfaces, Annals of Mathematics Studies, 61, Princeton University Press; University of Tokyo Press, 1968 | MR | Zbl
[34] Seifert circles and knot polynomials, Math. Proc. Camb. Philos. Soc., Volume 99 (1986) no. 1, pp. 107-109 | DOI | MR | Zbl
[35] On the genus of the alternating knot. I, II, J. Math. Soc. Japan, Volume 10 (1958), p. 94-105, 235–248 | MR | Zbl
[36] On a certain numerical invariant of link types, Trans. Am. Math. Soc., Volume 117 (1965), pp. 387-422 | DOI | MR | Zbl
[37] The skein polynomial of a planar star product of two links, Math. Proc. Camb. Philos. Soc., Volume 106 (1989) no. 2, pp. 273-276 | DOI | MR | Zbl
[38] Positive alternating links are positively alternating, J. Knot Theory Ramifications, Volume 9 (2000) no. 1, pp. 107-112 | DOI | MR | Zbl
[39] On the minimal crossing number and the braid index of links, Can. J. Math., Volume 45 (1993) no. 1, pp. 117-131 | DOI | MR | Zbl
[40] Closed incompressible surfaces in complements of positive knots, Comment. Math. Helv., Volume 77 (2002) no. 2, pp. 235-243 | DOI | MR | Zbl
[41] Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Not., Volume 11 (1994), pp. 445-454 | DOI | Zbl
[42] On the Casson knot invariant, J. Knot Theory Ramifications, Volume 10 (2001) no. 5, pp. 711-738 | DOI | MR | Zbl
[43] On the topological invariance of Murasugi special components of an alternating link, Math. Proc. Camb. Philos. Soc., Volume 137 (2004) no. 1, pp. 95-108 | MR | Zbl
[44] Quasipositivity as an obstruction to sliceness, Bull. Am. Math. Soc., Volume 29 (1993) no. 1, pp. 51-59 | DOI | MR | Zbl
[45] Positive links are strongly quasipositive, Proceedings of the Kirbyfest (Berkeley, CA, 1998) (Geometry and Topology Monographs), Volume 2 (1999), pp. 555-562 | MR | Zbl
[46] Gauß sum invariants, Vassiliev invariants and braiding sequences, J. Knot Theory Ramifications, Volume 9 (2000) no. 2, pp. 221-269 | DOI | Zbl
[47] Knots of genus one or on the number of alternating knots of given genus, Proc. Am. Math. Soc., Volume 129 (2001) no. 7, pp. 2141-2156 | DOI | MR | Zbl
[48] Some minimal degree Vassiliev invariants not realizable by the Homfly and Kauffman polynomial, C. R. Acad. Bulg. Sci., Volume 54 (2001) no. 4, pp. 9-14
[49] On the crossing number of positive knots and braids and braid index criteria of Jones and Morton–Williams–Franks, Trans. Am. Math. Soc., Volume 354 (2002) no. 10, pp. 3927-3954 | DOI | MR | Zbl
[50] On the coefficients of the link polynomials, Manuscr. Math., Volume 110 (2003) no. 2, pp. 203-236 | DOI | MR
[51] Positive knots, closed braids, and the Jones polynomial, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 2 (2003) no. 2, pp. 237-285 | Numdam | MR | Zbl
[52] Gauß diagram sums on almost positive knots, Compos. Math., Volume 140 (2004) no. 1, pp. 228-254 | DOI | Zbl
[53] Newton-like polynomials of links, Enseign. Math., Volume 51 (2005) no. 3-4, pp. 211-230 | MR | Zbl
[54] On some restrictions to the values of the Jones polynomial, Indiana Univ. Math. J., Volume 54 (2005) no. 2, pp. 557-574 | DOI | MR | Zbl
[55] A property of the skein polynomial with an application to contact geometry, J. Differ. Geom., Volume 77 (2007) no. 3, pp. 555-566 | MR | Zbl
[56] Square numbers and polynomial invariants of achiral knots, Math. Z., Volume 255 (2007) no. 4, pp. 703-719 | DOI | Zbl
[57] Knots of (canonical) genus two, Fundam. Math., Volume 200 (2008) no. 1, pp. 1-67 | DOI | MR | Zbl
[58] Application of braiding sequences I: On the characterization of Vassiliev and polynomial link invariants, Commun. Contemp. Math., Volume 12 (2010) no. 5, pp. 681-726 | DOI | MR | Zbl
[59] On the definition of graph index, J. Aust. Math. Soc., Volume 94 (2013) no. 3, pp. 417-429 | DOI | MR | Zbl
[60] On the crossing number of semiadequate links, Forum Math., Volume 26 (2014) no. 4, pp. 1187-1246 | MR | Zbl
[61] Application of braiding sequences II: polynomial invariants of positive knots, Proc. Edinb. Math. Soc., II. Ser., Volume 59 (2016) no. 4, pp. 1037-1064 | DOI | MR | Zbl
[62] Diagram genus, generators and applications, Monographs and Research Notes in Mathematics, CRC Press, 2016, xviii+173 pages | Zbl
[63] Properties of Closed 3-Braids and Braid Representations of Links, SpringerBriefs in Mathematics, Springer, 2017, x+110 pages | Zbl
[64] The canonical genus of a classical and virtual knot, Geom. Dedicata, Volume 95 (2002) no. 1, pp. 215-225 | DOI | MR | Zbl
[65] Counting alternating knots by genus, Math. Ann., Volume 333 (2005) no. 1, pp. 1-27 | DOI | MR | Zbl
[66] Maximal Bennequin numbers and Kauffman polynomials of positive links, Proc. Am. Math. Soc., Volume 127 (1999) no. 11, pp. 3427-3432 | DOI | MR | Zbl
[67] On the index of graphs: Index versus cycle index, J. Knot Theory Ramifications, Volume 3 (1994) no. 4, pp. 531-536 | DOI | MR | Zbl
[68] Reducing braid index by Murasugi–Przytycki operation, J. Knot Theory Ramifications, Volume 20 (2011) no. 1, pp. 223-229 | DOI | MR | Zbl
[69] Twist sequences and Vassiliev invariants, J. Knot Theory Ramifications, Volume 3 (1994) no. 3, pp. 391-405 | DOI | MR | Zbl
[70] Cohomology of knot spaces, Theory of Singularities and its Applications (Advances in Soviet Mathematics), Volume 1 (1990), pp. 23-69 | DOI | MR
[71] Algebraic structures on modules of diagrams, J. Pure Appl. Algebra, Volume 215 (2011) no. 6, pp. 1292-1339 (Université Paris VII preprint, June 1996) | DOI | MR | Zbl
[72] Polynomial invariants of positive links, Topology, Volume 31 (1992) no. 4, pp. 805-811 | DOI | MR | Zbl
[73] The rank of the trip matrix of a positive knot diagram, J. Knot Theory Ramifications, Volume 6 (1997) no. 2, pp. 299-301 | DOI | MR | Zbl
Cited by Sources: