Hostname: page-component-76c49bb84f-vcmr7 Total loading time: 0 Render date: 2025-07-12T06:55:38.680Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterization of finite type string link invariants of degree <5

Published online by Cambridge University Press:  16 March 2010

JEAN-BAPTISTE MEILHAN  and
AKIRA YASUHARA
JEAN-BAPTISTE MEILHAN
Affiliation:
Institut Fourier, Université Grenoble 1, 100 rue des Maths - BP 74, 38402 St Martin d'Hères, France. e-mail: jean-baptiste.meilhan@ujf-grenoble.fr
AKIRA YASUHARA
Affiliation:
Tokyo Gakugei University, Department of Mathematics, Koganeishi, Tokyo 184-8501, Japan. e-mail: yasuhara@u-gakugei.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a complete set of finite type string link invariants of degree <5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closures of (cabled) string links. We show that finite type invariants classify string links up to Ck-moves for k ≤ 5, which proves, at low degree, a conjecture due to Goussarov and Habiro. We also give a similar classification of string links up to Ck-moves and concordance for k ≤ 6.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 3 , May 2010 , pp. 439 - 472
Copyright
Copyright © Cambridge Philosophical Society 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Bar-Natan, D.On the Vassiliev knot invariants. Topology 34 (1995), 423–472.CrossRefGoogle Scholar
[2]Bar-Natan, D.Vassiliev homotopy string link invariants. J. Knot Theory Ram. 4, no. 1 (1995), 13–32.CrossRefGoogle Scholar
[3]Bar-Natan, D. Some computations related to Vassiliev invariants. (1996), available at http://www.math.toronto.edu/~drorbn/.Google Scholar
[4]Casson, A. J.Link cobordism and Milnor's invariant. Bull. London Math. Soc. 7 (1975), 39–40.CrossRefGoogle Scholar
[5]Cochran, T. D.Derivatives of link: Milnor's concordance invariants and Massey's products. Mem. Amer. Math. Soc. 84 (1990), No. 427.Google Scholar
[6]Conant, J. and Teichner, P.Grope cobordism of classical knots. Topology 43 (2004), 119–156.CrossRefGoogle Scholar
[7]Fleming, T. and Yasuhara, A.Milnor's invariants and self C k-equivalence. Proc. Amer. Math. Soc. 137 (2009) 761–770.CrossRefGoogle Scholar
[8]Garoufalidis, S. and Levine, J.Concordance and 1-loop clovers. Alg. Geom. Topol. 1 (2001), 687–697.CrossRefGoogle Scholar
[9]Gusarov, M. N.A new form of the Conway-Jones polynomial of oriented links (Russian). Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 193 (1991), Geom. i Topol. 1, 4–9, 161; translation in “Topology of manifolds and varieties”, 167–172 Adv. Soviet Math. 18 (Amer. Math. Soc., 1994).Google Scholar
[10]Gusarov, M.On n-equivalence of knots and invariants of finite degree. Topology of manifolds and varieties, 173–192, Adv. Soviet Math. 18 (Amer. Math. Soc., 1994).Google Scholar
[11]Gusarov, M. N.Variations of knotted graphs. The geometric technique of n-equivalence (Russian). Algebra i Analiz 12 (2000), no. 4, 79–125; translation in St. Petersburg Math. J. 12 (2001), no. 4, 569–604.Google Scholar
[12]Habegger, N. and Lin, X. S.The classification of links up to link-homotopy. J. Amer. Math. Soc. 3 (1990), 389–419.CrossRefGoogle Scholar
[13]Habegger, N. and Masbaum, G.The Kontsevich integral and Milnor's invariants. Topology 39 (2000), no. 6, 1253–1289.CrossRefGoogle Scholar
[14]Habiro, K.Claspers and finite type invariants of links. Geom. Topol. 4 (2000), 1–83.CrossRefGoogle Scholar
[15]Habiro, K. Replacing a graph clasper by tree claspers, preprint math.GT/0510459.Google Scholar
[16]Habiro, K. and Meilhan, J. B.Finite type invariants and Milnor invariants for Brunnian links. Int. J. Math. 19, no. 6 (2008), 747–766.CrossRefGoogle Scholar
[17]Horiuchi, S.The Jacobi diagram for a C n-move and the HOMFLY polynomial. J. Knot Theory Ram. 16, no. 2 (2007), 227–242.CrossRefGoogle Scholar
[18]Kanenobu, T. Finite type invariants of order 4 for 2-component links. from: Intelligence of Low Dimensional Topology 2006, (Carter, J. S. et al. editors), Ser. Knots Everything 40 (World Sci. Publ.; 2007), 109–116.CrossRefGoogle Scholar
[19]Kanenobu, T. and Miyazawa, Y.HOMFLY polynomials as Vassiliev link invariants. In Knot Theory 42 (1998), 165–185.Google Scholar
[20]Kanenobu, T., Miyazawa, Y. and Tani, A.Vassiliev link invariants of order three. J. Knot Theory Ram. 7 (1998), 433–462.CrossRefGoogle Scholar
[21]Kontsevich, M.Vassiliev's knot invariants. “I. M. Gel'fand Seminar”, 137–150, Adv. Soviet Math. 16, Part 2 (Amer. Math. Soc., 1993).Google Scholar
[22]Lickorish, W. B. R.An Introduction to Knot Theory, GTM 175 (Springer-Verlag, 1997).Google Scholar
[23]Lin, X. S. Power series expansions and invariants of links. In Geometric topology, AMS/IP Stud. Adv. Math. 2.1. (Amer. Math. Soc. 1997), 184–202.Google Scholar
[24]Massuyeau, G.Finite-type invariants of three-manifolds and the dimension subgroup problem. J. London Math. Soc. 75:3 (2007), 791–811.CrossRefGoogle Scholar
[25]Meilhan, J. B.On Vassiliev invariants of order two for string links. J. Knot Theory Ram. 14 (2005), No. 5, 665–687.CrossRefGoogle Scholar
[26]Meilhan, J. B. and Yasuhara, A.On Cn-moves for links. Pacific J. Math. 238 (2008), 119–143.CrossRefGoogle Scholar
[27]Milnor, J.Link groups. Ann. of Math. (2) 59 (1954), 177–195.CrossRefGoogle Scholar
[28]Milnor, J. Isotopy of links. Algebraic geometry and topology. A symposium in honor of S. Lefschetz. (Princeton University Press, 1957). pp. 280–306.CrossRefGoogle Scholar
[29]Murakami, H. and Nakanishi, Y.On a certain move generating link-homology. Math. Ann. 283 (1989), 75–89.CrossRefGoogle Scholar
[30]Ng, K. Y.Groups of ribbon knots. Topology 37 (1998), 441–458.CrossRefGoogle Scholar
[31]Ohyama, Y. and Yamada, H.A C n-move for a knot and the coefficients of the Conway polynomial. J. Knot Theory Ram. 17 (2008), no. 7, 771–785.CrossRefGoogle Scholar
[32]Robertello, R. A.An invariant of knot cobordism. Comm. Pure Appl. Math. 18 (1965), 543–555.CrossRefGoogle Scholar
[33]Sase, T.C k-concordant ni yoru string link no bunrui. Master's thesis, Tokyo Gakugei University. (2009).Google Scholar
[34]Stanford, T.Braid commutators and Vassiliev invariants. Pacific J. Math. 174 (1996), no. 1, 269–276.CrossRefGoogle Scholar
[35]Vassiliev, V. A.Cohomology of knot spaces. Theory of singularities and its applications. 23–69Adv. Soviet Math. 1 (Amer. Math. Soc., 1990).Google Scholar
[36]Yasuhara, A.Classification of string links up to self delta-moves and concordance. Alg. Geom. Topol. 9 (2009), 265–275.CrossRefGoogle Scholar
[37]Yasuhara, A.Self Delta-equivalence for links whose Milnor's isotopy invariants vanish. Trans. Amer. Math. Soc. 361 (2009), 4721–4749.CrossRefGoogle Scholar