Abstract
This paper is devoted to the solution of the integration problem for higher derivatives of finite-type invariants (so-called weight systems) up to real invariants defined on the space of knots.
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REFERENCES
J. W. Alexander, “Topological invariants of knots and links,” Trans. Amer. Math. Soc. 20, 275–306 (1923).
V. I. Arnold, “Plane curves, their invariants, perestroikas, and classification,” in: Singularities and Bifurcations. tAdv. Sov. Math., 21, Amer. Math. Soc., Providence, Rhode Island (1994), pp. 33–91.
V. I. Arnold, “Vassiliev's theory of discriminants and knots,” in: Proc. 1 Eur. Math. Congr. (Paris, July 1992), 1, Birkhauser, Basel-Boston-Berlin (1992), pp. 3–29.
D. Bar-Natan, “On the Vassiliev knot invariants,” Topology 34, 423–472 (1995).
D. Bar-Natan, S. Garoufalidis, L. Rozansky, and D. Thurston, “Wheels, wheeling, and the Kontsevich integral of the unknot,” Isr. J. Math., 119, 217–237 (2000).
J. Birman, Braids, links, and mapping class groups,” Ann. Math. Stud., 82, Princeton Univ. Press, Princeton, New Jersey (1974).
J. Birman, “New points of view in knot theory,” Bull. Amer. Math. Soc., 28, 253–287 (1993).
J. Birman and X. S. Lin, “Knot polynomials and Vassiliev's invariants,” Invent. Math., 111, 225–270 (1993).
J. Birman and M. D. Hirsch, “A new algorithm for recognizing the unknot,” Geom. Topol., 2, 175–220 (1998).
P. Cartier, “Construction combinatoire des invariants de Vassiliev-Kontsevich des noeuds,” C. R. Acad. Sci. Paris, 316, 1205–1210 (1993).
S. V. Chmutov, S. V. Duzhin, and S. K. Lando, “Vassiliev knot invariants,” Adv. Sov. Math., 21, 117–147 (1994).
S. V. Duzhin and S. V. Chmutov, “Knots and their invariants,” Mat. Prosv., Ser. 3, 3, 59–93 (1999).
S. V. Chmutov and A. N. Varchenko, “Remarks on the Vassiliev knot invariants coming from sl 2,” Topology, 36, 153–178 (1997).
O. T. Dasbach and S. Hougardy, “Does the Jones polynomial detect unknottedness?” J. Experimental Math., 6, No.1, 51–56 (1997).
M. Domergue and P. Donato, “Integrating a weight system of order n to an invariant of (n − 1)-singular knots,” J. Knot Theory Ramifications, 5, No.1, 23–35 (1996).
V. G. Drinfeld, “Quasi-Hopf algebras,” Algebra Analiz, 1, No.6, 114–148 (1989).
V. G. Drinfeld, “On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(Q/Q),” Algebra Analiz, 2, No.4, 149–181 (1990).
Encyclopedia of Mathematics [in Russian], Moscow (1985).
S. Gordan and J. Luecke, “Knots are determined by their complements,” J. Amer. Math. Soc., 2, 371–415 (1989).
M. N. Gusarov, “A new form of the Conway-Jones polynomial of oriented links,” in: Topology of Manifolds and Varieties (O. Viro, ed.), Amer. Math. Soc., Providence, Rhode Island (1994), pp. 167–172.
J. Hoste, M. Thistlethwaite, and J. Weeks, “The first 1,701,936 knots,” Math. Intelligencer, 20, 33–48 (1998).
J. Hoste, M. Thistlethwaite, and J. Weeks, KnotScape, http://www.math.utk.edu/∼morwen/
V. F. R. Jones, “Polynomial invariants of knots via von Neumann algebras,” Bull. Amer. Math. Soc., 12, 103–111 (1985).
V. F. R. Jones, “Hecke algebra representations of braid groups and link polynomials,” Ann. Math., 126, 335–388 (1987).
C. Kassel, Quantum Groups, Grad. Texts Math., 155, Springer-Verlag, New York-Heidelberg-Berlin (1995).
L. Kauffman, “State models and the Jones polynomial,” Topology, 26, 395–407 (1987).
L. Kauffman, Knots and Physics, World Scientific, Singapore (1993).
M. Kontsevich, “Vassiliev's knot invariants,” in: Adv. Sov. Math., Amer. Math. Soc., 16, No. 2, Providence, Rhode Island (1993), pp. 137–150.
C. Kosniowski, A First Course in Algebraic Topology, Cambridge Univ. Press, Cambridge (1980).
R. H. Crowell and R. H. Fox, Introduction to Knot Theory, Grad. Texts Math., 57. Springer-Verlag, New York-Heidelberg-Berlin (1977).
J. Lannes, “Sur les invariants de Vassiliev de degree inferieur ou egal a 3,” L'Enseignement Math., 39, 295–316 (1993).
T. Q. T. Le and J. Murakami, “Representations of the category of tangles by Kontsevich's iterated integral,” Commun. Math. Phys., 168, 535–562 (1995).
T. Q. T. Le and J. Murakami, “The universal Vassiliev-Kontsevich invariant for framed oriented links,” Comp. Math., 102, 41–64 (1996).
T. Q. T. Le and J. Murakami, “Kontsevich's integral for the Kauffman polynomial,” Nagoya Math. J., 142, 39–65 (1996).
S. V. Matveev, “Classification of ‘sufficiently large’ three-dimensional manifolds,” Usp. Mat. Nauk, 52, No.5, 147–174 ().
S. V. Matveev and A. T. Fomenko, Algorithmic and Computer Methods in Three-Dimensional Topology [in Russian], Nauka, Moscow (1998).
W. Menasco and M. Thistlethwaite, “A classification of alternating links,” Ann. Math., 138, 113–171 (1993).
K. Murasugi, “Jones polynomial and classical conjectures in knot theory,” Topology, 26, 184–194 (1987).
O.-P. Ostlund, “Invariants of knot diagrams and relations among Reidemeister moves,” J. Knot Theory Ramifications, 10, No.8, 1215–1227 (2001).
M. Polyak, O. Viro, “Gauss diagram formulas for Vassiliev invariants,” Int. Math. Res. Not., 11, 445–453 (1994).
M. Polyak, “On Milnor's triple linking number,” C. R. Acad. Sci. Paris, Ser. I, 325, 77–82 (1997).
V. V. Prasolov and A. B. Sossinsky, Knots, Links, Braids, and Three-Dimensional Manifolds [in Russian], Moscow (1997).
S. Piunikhin, “Combinatorial expression for universal Vassiliev link invariants,” Commun. Math. Phys., 168, 1–22 (1995).
S. Piunikhin, “Weights of Feynman diagrams, links polynomials, and Vassiliev knot invariants,” J. Knot Theory Ramifications, 4, No.1, 163–188 (1995).
V. V. Prasolov, “Seifert surfaces,” Mat. Prosveshchenie, Ser. 3, 3, 116–126 (1999).
M. Thistlethwaite, “A spanning tree expansion for the Jones polynomial,” Topology, 26, 297–309 (1987).
H. F. Trotter, “Non-invertible knots exist,” Topology, 2, 275–280 (1964).
S. D. Tyurina, “On formulas of the type of Lannes and Viro-Polyak for Vassiliev's invariants,” in: Proc. Int. Conf. Dedic. to the 70th Anniv. of L. S. Pontryagin [in Russian], Moscow State Univ., Moscow (1998), pp. 329–330.
S. D. Tyurina, “Explicit formulas for finite-degree invariants,” in: Proc. Int. Conf. “Monodromie et equations differentielles en theorie des singularites et des representations de groupes,” Luminy, France, CNRS-SMF (1999).
S. D. Tyurina, “On formulas of Lannes and Viro-Polyak type for knot invariants of finite order,” Mat. Zametki, 66, No.4 (1999).
S. D. Tyurina, “Diagrammatic formulae of Viro-Polyak type for knot invariants of finite order,” Russ. Math. Surv., 54, No.3, 658–659 (1999).
V. Vassiliev, “Topology of complements to discriminants and loop spaces,” Theory of Singularities and Applications, Adv. Sov. Math., 1, 9–21 (1990).
V. Vassiliev, “Cohomology of knot spaces,” in; Theory of Singularities and Applications, Adv. Soviet Math., 1, 23–69 (1990).
V. Vassiliev, Topology of Complements to Discriminants [in Russian], Moscow (1997).
O. Viro, First-degree invariants of generic curves on surfaces, Preprint, Uppsala University (1994).
W. Whitten, “Knot complements and groups,” Topology, 26, 41–44 (1987).
S. Willerton, “Vassiliev invariants and Hopf algebra of chord diagrams,” Math. Proc. Camb. Phil. Soc., 119, 55–65 (1996).
S. Willerton, “A combinatorial half-integration from weight systems to Vassiliev knot invariants,” J. Knot Theory Ramifications, 7, No.4, 519–526 (1998).
S. Willerton, Vassiliev Invariants as Polynomials, Banach Center Publ., 42, Warszawa (1998).
E. Witten, “Quantum field theory and the Jones polynomial,” Commun. Math. Phys., 121, 351–399 (1989).
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 19, Topology and Noncommutative Geometry, 2004.
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Tyurina, S.D. Diagram Invariants of Knots and the Kontsevich Integral. J Math Sci 134, 2017–2071 (2006). https://doi.org/10.1007/s10958-006-0095-9
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DOI: https://doi.org/10.1007/s10958-006-0095-9