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Combinatorial computation of combinatorial formulas for knot invariants


Author: V. A. Vassiliev
Translated by: the author
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 66 (2005).
Journal: Trans. Moscow Math. Soc. 2005, 1-83
MSC (2000): Primary 57M27, 57M25
DOI: https://doi.org/10.1090/S0077-1554-05-00148-2
Published electronically: October 28, 2005
MathSciNet review: 2193429
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Abstract | References | Similar Articles | Additional Information

Abstract: We construct a homology algebraic algorithm for computing combinatorial formulas of all finite degree knot invariants. Its input is an arbitrary weight system, i.e., a virtual principal part of a finite degree invariant, and the output is either a proof of the fact that this weight system actually does not correspond to any knot invariant or an effective description of some invariant with this principal part, i.e., a finite collection of easily described singular chains of full dimension in the space of spatial curves such that the value of this invariant on any generic knot is equal to the sum of multiplicities of these chains in a neighborhood of the knot. (In examples calculated by now, the former possibility never occurred.) This algorithm is formally realized over $ \mathbb{Z}_2$, but its generalization to the case of arbitrary coefficients is just a technical task. The algorithm is based on the study of a complex of chains in the space of smooth curves in the three-dimensional space with a fixed flag of directions, and also in the discriminant variety of this space of curves.


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  • 1. Bar-Natan, D. (1994- ) Bibliography of Vassiliev Invariants. Web publication http://www.math.toronto.edu/~drorbn/VasBib/index.html
  • 2. Bar-Natan, D. (1995) On the Vassiliev knot invariants, Topology, Vol. 34, 423-472. MR 1318886 (97d:57004)
  • 3. Birman, J. (1993) New points of view in knot theory. Bull. AMS (N.S.), 28:3, 253-287. MR 1191478 (94b:57007)
  • 4. Budney, R., Conant, J., Scannell, K., and Sinha, D. (2003) New perspectives of self-linking, Web publication math.GT/0303034.
  • 5. Fiedler, T. (2001) Gauss diagram invariants for knots and links, Mathematics and its Applications, Vol. 532, Kluwer Academic Publishers, Dordrecht. MR 1948012 (2003m:57031)
  • 6. Golubitsky, M., Guillemin, V. (1973) Stable Mappings and Their Singularities. Grad. Texts Math., vol. 14, Springer-Verlag, New York-Heidelberg-Berlin. MR 0341518 (49:6269)
  • 7. Goresky, M. and MacPherson, R., Stratified Morse Theory, Springer, Berlin, 1988 MR 0932724 (90d:57039)
  • 8. Goussarov, M., Polyak, M., and Viro, O. (2000) Finite-type invariants of classical and virtual knots, Topology 39:5, 1045-1068. MR 1763963 (2001i:57017)
  • 9. Hatcher, A., Spaces of knots, http://math.cornell.edu/~hatcher
  • 10. Kauffman, L.H. (1999) Virtual Knot Theory, European Journal of Combinatorics, 20:7, 663-690. MR 1721925 (2000i:57011)
  • 11. Kontsevich, M., Vassiliev's knot invariants, in Adv. in Sov. Math., 16:2, (1993) AMS, Providence RI, 137-150. MR 1237836 (94k:57014)
  • 12. Lannes, J. (1993) Sur les invariants de Vassiliev de degré inférieur ou égal à 3. L'Enseignement Mathématique 39 (3-4), 295-316. MR 1252070 (94i:57015)
  • 13. Merkov, A.B. (2003) Vassiliev invariants classify plane curves and doodles, Math. Sbornik, 194:9-10, 1301-1330. MR 2037502 (2005c:57013)
  • 14. Merkov, A.B. (2000) Segment-arrow diagrams and invariants of ornaments, Mat. Sbornik, 191:11, 47-78. MR 1827512 (2001m:57013)
  • 15. Polyak, M. and Viro, O. (1994) Gauss diagram formulas for Vassiliev invariants, Internat. Math. Res. Notes 11, 445-453. MR 1316972 (95k:57012)
  • 16. Tyurina, S.D. (1999) On the Lannes and Viro-Polyak type formulas for finite type invariants, Matem. Zametki 66:4, 635-640; Engl. transl. in Math. Notes, 66, No.3-4, 525-530. MR 1747094 (2001b:57031)
  • 17. Vassiliev, V.A. (1990) Cohomology of knot spaces, in: Theory of Singularities and its Applications (V. I. Arnold, ed.), Advances in Soviet Math. Vol. 1, 23-69 (AMS, Providence, RI). MR 1089670 (92a:57016)
  • 18. Vassiliev, V.A. (1999) Topological order complexes and resolutions of discriminant sets, Publications de l'Institut Mathématique Belgrade, Nouvelle série 66(80), 165-185. MR 1765045 (2002f:55040)
  • 19. Vassiliev, V.A., Complexes of connected graphs, in: L. Corvin, I. Gel$ '$fand, J. Lepovsky (eds.), The I. M. Gel$ '$fands mathematical seminars 1990-1992, 1993, Birkhäuser, Basel, 223-235. MR 1247293 (94h:55032)
  • 20. Vassiliev, V.A. (1994) Complements of discriminants of smooth maps: topology and applications, Revised ed., Translations of Math. Monographs 98, AMS, Providence, RI. MR 1168473 (94i:57020)
  • 21. Vassiliev, V.A. (1997) Topology of complements of discriminants, FAZIS, Moscow. (Russian) MR 1642095 (99m:57011)
  • 22. Vassiliev, V.A. (1999) Homology of $ i$-connected graphs and invariants of knots, plane arrangements, etc. Proc. of the Arnoldfest Conference, Fields Inst. Communications, Vol. 24, AMS, Providence, RI, pp. 451-469. MR 1733588 (2000j:57005)
  • 23. Vassiliev, V.A. (1993) Lagrange and Legendre Characteristic classes, 2nd edition. Gordon and Breach Publ., New York, 265 pp. MR 1065996 (91k:57034)
  • 24. Vassiliev, V.A. (2001) On combinatorial formulas for cohomology of spaces of knots, Moscow Math. J. 1:1, 91-123. MR 1852936 (2002g:55028)
  • 25. Vassiliev V.A. (2001) Homology of spaces of knots in any dimensions. Philos. Transact. of the London Royal Society, 359:1784, 1343-1364 (Proceedings of the Discussion Meeting of the London Royal Society ``Topological Methods in the Physical Sciences"). MR 1853624 (2002g:55029)
  • 26. Vassiliev V.A. (2001) Topology of plane arrangements and their complements, Russian Math. Surveys, 56:2, 167-203. MR 1859709 (2002g:55030)
  • 27. Vassiliev, V.A., Combinatorial formulas for cohomology of spaces of knots, in: Advances in Topological Quantum Field Theory (J. Bryden, ed.), Kluwer, Dordrecht, 2004, MR 2147414
  • 28. Ziegler, G.M., Zivaljevic, R.T. (1993) Homotopy types of subspace arrangements via diagrams of spaces. Math. Ann., 295, 527-548. MR 1204836 (94c:55018)

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Additional Information

V. A. Vassiliev
Affiliation: Steklov Mathematical Institute and Poncelet Laboratory (UMI 2615 of CNRS and Independent University of Moscow)

DOI: https://doi.org/10.1090/S0077-1554-05-00148-2
Keywords: Knot, invariant, combinatorial formula, discriminant, combinatorial algorithm, spectral sequence, subalgebraic chain, simplicial resolution.
Published electronically: October 28, 2005
Additional Notes: Supported in part by grants RFBR-01-01-00660, INTAS–00-0259, grant 1972.2003.01 of President of Russia, and Program “Contemporary Mathematics" of the Mathematical division of Russian Ac. Sci.
Article copyright: © Copyright 2005 American Mathematical Society

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