Geometric aspects of invariants of finite 
type of knots and links in $S^3$

L. P. Plachta

In this paper we briefly review some well-known results on geometric properties of finite type invariants of knots and links in $S^3$ and announce several new results. We also formulate several open problems on the topics discussed.

1. D.Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423-472.

2. D.Bar-Natan, Vassiliev Homotopy String Link Invariants, J. Knot Theory Ramif. 4 (1995), 13-32.

3. D.Bar-Natan, S.Garoufalidis, L.Rozansky and D.Thurston, The Aarhus Integral of rational Homology 3-Spheres II: Invariance and Universality, preprint 1998.

4. J.S.Birman and X.-S.Lin, Knot polynomials and Vassiliev's knot invariants, Inventiones mathematicae 111 (1993), 225-270.

5. T.D.Cochran, Geometric invariants of link cobordism, Comm. Math. Helv. 60 (1985), 291-311.

6. J.Conant and P.Teichner, Groupe cobordism of classical knots, Preprint 2001, math.GT/0101047.

7. M.N.Gusarov, On $n$-equivalence of knots and invariants of finite degree, in: Topology of manifolds and varieties (ed. O.Viro), Advances in Soviet Mathematics 18, 1974, pp. 173-192.

8. K.Habiro, Claspers and finite type invariants of links, Geom. and Top. 4 (2000), 1-83.

9. E.Kalfagianni and X.-S.Lin, Regular Seifert surfaces and Vassiliev knot invariants, Preprint 1998, math. GT/9804032S.

10. P.Kirk and C.Livingston, Vasiliev invariants of two component links and the Casson-Walker invariant, Topology, 36 (1997), 1333-1353.

11. T.T.Q. Le and J.Murakami, The universal Vassiliev-Kontsevich invariant for framed oriented links, Compositio Math., 102 (1996) 41-64.

12. X.-S.Lin, Null $k$-cobordant links in $S^3$, Comm. Math. Helv. 66 (1991), 333-339.

13. X.-S.Lin, Link homotopy invariants of finite type, Preprint 2000, math.GT/0012096.

14. B.Mellor, Finite Type Link Homotopy Invariants II : Milnor's ${\bar\mu}$-invariants, J. Knot Theory Ramif. V.9, № 6 735--758 (2000).

15. B.Mellor and D.Thurston, On the existence of finite type link homotopy invariants, Preprint 2000, math.GT/0010206.

16. S.A.Melikhov and D.Repovs, A geometric filtration of links modulo knots: I. Question of nilpotence, Preprint 2001, math.GT/0103113.

17. S.A.Melikhov and D.Repovs, A geometric filtration of links modulo knots: II. Comparison, Preprint 2001, math.GT/0103113.

18. H.Murakami and T.Ohtsuki, Finite type invariants of knots via their Seifert matrices, Asian J.Math. 5, (2001) 379-386.

19. K.Y.Ng, Groups of ribbon knots, Topology 37 (1998), 441-458.

20. K.Y. Ng and T. Stanford, On Gusarov's groups of knots, Math. Proc. Camb. Phil. Soc. 126 (1998), 63-76.

21. L.Plachta, $n$-trivial knots and the Alexander polynomial (to appear in Visnyk of the Lviv University).

22. L.Plachta, $C_n$-moves, braid commutators and Vassiliev knot invariants (submit. to J. Knot Theory Ramif.)

23. L.Plachta, Double trivalent diagrams and $n$-hyperbolic knots (submit. to Methods of Func. Analysis and Topology).

24. T.Stanford, Vassiliev invariants and knots modulo pure braid subgroups, Preprint 1998, math.GT/9805092.

	Geometric aspects of invariants of finite type of knots and links in $S^3$
Author	L. P. Plachta Institute of Applied Problems of Mechanics and,Mathematics ,Naukova 3b, 79000, Lviv, Ukraine
Abstract	In this paper we briefly review some well-known results on geometric properties of finite type invariants of knots and links in $S^3$ and announce several new results. We also formulate several open problems on the topics discussed.
Keywords	geometric properties, finite type invariants, open problems
DOI	doi:10.30970/ms.18.2.213-222
Reference	1. D.Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423-472. 2. D.Bar-Natan, Vassiliev Homotopy String Link Invariants, J. Knot Theory Ramif. 4 (1995), 13-32. 3. D.Bar-Natan, S.Garoufalidis, L.Rozansky and D.Thurston, The Aarhus Integral of rational Homology 3-Spheres II: Invariance and Universality, preprint 1998. 4. J.S.Birman and X.-S.Lin, Knot polynomials and Vassiliev's knot invariants, Inventiones mathematicae 111 (1993), 225-270. 5. T.D.Cochran, Geometric invariants of link cobordism, Comm. Math. Helv. 60 (1985), 291-311. 6. J.Conant and P.Teichner, Groupe cobordism of classical knots, Preprint 2001, math.GT/0101047. 7. M.N.Gusarov, On $n$-equivalence of knots and invariants of finite degree, in: Topology of manifolds and varieties (ed. O.Viro), Advances in Soviet Mathematics 18, 1974, pp. 173-192. 8. K.Habiro, Claspers and finite type invariants of links, Geom. and Top. 4 (2000), 1-83. 9. E.Kalfagianni and X.-S.Lin, Regular Seifert surfaces and Vassiliev knot invariants, Preprint 1998, math. GT/9804032S. 10. P.Kirk and C.Livingston, Vasiliev invariants of two component links and the Casson-Walker invariant, Topology, 36 (1997), 1333-1353. 11. T.T.Q. Le and J.Murakami, The universal Vassiliev-Kontsevich invariant for framed oriented links, Compositio Math., 102 (1996) 41-64. 12. X.-S.Lin, Null $k$-cobordant links in $S^3$, Comm. Math. Helv. 66 (1991), 333-339. 13. X.-S.Lin, Link homotopy invariants of finite type, Preprint 2000, math.GT/0012096. 14. B.Mellor, Finite Type Link Homotopy Invariants II : Milnor's ${\bar\mu}$-invariants, J. Knot Theory Ramif. V.9, № 6 735--758 (2000). 15. B.Mellor and D.Thurston, On the existence of finite type link homotopy invariants, Preprint 2000, math.GT/0010206. 16. S.A.Melikhov and D.Repovs, A geometric filtration of links modulo knots: I. Question of nilpotence, Preprint 2001, math.GT/0103113. 17. S.A.Melikhov and D.Repovs, A geometric filtration of links modulo knots: II. Comparison, Preprint 2001, math.GT/0103113. 18. H.Murakami and T.Ohtsuki, Finite type invariants of knots via their Seifert matrices, Asian J.Math. 5, (2001) 379-386. 19. K.Y.Ng, Groups of ribbon knots, Topology 37 (1998), 441-458. 20. K.Y. Ng and T. Stanford, On Gusarov's groups of knots, Math. Proc. Camb. Phil. Soc. 126 (1998), 63-76. 21. L.Plachta, $n$-trivial knots and the Alexander polynomial (to appear in Visnyk of the Lviv University). 22. L.Plachta, $C_n$-moves, braid commutators and Vassiliev knot invariants (submit. to J. Knot Theory Ramif.) 23. L.Plachta, Double trivalent diagrams and $n$-hyperbolic knots (submit. to Methods of Func. Analysis and Topology). 24. T.Stanford, Vassiliev invariants and knots modulo pure braid subgroups, Preprint 1998, math.GT/9805092.
Pages	213-222
Volume	18
Issue	2
Year	2002
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Geometric aspects of invariants of finite type of knots and links in $S^3$