No Access

SEIFERT SURFACES, COMMUTATORS AND VASSILIEV INVARIANTS

E. KALFAGIANNI

Department of Mathematics, Michigan State University, E. Lansing, MI, 48823, USA

Search for more papers by this author

and

XIAO-SONG LIN

Department of Mathematics, University of California at Riverside, Riverside, CA 92521-0135, USA

X.-S. Lin passed away on January 14, 2007.

Search for more papers by this author

https://doi.org/10.1142/S0218216507005816Cited by:1 (Source: Crossref)

Abstract

We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group. We also conjecture a characterization of knots whose invariants of all orders vanish in terms of their Seifert surfaces.

To L. Kauffman on the occasion of his 60th birthday

Keywords:

AMSC: 57M25, 57N10

References

E. Kalfagianni and N. Askitas, Topol. Appl. 126(1–2), 63 (2002). Web of Science, Google Scholar
J. S. Birman, Bull. Amer. Math. Soc. 28, 253 (1993), DOI: 10.1090/S0273-0979-1993-00389-6. Crossref, Web of Science, Google Scholar
D. Bar-Natan, Topology 3, 423 (1995). Web of Science, Google Scholar
J. Conant and P. Teichner, Topology 43, 119 (2004), DOI: 10.1016/S0040-9383(03)00031-4. Crossref, Web of Science, Google Scholar
D. Bar-Natan, J. Knot. Theory Ramifications 4, 13 (1995), DOI: 10.1142/S021821659500003X. Link, Web of Science, Google Scholar
M. N. Gusarov, Topology of Manifolds and Varieties, Advances in Soviet Mathematics 18 (American Mathematical Society, 1994) pp. 173–192. Crossref, Google Scholar
K. Habiro, Geom. Topol. 4, 1 (2000), DOI: 10.2140/gt.2000.4.1. Crossref, Web of Science, Google Scholar
J. Hempel , 3-Manifolds , Annals of Mathematics Studies 86 ( Princeton University Press , 1976 ) . Google Scholar
E. Kalfagianni and X.-S. Lin, Regular Seifert surfaces and Vassiliev knot invariants , arXiv:math.GT/9804032 . Google Scholar
A. Karras , W. Magnus and D. Solitar , Combinatorial Group Theory , Pure and Applied Mathematics XIII ( Interscience , NY , 1966 ) . Google Scholar
K.-Y. Ng and T. Stanford, Math. Proc. Cambridge Phil. Soc. 126(1), 63 (1999). Crossref, Web of Science, Google Scholar
L. Plachta, Methods Funct. Anal. Topology 10(2), 43 (2004). Google Scholar
L. Plachta, Mat. Stud. 18(2), 213 (2002). Google Scholar
T. Stanford, Pacific J. Math. 174, (1996). Google Scholar
T. Stanford, Vassiliev invariants and knots modulo pure braid subgroups , arXiv:math.GT/9805092 . Google Scholar
D. Rolfsen , Knots and Links ( Publish or Perish , 1976 ) . Google Scholar
L. Rudolph, Math. Proc. Cambridge Phil. Soc. 107, 319 (1990). Crossref, Web of Science, Google Scholar