Available in electronic format
Available in print format
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
     

Quantum deformations of fundamental groups of oriented $3$-manifolds

Author(s): Uwe Kaiser
Journal: Trans. Amer. Math. Soc. 356 (2004), 3869-3880.
MSC (2000): Primary 57M25, 57M35, 57R42
Posted: November 25, 2003
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We compute two-term skein modules of framed oriented links in oriented $3$-manifolds. They contain the self-writhe and total linking number invariants of framed oriented links in a universal way. The relations in a natural presentation of the skein module are interpreted as monodromies in the space of immersions of circles into the $3$-manifold.


References:

1.
V. Chernov, Framed knots in $3$-manifolds preprint math.GT/0105139, http://xxx.lanl.gov, 2001.

2.
D. Gabai, Foliations and the topology of $3$-manifolds, J. Differential Geometry 18, 1983, 445-503. MR 86a:57009

3.
M. W. Hirsch, Immersions of manifolds Transactions of the AMS 93, 1959, 242-276. MR 22:9980

4.
J. Hoste, J. Przytycki, Homotopy skein modules of orientable $3$-manifolds, Math. Proc. Camb. Phil. Soc. (1990), 108, 475-488.MR 91g:57006

5.
U. Kaiser, Presentations of $q$-homotopy skein modules of oriented $3$-manifolds, Journal of Knot Theory and its Ramifications, no. 3, 2001, 461-491.MR 2002g:57011

6.
U. Kaiser, Link theory in manifolds, Lecture Notes in Mathematics 1669, Springer Verlag 1997.MR 98j:57010

7.
L. Kauffman, On Knots, Annals of Mathematical Studies 115, Princeton University Press 1987.MR 89c:57005

8.
E. Kalfagianni, X. S. Lin, The HOMFLY polynomial for links in rational homology spheres, Topology 38, no. 1, 1999, 95-115.MR 99h:57011

9.
R. Kirby. Problems in low dimensional topology, in 1993 Georgia International Topology Conference Proceedings, Part II, ed. by W. Kazez, AMS/IP Studies in Advanced Mathematics 1997.

10.
U. Koschorke, Vector Fields and Other Vector Bundle Morphisms - A Singularity Approach, Lecture Notes in Mathematics 847, 1981.MR 82i:57026

11.
R. K. Lashof, S. Smale, Self-intersections of immersed manifolds, Journal of Mathematics and Mechanics, Vol. 8, No. 1, 1959, 143-157.MR 21:332

12.
X. S. Lin, Finite type link invariants of $3$-manifolds, Topology 33, no. 1, 1994, 45-71. MR 94m:57020

13.
J. Przytycki, Algebraic topology based on knots: an introduction, in Proceedings of Knots 96, edited by Shin'ichi Suzuki, 1997, 279-297. MR 99k:57030

14.
J. Przytycki, Skein modules of $3$-manifolds, Bull. Ac. Pol. Math. 39(1-2), 1991, 91-100. MR 94g:57011

15.
J. Przytycki, A $q$-analogue of the first homology group of a $3$-manifold, Contemporary Mathematics Volume 214, 1998, 135-143.MR 98m:57011

16.
V. Vassiliev, On invariants and homology of spaces of knots in arbitrary manifolds, American Math. Soc. Translations (2), Vol. 185, 1998.MR 2000j:57030

17.
V. Vassiliev, Complements of discriminants of smooth maps: topology and applications, Transl. Math. Monographs, vol. 98, AMS, Providence, RI 1994.MR 94i:57020

18.
G. W. Whitehead, Elements of Homotopy Theory, Graduate Texts in Mathematics Vol. 61, Springer Verlag 1978. MR 80b:55001

19.
H. Whitney, The self-intersections of a smooth $n$-manifold in $2n$-space, Ann. Math. (1944), 220-240. MR 5:273g


Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57M25, 57M35, 57R42

Retrieve articles in all Journals with MSC (2000): 57M25, 57M35, 57R42


Additional Information:

Uwe Kaiser
Affiliation: Department of Mathematics, Boise State University, 1910 University Drive, Boise, Idaho 83725-1555
Email: kaiser@math.boisestate.edu

DOI: 10.1090/S0002-9947-03-03424-X
PII: S 0002-9947(03)03424-X
Received by editor(s): March 12, 2002
Received by editor(s) in revised form: May 1, 2003
Posted: November 25, 2003
Copyright of article: Copyright 2003, American Mathematical Society


  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement
Search the AMSPowered by Google