Quantum deformations of fundamental groups of oriented -manifolds

Author:
Uwe Kaiser

Journal:
Trans. Amer. Math. Soc. **356** (2004), 3869-3880

MSC (2000):
Primary 57M25, 57M35, 57R42

DOI:
https://doi.org/10.1090/S0002-9947-03-03424-X

Published electronically:
November 25, 2003

MathSciNet review:
2058509

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We compute two-term skein modules of framed oriented links in oriented -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 -manifold.

**1.**V. Chernov,*Framed knots in -manifolds*preprint math.GT/0105139, http://xxx.lanl.gov, 2001.**2.**D. Gabai,*Foliations and the topology of -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 -manifolds*, Math. Proc. Camb. Phil. Soc. (1990), 108, 475-488.MR**91g:57006****5.**U. Kaiser,*Presentations of -homotopy skein modules of oriented -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 -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 -manifolds*, Bull. Ac. Pol. Math. 39(1-2), 1991, 91-100. MR**94g:57011****15.**J. Przytycki,*A -analogue of the first homology group of a -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 -manifold in -space*, Ann. Math. (1944), 220-240. MR**5:273g**

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:
https://doi.org/10.1090/S0002-9947-03-03424-X

Received by editor(s):
March 12, 2002

Received by editor(s) in revised form:
May 1, 2003

Published electronically:
November 25, 2003

Article copyright:
© Copyright 2003
American Mathematical Society