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

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- by Uwe Kaiser PDF
- Trans. Amer. Math. Soc.
**356**(2004), 3869-3880 Request permission

## 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

- V. Chernov,
*Framed knots in $3$-manifolds*preprint math.GT/0105139, http://xxx.lanl.gov, 2001. - David Gabai,
*Foliations and the topology of $3$-manifolds*, J. Differential Geom.**18**(1983), no. 3, 445–503. MR**723813** - Morris W. Hirsch,
*Immersions of manifolds*, Trans. Amer. Math. Soc.**93**(1959), 242–276. MR**119214**, DOI 10.1090/S0002-9947-1959-0119214-4 - Jim Hoste and Józef H. Przytycki,
*Homotopy skein modules of orientable $3$-manifolds*, Math. Proc. Cambridge Philos. Soc.**108**(1990), no. 3, 475–488. MR**1068450**, DOI 10.1017/S0305004100069371 - Uwe Kaiser,
*Presentations of homotopy skein modules of oriented 3-manifolds*, J. Knot Theory Ramifications**10**(2001), no. 3, 461–491. MR**1825969**, DOI 10.1142/S0218216501000962 - Uwe Kaiser,
*Link theory in manifolds*, Lecture Notes in Mathematics, vol. 1669, Springer-Verlag, Berlin, 1997. MR**1479639**, DOI 10.1007/BFb0092686 - Louis H. Kauffman,
*On knots*, Annals of Mathematics Studies, vol. 115, Princeton University Press, Princeton, NJ, 1987. MR**907872** - Efstratia Kalfagianni and Xiao-Song Lin,
*The HOMFLY polynomial for links in rational homology $3$-spheres*, Topology**38**(1999), no. 1, 95–115. MR**1644083**, DOI 10.1016/S0040-9383(98)00005-6 - 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. - Ulrich Koschorke,
*Vector fields and other vector bundle morphisms—a singularity approach*, Lecture Notes in Mathematics, vol. 847, Springer, Berlin, 1981. MR**611333**, DOI 10.1007/BFb0090440 - R. K. Lashof and S. Smale,
*Self-intersections of immersed manifolds*, J. Math. Mech.**8**(1959), 143–157. MR**0101522**, DOI 10.1512/iumj.1959.8.58010 - Xiao-Song Lin,
*Finite type link invariants of $3$-manifolds*, Topology**33**(1994), no. 1, 45–71. MR**1259514**, DOI 10.1016/0040-9383(94)90034-5 - Józef H. Przytycki,
*Algebraic topology based on knots: an introduction*, KNOTS ’96 (Tokyo), World Sci. Publ., River Edge, NJ, 1997, pp. 279–297. MR**1664968** - Józef H. Przytycki,
*Skein modules of $3$-manifolds*, Bull. Polish Acad. Sci. Math.**39**(1991), no. 1-2, 91–100. MR**1194712** - Józef H. Przytycki,
*A $q$-analogue of the first homology group of a $3$-manifold*, Perspectives on quantization (South Hadley, MA, 1996) Contemp. Math., vol. 214, Amer. Math. Soc., Providence, RI, 1998, pp. 135–144. MR**1601241**, DOI 10.1090/conm/214/02910 - V. A. Vassiliev,
*On invariants and homology of spaces of knots in arbitrary manifolds*, Topics in quantum groups and finite-type invariants, Amer. Math. Soc. Transl. Ser. 2, vol. 185, Amer. Math. Soc., Providence, RI, 1998, pp. 155–182. MR**1736171**, DOI 10.1090/trans2/185/11 - V. A. Vassiliev,
*Complements of discriminants of smooth maps: topology and applications*, Translations of Mathematical Monographs, vol. 98, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by B. Goldfarb. MR**1168473**, DOI 10.1090/conm/478 - George W. Whitehead,
*Elements of homotopy theory*, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR**516508**, DOI 10.1007/978-1-4612-6318-0 - Albert Eagle,
*Series for all the roots of a trinomial equation*, Amer. Math. Monthly**46**(1939), 422–425. MR**5**, DOI 10.2307/2303036

## Additional Information

**Uwe Kaiser**- Affiliation: Department of Mathematics, Boise State University, 1910 University Drive, Boise, Idaho 83725-1555
- Email: kaiser@math.boisestate.edu
- Received by editor(s): March 12, 2002
- Received by editor(s) in revised form: May 1, 2003
- Published electronically: November 25, 2003
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 2058509