-character varieties as spaces of graphs

Author:
Adam S. Sikora

Journal:
Trans. Amer. Math. Soc. **353** (2001), 2773-2804

MSC (1991):
Primary 20C15, 57M27

DOI:
https://doi.org/10.1090/S0002-9947-01-02700-3

Published electronically:
March 15, 2001

MathSciNet review:
1828473

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Abstract | References | Similar Articles | Additional Information

An -character of a group is the trace of an -representation of We show that all algebraic relations between -characters of can be visualized as relations between graphs (resembling Feynman diagrams) in any topological space with We also show that all such relations are implied by a single local relation between graphs. In this way, we provide a topological approach to the study of -representations of groups.

The motivation for this paper was our work with J. Przytycki on invariants of links in 3-manifolds which are based on the Kauffman bracket skein relation. These invariants lead to a notion of a skein module of which, by a theorem of Bullock, Przytycki, and the author, is a deformation of the -character variety of This paper provides a generalization of this result to all -character varieties.

**[BHMV-1]**C. Blanchet, N. Habegger, G. Masbaum, P. Vogel, Three-manifold invariants derived from the Kauffman bracket,*Topology,***31**(1992), no. 4, 685-699. MR**94a:57010****[BHMV-2]**C. Blanchet, N. Habegger, G. Masbaum, P. Vogel, Topological quantum field theories derived from the Kauffman bracket,*Topology,***34**(1995), no. 4, 883-927. MR**96i:57015****[B-H]**G. W. Brumfiel, H. M. Hilden, Representations of Finitely Presented Groups,*Contemp. Math.***187**(1995). MR**96g:20004****[B-1]**D. Bullock, Estimating a skein module with characters,*Proc. Amer. Math. Soc.***125**(6), 1997, 1835-1839. MR**97g:57018****[B-2]**D. Bullock, Rings of -characters and the Kauffman bracket skein module,*Comment. Math. Helv.***72,**1997, 521-542. MR**98k:57008****[Co]**J. H. Conway, An enumeration of knots and links, and some of their algebraic properties. Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), pp. 329-358, Pergamon, Oxford, 1970. MR**41:2661****[Cv]**P. Cvitanovic, Group Theory, Part I, Nordita Notes, January 1984.**[Fog]**J. Fogarty, Invariant Theory, W. A. Benjamin, Inc. 1969. MR**29:1458****[For]**E. Formanek, The Polynomial Identities and Invariants of Matrices,*CBMS Regional Conf. Ser.***78**AMS, 1991. MR**92g:16031****[FYHLMO]**P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millett, A. Ocneanu, A new polynomial invariant of knots and links.*Bull. Amer. Math. Soc. (N.S.)***12**(1985), no. 2, 239-246. MR**86e:57007****[H-P]**J. Hoste, J.H. Przytycki, A survey of skein modules of 3-manifolds; in Knots 90,*Proceedings of the International Conference on Knot Theory and Related Topics*, Osaka (Japan), August 15-19, 1990, Editor A. Kawauchi, Walter de Gruyter (1992), 363-379. MR**93m:57018****[Jo]**V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials.*Ann. of Math.***126**(1987), no. 2, 335-388. MR**89c:46092****[KM]**M. Kapovich, J. J. Millson, On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties,*Inst. Hautes Études Sci. Publ. Math.***88**, (1998), 5-95 (1999). CMP**2000:07****[Ka]**L. H. Kauffman, An invariant of regular isotopy,*Trans. Amer. Math. Soc.***318**(1990), no. 2, 417-471. MR**90g:57007****[L1]**W. B. R. Lickorish, Three-manifolds and the Temperley-Lieb algebra,*Math. Ann.***290**(1991), no. 4, 657-670. MR**92e:57014****[L2]**W. B. R. Lickorish, Calculations with the Temperley-Lieb algebra,*Comment. Math. Helv.***67**(1992), no. 4, 571-591. MR**94c:57032****[L3]**W. B. R. Lickorish, The skein method for three-manifold invariants,*J. Knot Theory Ramifications***2**(1993), no. 2, 171-194. MR**94g:57006****[L-M]**A. Lubotzky, A. Magid, Varieties of representations of finitely generated groups,*Memoirs of the AMS***336**(1985). MR**87c:20021****[MFK]**D. Mumford, J. Fogarty, F. Kirwan Geometric Invariant Theory, Springer-Verlag 1994. MR**95m:14012****[Ny]**L. Nyssen, Pseudo-représentations,*Mathematische Annalen***306**(1996), 257-283. MR**98a:20013****[Pro-1]**C. Procesi, The invariant theory of matrices,*Adv. in Math.***19**(1976), 301-381. MR**54:7512****[Pro-2]**C. Procesi, A formal inverse to the Cayley-Hamilton theorem,*J. of Alg.***107**(1987), 63-74. MR**88b:16033****[PS-1]**J.H. Przytycki, A.S. Sikora, Skein algebra of a group, Proceedings of an International Conference in Knot Theory, Warsaw 1995,*Banach Center Publications,***42**, Polish Acad. Sci., Warsaw, 1998, pp. 297-306. MR**99e:57019****[PS-2]**J.H. Przytycki, A.S. Sikora, On Skein Algebras and -character varieties,*Topology.***39**(2000), 115-148. MR**2000g:57026****[PS-3]**J.H. Przytycki, A.S. Sikora, Skein algebras of surfaces, Preprint.**[P-T]**J. H. Przytycki, P. Traczyk, Invariants of links of Conway type,*Kobe J. Math.***4**(1988), no. 2, 115-139. MR**89h:57006****[Ra]**Y. P. Razmyslov, Identities of algebras and their representations, Translations of Mathematical Monographs,**138,**AMS 1994. MR**95i:16022****[R-T]**N. Reshetikhin, V. G. Turaev, Invariants of -manifolds via link polynomials and quantum groups,*Invent. Math.***103**(1991), no. 3, 547-597. MR**92b:57024****[Ro-1]**J. Roberts, Skein theory and Turaev-Viro invariants,*Topology***34**(1995), no. 4, 771-787. MR**97g:57014****[Ro-2]**J. Roberts, Skeins and mapping class groups,*Math. Proc. Cambridge Philos. Soc.***115**(1994), no. 1, 53-77. MR**94m:57035****[Ro]**R. Rouquier, Caractérisation des caractèrs et Pseudo-caractères,*J. of Algebra***180**, no. 0083, (1996) 571-586. MR**97a:20010****[Sa-1]**K. Saito, Representation varieties of a finitely generated group into or RIMS Publications**958**, Kyoto University (1993).**[Sa-2]**K. Saito, Character variety of representations of a finitely generated group in Topology and Teichmüller Spaces (Proc. Taniguchi Sympos., Katinkulta, Finland, 1995; S. Kojima*et al*., eds.), World Sci. Publ., Singapore, 1996, pp. 253-264. MR**99k:20011****[Si]**A.S. Sikora, Spin networks and character varieties, in preparation.**[Ta]**R. Taylor, Galois representations associated to Siegel modular forms of low weight,*Duke Math. Journal***63**(1991), no. 2, 281-332. MR**92j:11044****[We]**H. Weyl, The Classical Groups, Princeton Univ. Press, 1939. MR**1:42c****[Yo]**Y. Yokota, Skeins and quantum invariants of -manifolds.*Math. Ann.***307**(1997), no. 1, 109-138. MR**98h:57039**

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Additional Information

**Adam S. Sikora**

Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742

Email:
asikora@math.umd.edu

DOI:
https://doi.org/10.1090/S0002-9947-01-02700-3

Keywords:
Character,
character variety,
skein module

Received by editor(s):
May 18, 1999

Received by editor(s) in revised form:
August 9, 2000

Published electronically:
March 15, 2001

Additional Notes:
Partially supported by NSF grant DMS93-22675

Article copyright:
© Copyright 2001
American Mathematical Society