The Apolynomial from the noncommutative viewpoint
Authors:
Charles Frohman, Razvan Gelca and Walter LoFaro
Journal:
Trans. Amer. Math. Soc. 354 (2002), 735747
MSC (1991):
Primary 57M25, 58B30, 46L87
Published electronically:
October 3, 2001
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Abstract: The paper introduces a noncommutative generalization of the Apolynomial of a knot. This is done using the Kauffman bracket skein module of the knot complement, and is based on the relationship between skein modules and character varieties. The construction is possible because the Kauffman bracket skein algebra of the cylinder over the torus is a subalgebra of the noncommutative torus. The generalized version of the Apolynomial, called the noncommutative Aideal, consists of a finitely generated ideal of polynomials in the quantum plane. Some properties of the noncommutative Aideal and its relationships with the Apolynomial and the Jones polynomial are discussed. The paper concludes with the description of the examples of the unknot, and the right and lefthanded trefoil knots.
 [AL]
William
W. Adams and Philippe
Loustaunau, An introduction to Gröbner bases, Graduate
Studies in Mathematics, vol. 3, American Mathematical Society,
Providence, RI, 1994. MR 1287608
(95g:13025)
 [B]
Doug
Bullock, A finite set of generators for the Kauffman bracket skein
algebra, Math. Z. 231 (1999), no. 1,
91–101. MR
1696758 (2000d:57013), http://dx.doi.org/10.1007/PL00004727
 [BH]
G.
W. Brumfiel and H.
M. Hilden, 𝑆𝐿(2) representations of finitely
presented groups, Contemporary Mathematics, vol. 187, American
Mathematical Society, Providence, RI, 1995. MR 1339764
(96g:20004)
 [BP]
D. Bullock, J.H. Przytycki, Kauffman bracket skein module quantization of symmetric algebra and , preprint.
 [C]
P.
M. Cohn, Free rings and their relations, Academic Press,
London, 1971. London Mathematical Society Monographs, No. 2. MR 0371938
(51 #8155)
 [CCGLS]
D.
Cooper, M.
Culler, H.
Gillet, D.
D. Long, and P.
B. Shalen, Plane curves associated to character varieties of
3manifolds, Invent. Math. 118 (1994), no. 1,
47–84. MR
1288467 (95g:57029), http://dx.doi.org/10.1007/BF01231526
 [CL]
D.
Cooper and D.
D. Long, Representation theory and the 𝐴polynomial of a
knot, Chaos Solitons Fractals 9 (1998), no. 45,
749–763. Knot theory and its applications. MR 1628754
(99c:57013), http://dx.doi.org/10.1016/S09600779(97)001021
 [Co]
Alain
Connes, Noncommutative geometry, Academic Press Inc., San
Diego, CA, 1994. MR 1303779
(95j:46063)
 [CS]
Marc
Culler and Peter
B. Shalen, Varieties of group representations and splittings of
3manifolds, Ann. of Math. (2) 117 (1983),
no. 1, 109–146. MR 683804
(84k:57005), http://dx.doi.org/10.2307/2006973
 [FG]
Charles
Frohman and Răzvan
Gelca, Skein modules and the noncommutative
torus, Trans. Amer. Math. Soc.
352 (2000), no. 10, 4877–4888. MR 1675190
(2001b:57014), http://dx.doi.org/10.1090/S0002994700025125
 [Ge]
R. Gelca, Noncommutative trigonometry and the Apolynomial of the trefoil knot, to appear, Proceedings of Cambridge Philosophical Society.
 [HP]
Jim
Hoste and Józef
H. Przytycki, The (2,∞)skein module of Whitehead
manifolds, J. Knot Theory Ramifications 4 (1995),
no. 3, 411–427. MR 1347362
(97f:57019), http://dx.doi.org/10.1142/S021821659500020X
 [K]
Christian
Kassel, Quantum groups, Graduate Texts in Mathematics,
vol. 155, SpringerVerlag, New York, 1995. MR 1321145
(96e:17041)
 [KM]
Michael
Kapovich and John
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). MR 1733326
(2001d:14024)
 [Li]
W.
B. Raymond Lickorish, An introduction to knot theory, Graduate
Texts in Mathematics, vol. 175, SpringerVerlag, New York, 1997. MR 1472978
(98f:57015)
 [LM]
Michel
Broué, Les 𝑙blocs des groups
𝐺𝐿(𝑛,𝑞) et
𝑈(𝑛,𝑞²) et leurs structures locales,
Astérisque 133134 (1986), 159–188 (French).
Seminar Bourbaki, Vol. 1984/85. MR 837219
(87e:20021)
 [PS]
Józef
H. Przytycki and Adam
S. Sikora, On skein algebras and
𝑆𝑙₂(𝐶)character varieties, Topology
39 (2000), no. 1, 115–148. MR 1710996
(2000g:57026), http://dx.doi.org/10.1016/S00409383(98)000627
 [S1]
Adam Sikora, A geometric method in the theory of representations of groups, Preprin, xxx.lanl.gov/ math.RT9806016, (1998).
 [Ri1]
Marc
A. Rieffel, 𝐶*algebras associated with irrational
rotations, Pacific J. Math. 93 (1981), no. 2,
415–429. MR
623572 (83b:46087)
 [Ri2]
Marc
A. Rieffel, Deformation quantization of Heisenberg manifolds,
Comm. Math. Phys. 122 (1989), no. 4, 531–562.
MR
1002830 (90e:46060)
 [Ri3]
Marc
A. Rieffel, Noncommutative tori—a case study of
noncommutative differentiable manifolds, Geometric and topological
invariants of elliptic operators (Brunswick, ME, 1988), Contemp. Math.,
vol. 105, Amer. Math. Soc., Providence, RI, 1990,
pp. 191–211. MR 1047281
(91d:58012), http://dx.doi.org/10.1090/conm/105/1047281
 [Ro1]
Justin
Roberts, Skeins and mapping class groups, Math. Proc.
Cambridge Philos. Soc. 115 (1994), no. 1,
53–77. MR
1253282 (94m:57035), http://dx.doi.org/10.1017/S0305004100071917
 [Ro2]
Justin
Roberts, Kirby calculus in manifolds with boundary, Turkish J.
Math. 21 (1997), no. 1, 111–117. MR 1456165
(99c:57045)
 [We]
Alan
Weinstein, Contact surgery and symplectic handlebodies,
Hokkaido Math. J. 20 (1991), no. 2, 241–251. MR 1114405
(92g:53028)
 [AL]
 William Adams, Phillippe Loustaunau, An introduction to Gröbner bases, Graduate Studies in Mathematics, ISSN 10657339; 3, AMS, Providence RI, 1994. MR 95g:13025
 [B]
 D. Bullock,Rings of characters and the Kauffman bracket skein module, Comment. Math. Helv., 72 (1997), 521542. MR 2000d:57013
 [BH]
 G. W. Brumfiel, H. M. Hilden, Representations of Finitely Presented Groups, Contemp. Math. 187 (1995). MR 96g:20004
 [BP]
 D. Bullock, J.H. Przytycki, Kauffman bracket skein module quantization of symmetric algebra and , preprint.
 [C]
 P.M. Cohn, Free Rings and their Relations, Academic Press, 1971. MR 51:8155
 [CCGLS]
 D. Cooper, M. Culler, H. Gillett, D.D. Long, P.B. Shalen, Plane Curves associated to character varieties of 3manifolds, Inventiones Math. 118, pp. 4784 (1994). MR 95g:57029
 [CL]
 D. Cooper, D. Long, Representation theory and the Apolynomial of a knot, Chaos, Solitons and Fractals, 9 (1998) no 4/5, 749763. MR 99c:57013
 [Co]
 A. Connes, Noncommutative Geometry, Academic Press, London, 1994. MR 95j:46063
 [CS]
 M. Culler and P.B. Shalen, Varieties of group representations and splittings of 3manifolds, Ann. of Math. 117 (1983), 109146. MR 84k:57005
 [FG]
 C. Frohman, R. Gelca, Skein Modules and the Noncommutative Torus, Transactions of the AMS, 352 (2000), 48774888. MR 2001b:57014
 [Ge]
 R. Gelca, Noncommutative trigonometry and the Apolynomial of the trefoil knot, to appear, Proceedings of Cambridge Philosophical Society.
 [HP]
 J. Hoste, J.H. Przytycki, The skein module of lens spaces; a generalization of the Jones polynomial, J. Knot Theor. Ramif., 2 (1993), no. 3. 321333. MR 97f:57019
 [K]
 Ch. Kassel, Quantum Groups, Springer Verlag, 1995. MR 96e:17041
 [KM]
 M. Kapovich, J.J. Millson, On representation varieties of Artin groups, projective arrangements and fundamental groups of smooth complex algebraic varieties, Inst. Hautes Etude Sci. Publ. Math.No. 88 (1998), 595. MR 2001d:14024
 [Li]
 W. B. R. Lickorish, An Introduction to Knot Theory, Springer, GTM 175, 1997. MR 98f:57015
 [LM]
 A. Lubotzky, A. Magid, Varieties of representations of finitely generated groups, Memoirs of the AMS 336 (1985). MR 87e:20021
 [PS]
 J.H. Przytycki, A.S. Sikora, On Skein Algebras and Character Varieties, Topology 39 (2000), 115148. MR 2000g:57026
 [S1]
 Adam Sikora, A geometric method in the theory of representations of groups, Preprin, xxx.lanl.gov/ math.RT9806016, (1998).
 [Ri1]
 M. Rieffel, C*algebras associated with irrational rotations Pac. J. Math. 93, 415429 (1981). MR 83b:46087
 [Ri2]
 M. Rieffel, Deformation Quantization of Heisenberg Manifolds, Commun. Math. Phys. 122 (1989), 531562. MR 90e:46060
 [Ri3]
 M. Rieffel, Noncommutative tori  a case study of noncommutative differentiable manifolds, in Geometric and topological invariants of elliptic operators, Proc. AMSIMSSIAM Jt. Summer Res. Conf., Brunswick/ME (USA) 1988, Contemp. Math. 105, 191211 (1990). MR 91d:58012
 [Ro1]
 J. Roberts Skeins and Mapping Class Groups, Math. Proc. Camb. Phil. Soc. 115 (1994), 5377. MR 94m:57035
 [Ro2]
 J. Roberts, Kirby Calculus in Manifolds with Boundary, Turkish J. Math. 21 (1997), 111117. MR 99c:57045
 [We]
 A. Weinstein, Symplectic groupoids, geometric quantization, and irrational rotation algebras, in Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989), 281290, Springer, New York, 1991. MR 92g:53028
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Additional Information
Charles Frohman
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
Email:
frohman@math.uiowa.edu
Razvan Gelca
Affiliation:
Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409 and Institute of Mathematics of The Romanian Academy, Bucharest, Romania
Email:
rgelca@math.ttu.edu
Walter LoFaro
Affiliation:
Department of Mathematics and Computing, University of WisconsinStevens Point, Stevens Point, Wisconsin 54481
Email:
Walter.LoFaro@uwsp.edu
DOI:
http://dx.doi.org/10.1090/S0002994701028896
PII:
S 00029947(01)028896
Keywords:
Kauffman bracket,
skein modules,
Apolynomial,
character varieties,
noncommutative geometry
Received by editor(s):
March 14, 2001
Received by editor(s) in revised form:
May 7, 2001
Published electronically:
October 3, 2001
Article copyright:
© Copyright 2001 American Mathematical Society
