JOURNAL BROWSE
Search
Advanced SearchSearch Tips
THE QUANTUM sl(n, ℂ) REPRESENTATION THEORY AND ITS APPLICATIONS
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
THE QUANTUM sl(n, ℂ) REPRESENTATION THEORY AND ITS APPLICATIONS
Jeong, Myeong-Ju; Kim, Dong-Seok;
  PDF(new window)
 Abstract
In this paper, we study the quantum sl() representation category using the web space. Specially, we extend sl() web space for as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial specialized to a one variable polynomial can be computed by a linear expansion with respect to a presentation of the quantum representation category of sl(). Moreover, we correct the false conjecture [30] given by Chbili, which addresses the relation between some link polynomials of a periodic link and its factor link such as Alexander polynomial () and Jones polynomial () and prove the corrected conjecture not only for HOMFLY polynomial but also for the colored HOMFLY polynomial specialized to a one variable polynomial.
 Keywords
quantum sl() representation theory;colored HOMFLY polynomial specialized to a one variable polynomial;periodic links;web spaces;
 Language
English
 Cited by
1.
On skein relations in class S theories, Journal of High Energy Physics, 2015, 2015, 6  crossref(new windwow)
2.
Webs and quantum skew Howe duality, Mathematische Annalen, 2014, 360, 1-2, 351  crossref(new windwow)
 References
1.
C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, New York, W. H. Freeman, 1994.

2.
C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology 34 (1995), no. 4, 883-927. crossref(new window)

3.
S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves I. The SL(2) case, Duke Math. J. 142 (2008), no. 3, 511-588. crossref(new window)

4.
N. Chbili, The quantum SU(3) invariant of links and Murasugi's congruence, Topology Appl. 122 (2002), no. 3, 479-485.

5.
N. Chbili, Quantum invariants and finite group actions on three-manifolds, Topology Appl. 136 (2004), no. 1-3, 219-231. crossref(new window)

6.
Q. Chen and T. Le, Quantum invariants of periodic links and periodic 3-manifolds, Fund. Math. 184 (2004), 55-71. crossref(new window)

7.
I. Frenkel and M. Khovanov, Canonical bases in tensor products and graphical calculus for $U_{q}$(sl2), Duke Math. J. 87 (1997), no. 3, 409-480. crossref(new window)

8.
W. Fulton and J. Harris, Representation Theory, Graduate Texts in Mathematics, 129, Springer-Verlag, New York-Heidelberg-Berlin, 1991.

9.
V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1-25. crossref(new window)

10.
V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987), no. 2, 335-388. crossref(new window)

11.
M.-J. Jeong and C.-Y. Park, Lens knots, periodic links and Vassiliev invariants, J. Knot Theory Ramifications 13 (2004), no. 8, 1041-1056. crossref(new window)

12.
C. Kassel, M. Rosso, and V. Turaev, Quantum Groups and Knot Invariants, Panoramas et Syntheses, 5, Societe Mathematique de France, 1997.

13.
M. Khovanov, sl(3) link homology, Algebr. Geom. Topol. 4 (2004), 1045-1081. crossref(new window)

14.
M. Khovanov, Categorifications of the colored Jones polynomial, J. Knot Theory Ramifications 14 (2005), no. 1, 111-130. crossref(new window)

15.
M. Khovanov, private communication.

16.
M. Khovanov and L. Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008), no. 1, 1-91. crossref(new window)

17.
D. Kim, Graphical Calculus on Representations of Quantum Lie Algebras, Thesis, UC-Davis, 2003, arXiv:math.QA/0310143.

18.
D. Kim and J. Lee, The quantum sl(3) invariants of cubic bipartite planar graphs, J. Knot Theory Ramifications 17 (2008), no. 3, 361-375. crossref(new window)

19.
R. Kirby and P. Melvin, The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2), Invent. Math. 105 (1991), no. 3, 473-545. crossref(new window)

20.
G. Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996), no. 1, 109-151. crossref(new window)

21.
T. Le, Integrality and symmetry of quantum link invariants, Duke Math. J. 102 (2000), no. 2, 273-306. crossref(new window)

22.
W. Lickorish, Distinct 3-manifolds with all SU(2)q invariants the same, Proc. Amer. Math. Soc. 117 (1993), no. 1, 285-292.

23.
S. Morrison, A Diagrammatic Category for the Representation Theory of $U_{q}$($sl_{n}$), UC Berkeley Ph.D. thesis, arXiv:0704.1503.

24.
K. Murasugi, On periodic knots, Comment. Math. Helv. 46 (1971), 162-174. crossref(new window)

25.
K. Murasugi, Jones polynomials of periodic links, Pacific J. Math. 131 (1988), no. 2, 319-329. crossref(new window)

26.
H. Murakami, Asymptotic Behaviors of the colored Jones polynomials of a torus knot, Internat. J. Math. 15 (2004), no. 6, 547-555. crossref(new window)

27.
H.Murakami, T. Ohtsuki, and S. Yamada, Homfly polynomial via an invariant of colored plane graphs, Enseign. Math. (2) 44 (1998), no. 3-4, 325-360.

28.
T. Ohtsuki and S. Yamada, Quantum SU(3) invariant of 3-manifolds via linear skein theory, J. Knot Theory Ramifications 6 (1997), no. 3, 373-404. crossref(new window)

29.
J. H. Przytycki, On Murasugi's and Traczyk's criteria for periodic links, Math. Ann. 283 (1989), no. 3, 465-478. crossref(new window)

30.
J. Przytycki and A. Sikora, $SU_{n}$-quantum invariants for periodic links, Diagrammatic morphisms and applications (San Francisco, CA, 2000), 199-205, Contemp. Math., 318, Amer. Math. Soc., Providence, RI, 2003.

31.
J. Przytycki and A. Sikora, On skein algebras and $Sl_{2}$(C)-character varieties, Topology 39 (2000), no. 1, 115-148. crossref(new window)

32.
N. Yu. Reshetikhin and V. G. Turaev, Ribbob graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), no. 1, 1-26. crossref(new window)

33.
N. Yu. Reshetikhin and V. G. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547-597. crossref(new window)

34.
A. Sikora and B. Westbury, Confluence theory for graphs, Algebr. Geom. Topol. 7 (2007), 439-478. crossref(new window)

35.
P. Traczyk, A criterion for knots of period 3, Topology Appl. 36 (1990), no. 3, 275-281. crossref(new window)

36.
V. G. Turaev, The Conway and Kauffman modules of a solid torusa, (translation) J. Soviet Math. 52 (1990), no. 1, 2799-2805. crossref(new window)

37.
T. Van Zandt, PSTricks: PostScript macros for generic $T_{E}X$, Available at ftp://ftp.princeton.edu/pub/tvz/.

38.
M. Vybornov, Solutions of the Yang-Baxter equation and quantum sl(2), J. Knot Theory Ramifications 8 (1999), no. 7, 953-961. crossref(new window)

39.
H. Wenzl, On sequences of projections, C. R. Math. Rep. Acad. Sci. Canada 9 (1987), no. 1, 5-9.

40.
B. Westbury, Invariant tensors for the spin representation of so(7), Math. Proc. Cambridge Philos. Soc. 144 (2008), no. 1, 217-240.

41.
E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351-399. crossref(new window)

42.
Y. Yokota, The skein polynomial of periodic knots, Math. Ann. 291 (1991), no. 2, 281-291. crossref(new window)

43.
Y. Yokota, The Jones polynomial of periodic knots, Proc. Amer. Math. Soc. 113 (1991), no. 3, 889-894. crossref(new window)

44.
Y. Yokota, The Kauffman polynomial of periodic knots, Topology 32 (1993), no. 2, 309-324. crossref(new window)

45.
Y. Yokota, Skein and quantum SU(N) invariants of 3-manifolds, Math. Ann. 307 (1997), no. 1, 109-138. crossref(new window)