Geodesic
Random walk invariants of string links from R–matrices
Kerler, Thomas1 ; Wang, Yilong2
1Department of Mathematics, Ohio State University, Columbus, OH 43210, USA
2Department of Mathematics, Ohio State University, Columbus, OH 43210, uSA
Algebraic and Geometric Topology, Tome 16 (2016) no. 1, pp. 569-596
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

We show that the exterior powers of the matrix valued random walk invariant of string links, introduced by Lin, Tian, and Wang, are isomorphic to the graded components of the tangle functor associated to the Alexander polynomial by Ohtsuki divided by the zero graded invariant of the functor. Several resulting properties of these representations of the string link monoids are discussed.

DOI : 10.2140/agt.2016.16.569
Classification : 57M27, 57M25, 20F36, 57R56, 15A75, 17B37
Keywords: string links, tangles, R-matrices, Burau representation, Alexander polynomial, random walk

Kerler, Thomas  1   ; Wang, Yilong  2

1 Department of Mathematics, Ohio State University, Columbus, OH 43210, USA
2 Department of Mathematics, Ohio State University, Columbus, OH 43210, uSA 
@article{10_2140_agt_2016_16_569,
     author = {Kerler, Thomas and Wang, Yilong},
     title = {Random walk invariants of string links from {R{\textendash}matrices}},
     journal = {Algebraic and Geometric Topology},
     pages = {569--596},
     year = {2016},
     volume = {16},
     number = {1},
     doi = {10.2140/agt.2016.16.569},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2016.16.569/}
}
TY  - JOUR
AU  - Kerler, Thomas
AU  - Wang, Yilong
TI  - Random walk invariants of string links from R–matrices
JO  - Algebraic and Geometric Topology
PY  - 2016
SP  - 569
EP  - 596
VL  - 16
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2016.16.569/
DO  - 10.2140/agt.2016.16.569
ID  - 10_2140_agt_2016_16_569
ER  - 
%0 Journal Article
%A Kerler, Thomas
%A Wang, Yilong
%T Random walk invariants of string links from R–matrices
%J Algebraic and Geometric Topology
%D 2016
%P 569-596
%V 16
%N 1
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2016.16.569/
%R 10.2140/agt.2016.16.569
%F 10_2140_agt_2016_16_569
Kerler, Thomas; Wang, Yilong. Random walk invariants of string links from R–matrices. Algebraic and Geometric Topology, Tome 16 (2016) no. 1, pp. 569-596. doi: 10.2140/agt.2016.16.569

[1] D Cimasoni, V Turaev, A Lagrangian representation of tangles, Topology 44 (2005) 747

[2] S Garoufalidis, M Loebl, Random walks and the colored Jones function, Combinatorica 25 (2005) 651

[3] C Jackson, Braid group representations, Master’s thesis, Ohio State University (2001)

[4] V F R Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987) 335

[5] L H Kauffman, H Saleur, Free fermions and the Alexander–Conway polynomial, Comm. Math. Phys. 141 (1991) 293

[6] P Kirk, C Livingston, Z Wang, The Gassner representation for string links, Commun. Contemp. Math. 3 (2001) 87

[7] X S Lin, F Tian, Z Wang, Burau representation and random walks on string links, Pacific J. Math. 182 (1998) 289

[8] X S Lin, Z Wang, Random walk on knot diagrams, colored Jones polynomial and Ihara–Selberg zeta function, from: "Knots, braids, and mapping class groups — Papers dedicated to Joan S Birman" (editors J Gilman, W W Menasco, X S Lin), AMS/IP Stud. Adv. Math. 24, Amer. Math. Soc. (2001) 107

[9] J Murakami, The free-fermion model in presence of field related to the quantum group Uq(sl2) of affine type and the multi-variable Alexander polynomial of links, from: "Infinite Analysis, Part A, B" (editors A Tsuchiya, T Eguchi, M Jimbo), Adv. Ser. Math. Phys. 16, World Scientific (1992) 765

[10] T Ohtsuki, Quantum invariants : A study of knots, 3–manifolds, and their sets, 29, World Scientific (2002)

[11] V V Prasolov, A B Sossinsky, Knots, links, braids and 3–manifolds : An introduction to the new invariants in low-dimensional topology, 154, Amer. Math. Soc. (1997)

[12] N Y Reshetikhin, V G Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990) 1

[13] D S Silver, S G Williams, A generalized Burau representation for string links, Pacific J. Math. 197 (2001) 241

[14] V G Turaev, Operator invariants of tangles, and R–matrices, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989) 1073, 1135

[15] O Y Viro, Quantum relatives of the Alexander polynomial, Algebra i Analiz 18 (2006) 63

Cité par Sources :