# zbMATH — the first resource for mathematics

Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux. (English) Zbl 1328.81123
Summary: If a knot is represented by an $$m$$-strand braid, then HOMFLY polynomial in representation $$R$$ is a sum over characters in all representations $$Q\in R^{\oplus m}$$. Coefficients in this sum are traces of products of quantum $$\widehat{\mathscr{R}}$$-matrices along the braid, but these matrices act in the space of intertwiners, and their size is equal to the multiplicity $$M_{QR}$$ of $$Q$$ in $$R^{\oplus m}$$. If $$R$$ is the fundamental representation $$R=[1]=\square$$, then $$M_{\square Q}$$ is equal to the number of paths in representation graph, which lead from the fundamental vertex $$\square$$ to the vertex $$Q$$. In the basis of paths the entries of the $$m-1$$ relevant $$\widehat{\mathscr{R}}$$-matrices are associated with the pairs of paths and are nonvanishing only when the two paths either coincide or differ by at most one vertex, as a corollary $$\widehat{\mathscr{R}}$$-matrices consist of just $$1\times1$$ and $$2\times2$$ blocks, given by very simple explicit expressions. If cabling method is used to color the knot with the representation $$R$$, then the braid has as many as $$m|R|$$ strands; $$Q$$ have a bigger size $$m|R|$$, but only paths passing through the vertex $$R$$ are included into the sums over paths which define the products and traces of the $$m|R|-1$$ relevant $$\widehat{\mathscr{R}}$$-matrices. In the case of $$SU(N)$$, this path sum formula can also be interpreted as a multiple sum over the standard Young tableaux. By now it provides the most effective way for evaluation of the colored HOMFLY polynomials, conventional or extended, for arbitrary braids.

##### MSC:
 81R05 Finite-dimensional groups and algebras motivated by physics and their representations 57M27 Invariants of knots and $$3$$-manifolds (MSC2010)
Full Text:
##### References:
 [1] DOI: 10.2307/1971013 · Zbl 0283.53036 · doi:10.2307/1971013 [2] DOI: 10.2307/1989123 · JFM 54.0603.03 · doi:10.2307/1989123 [3] DOI: 10.1007/BF01389127 · Zbl 0508.46040 · doi:10.1007/BF01389127 [4] 26 (3) pp 395– (1987) [5] DOI: 10.1090/S0273-0979-1985-15361-3 · Zbl 0572.57002 · doi:10.1090/S0273-0979-1985-15361-3 [6] 4 (2) pp 115– (1988) [7] DOI: 10.1007/BF01217730 · Zbl 0667.57005 · doi:10.1007/BF01217730 [8] DOI: 10.1007/3-540-53503-9_51 · doi:10.1007/3-540-53503-9_51 [9] 235 (3-4) pp 275– (1990) [10] DOI: 10.1007/BF02096491 · Zbl 0768.57003 · doi:10.1007/BF02096491 [11] DOI: 10.1016/S0550-3213(96)00689-X · Zbl 0925.57007 · doi:10.1016/S0550-3213(96)00689-X [12] 220 (3) pp 422– (1989) [13] DOI: 10.1142/S0217732390001554 · Zbl 1020.81704 · doi:10.1142/S0217732390001554 [14] DOI: 10.1016/0370-2693(89)91289-6 · doi:10.1016/0370-2693(89)91289-6 [15] 16 pp 594– (1990) [16] 220 (1-2) pp 142– (1989) [17] 2 (2) pp 231– (1998) [18] DOI: 10.1016/S0370-2693(98)00377-3 · Zbl 1355.81126 · doi:10.1016/S0370-2693(98)00377-3 [19] 2 (2) pp 253– (1998) [20] DOI: 10.1007/s11005-010-0369-5 · Zbl 1185.81111 · doi:10.1007/s11005-010-0369-5 [21] DOI: 10.1088/1126-6708/2009/11/002 · doi:10.1088/1126-6708/2009/11/002 [24] DOI: 10.1088/0264-9381/27/12/125005 · Zbl 1190.83045 · doi:10.1088/0264-9381/27/12/125005 [26] (1) pp 097, 18– (2010) [28] DOI: 10.1016/j.geomphys.2011.01.012 · Zbl 1215.81092 · doi:10.1016/j.geomphys.2011.01.012 [31] DOI: 10.1016/0550-3213(94)90124-4 · Zbl 0996.81510 · doi:10.1016/0550-3213(94)90124-4 [32] 431 (3) pp 484– (1994) [33] DOI: 10.1016/0550-3213(95)00376-4 · Zbl 0925.81347 · doi:10.1016/0550-3213(95)00376-4 [36] DOI: 10.1016/0550-3213(96)00271-4 · Zbl 0925.81358 · doi:10.1016/0550-3213(96)00271-4 [39] DOI: 10.1016/S0550-3213(98)00635-X · Zbl 0948.81641 · doi:10.1016/S0550-3213(98)00635-X [40] DOI: 10.1007/JHEP08(2012)067 · Zbl 1397.81365 · doi:10.1007/JHEP08(2012)067 [43] 240 (3) pp 312– (1984) [44] 265 (1-2) pp 99– (1991) [45] 404 (3) pp 717– (1993) [47] 123 (6) pp 957– (2010) [53] DOI: 10.1016/j.nuclphysb.2010.10.016 · Zbl 1207.81146 · doi:10.1016/j.nuclphysb.2010.10.016 [54] DOI: 10.1007/s11005-011-0503-z · Zbl 1242.81119 · doi:10.1007/s11005-011-0503-z [55] DOI: 10.1016/j.nuclphysb.2011.04.014 · Zbl 1215.81096 · doi:10.1016/j.nuclphysb.2011.04.014 [56] 101 (3) pp 359– (2000) [57] DOI: 10.2140/agt.2002.2.337 · Zbl 0998.57016 · doi:10.2140/agt.2002.2.337 [61] (1) pp 065, front matter + 46– (2013) [62] DOI: 10.1007/s11005-005-0008-8 · Zbl 1105.57011 · doi:10.1007/s11005-005-0008-8 [63] 15 (2) pp 129– (2006) [78] 114 pp 127– (1998) [79] DOI: 10.1016/0550-3213(92)90524-F · Zbl 0938.81553 · doi:10.1016/0550-3213(92)90524-F [80] DOI: 10.1016/0550-3213(93)90652-6 · Zbl 0941.57500 · doi:10.1016/0550-3213(93)90652-6 [81] 422 (1-2) pp 291– (1994) [82] 03 pp 034– (2012) [83] DOI: 10.1016/j.nuclphysb.2012.11.006 · Zbl 1262.81073 · doi:10.1016/j.nuclphysb.2012.11.006 [85] DOI: 10.1016/j.nuclphysb.2010.03.012 · Zbl 1204.81097 · doi:10.1016/j.nuclphysb.2010.03.012 [89] (3) pp 034– (2012) [90] DOI: 10.1016/S0550-3213(00)00761-6 · Zbl 1097.81742 · doi:10.1016/S0550-3213(00)00761-6 [97] (1994) [99] 120 pp 92– (1982) [100] 34 pp 1948– (1986) [101] Lecture Notes in Physics 151, in: Quantum spectral transform method. Recent developments (1982) [102] DOI: 10.1007/BF00420302 · Zbl 0642.17015 · doi:10.1007/BF00420302 [103] DOI: 10.1142/S0218216593000064 · Zbl 0787.57006 · doi:10.1142/S0218216593000064 [104] DOI: 10.1090/S0002-9947-09-04691-1 · Zbl 1193.57006 · doi:10.1090/S0002-9947-09-04691-1 [105] DOI: 10.1007/s00023-010-0058-z · Zbl 1208.81149 · doi:10.1007/s00023-010-0058-z [109] DOI: 10.1142/S0217751X12300013 · Zbl 1247.81397 · doi:10.1142/S0217751X12300013 [110] DOI: 10.1088/1751-8113/45/35/355202 · Zbl 1252.81101 · doi:10.1088/1751-8113/45/35/355202 [114] DOI: 10.1016/j.nuclphysb.2010.10.016 · Zbl 1207.81146 · doi:10.1016/j.nuclphysb.2010.10.016 [116] (12) pp 116– (2012) [118] DOI: 10.1007/s00023-012-0171-2 · Zbl 1256.81086 · doi:10.1007/s00023-012-0171-2 [119] DOI: 10.1017/S0305004102006047 · Zbl 1017.57002 · doi:10.1017/S0305004102006047 [120] DOI: 10.1142/S021821650300238X · Zbl 1034.57005 · doi:10.1142/S021821650300238X [121] DOI: 10.1007/s00220-005-1312-y · Zbl 1115.57009 · doi:10.1007/s00220-005-1312-y [122] 1483 pp 189– (2012)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.