2011/12/31 by A. Mironov, A. Morozov, And. Morozov +1 · 4 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Braid #Braid group #Braid theory #Geometric and Algebraic Topology #HOMFLY polynomial #Knot (papermaking) #Knot polynomial #Knot theory #Simple (philosophy) #Skein relation #hep-th #math.QA
paper · pdf · doi:10.1007/jhep03(2012)034
published as JHEP 03 (2012) 034 · 33 pages
arxiv created 2012/01/04 · openalex publication_date 2012/03/01 · arxiv updated 2015/06/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Character expansion is introduced and explicitly constructed for the (non-colored) HOMFLY polynomials of the simplest knots. Expansion coefficients are not the knot invariants and can depend on the choice of the braid realization. However, the method provides the simplest systematic way to construct HOMFLY polynomials directly in terms of the variable A=qN: a much better way than the standard approach making use of the skein relations. Moreover, representation theory of the simplest quantum group SUq(2) is sufficient to get the answers for all braids with m<5 strands. Most important we reveal a hidden hierarchical structure of expansion coefficients, what allows one to express all of them through extremely simple elementary constituents. Generalizations to arbitrary knots and arbitrary representations is straightforward.