Colored HOMFLY-PT for hybrid weaving knot W ? 3 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ {hat{mathrm{W}}}_3 $$end{document} (m , n)

摘要

A bstract Weaving knots W ( p, n ) of type ( p, n ) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known ( p, n ) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W ( p, n ). In this paper, we confine to a hybrid generalization of W (3 , n ) which we denote as W ? 3 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ {hat{W}}_3 $$end{document} ( m, n ) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving ? documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ mathrm{mathcal{R}} $$end{document} -matrices. Further, we also compute [ r ]-colored HOMFLY-PT for W (3 , n ). Surprisingly, we observe that trace of the product of two dimensional ? ? documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ hat{mathrm{mathcal{R}}} $$end{document} -matrices can be written in terms of infinite family of Laurent polynomials V n , t q documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ {mathcal{V}}_{n,t}left[qight] $$end{document} whose absolute coefficients has interesting relation to the Fibonacci numbers ? n documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ {mathrm{mathcal{F}}}_n $$end{document} . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W (3 , n ) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.