Pretzel Knots and q-Series

The tail of the colored Jones polynomial of an alternating link is a $q$-series invariant whose first $n$ terms coincide with the first $n$ terms of the $n$-th colored Jones polynomial. Recently, it has been shown that the tail of the colored Jones polynomial of torus knots give rise to Ramanujan type identities. In this paper, we study $q$-series identities coming from the colored Jones polynomial of pretzel knots. We prove a false theta function identity that goes back to Ramanujan and we give a natural generalization of this identity using the tail of the colored Jones polynomial of Pretzel knots. Furthermore, we compute the tail for an infinite family of Pretzel knots and relate it to false theta function-type identities.

[1]  Kenneth C. Millett,et al.  A new polynomial invariant of knots and links , 1985 .

[2]  V. Jones.  A polynomial invariant for knots via von Neumann algebras , 1985 .

[3]  Louis H. Kauffman,et al.  State Models and the Jones Polynomial , 1987 .

[4]  Vaughan F. R. Jones,et al.  Hecke algebra representations of braid groups and link polynomials , 1987 .

[5]  Jozef H. Przytycki,et al.  Invariants of links of Conway type , 1988, 1610.06679.

[6]  George E. Andrews,et al.  The lost notebook and other unpublished papers , 1988 .

[7]  Vladimir Turaev,et al.  Invariants of 3-manifolds via link polynomials and quantum groups , 1991 .

[8]  N. Habegger,et al.  Three-manifold invariants derived from the Kauffman bracket , 1992 .

[9]  W. Lickorish.  Calculations with the temperley—Lieb algebra , 1992 .

[10]  Vladimir Turaev,et al.  State sum invariants of 3 manifolds and quantum 6j symbols , 1992 .

[11]  H. Morton.  Invariants of Links and 3-Manifolds From Skein Theory and From Quantum Groups , 1993 .

[12]  V. Turaev,et al.  QUANTUM INVARIANTS OF 3-MANIFOLDS ASSOCIATED WITH CLASSICAL SIMPLE LIE ALGEBRAS , 1993 .

[13]  G. Masbaum,et al.  3-valent graphs and the Kauffman bracket , 1994 .

[14]  J. Przytycki.  Fundamentals of Kauffman bracket skein modules , 1998, math/9809113.

[15]  J. Murakami,et al.  The colored Jones polynomials and the simplicial volume of a knot , 1999, math/9905075.

[16]  Kazuhiro Hikami,et al.  Volume Conjecture and Asymptotic Expansion of q-Series , 2003, Exp. Math..

[17]  George E. Andrews,et al.  Ramanujan's Lost Notebook: Part I , 2005 .

[18]  Oliver T. Dasbach,et al.  On the head and the tail of the colored Jones polynomial , 2006, Compositio Mathematica.

[19]  A volumish theorem for the Jones polynomial of alternating knots , 2004, math/0403448.

[20]  Antonio C. de A. Campello,et al.  On sequences of projections of the cubic lattice , 2011, ArXiv.

[21]  Cody Armond.  Walks along braids and the colored Jones polynomial , 2011, 1101.3810.

[22]  Oliver T. Dasbach,et al.  Rogers-Ramanujan type identities and the head and tail of the colored Jones polynomial , 2011, 1106.3948.

[23]  Thang T. Q. Lê,et al.  Nahm sums, stability and the colored Jones polynomial , 2011, 1112.3905.

[24]  Ramanujan's lost notebook , 2012 .

[25]  M. Hajij,et al.  The tail of a quantum spin network , 2013, 1308.2369.

[26]  Cody Armond.  The head and tail conjecture for alternating knots , 2011, 1112.3995.

[27]  The head and tail conjecture for alternating knots , 2013 .

[28]  M. Hajij,et al.  The colored Kauffman skein relation and the head and tail of the colored Jones polynomial , 2014, 1401.4537.

[29]  R. Osburn,et al.  Rogers–Ramanujan type identities for alternating knots , 2014, 1407.3482.