Stavros Garoufalidis and Xinyu Sun
Abstract: In an earlier paper, the first author introduced the noncommutative $A$-polynomial of a knot. By definition, it is the characteristic polynomial of the colored Jones function of the knot, and its specialization at $q=1$ is conjectured to coincide with the better-known $A$-polynomial of a knot. The aim of the paper is to introduce the $C$-polynomial of a knot and its noncommutative cousin. By its definition, the noncommutative $C$-polynomial of a knot is the characteristic polynomial of the cyclotomic function of the knot (the latter, introduced by Habiro), and as a result, it is entirely determined by the colored Jones function of the knot. A goal of the paper is to give explicit formulas for the noncommutative $C$-polynomial of all twist knots. Since the cyclotomic and the colored Jones function of a knot are related by an explicit transformation, we conjecture that the $C$-polynomial and the $A$-polynomial of a knot are likewise related by an explicit rational map of degree $2$. Using our explicit formulas, we verify this conjecture for all twist knots, and we observe that the genus of the $C$-polynomial of twist knots is zero. Our computation of the noncommutative $C$-polynomial of twist knots utilizes explicit single-sum formulas for the cyclotomic function, and Zeilberger's theory of recursion relations for sums of $q$-hypergeometric terms.
Key words: WZ Algorithm, Creative Telescoping, Colored Jones Function, Gosper's Algorithm, Cyclotomic Function, holonomic functions, characteristic varieties, $A$-polynomial, $C$-polynomial.
Notes: 19 pages, 1 figure.