The non-commutative $A$-polynomial of $(-2,3,n)$ pretzel knots

Details The non-commutative $A$-polynomial of $(-2,3,n)$ pretzel knots The non-commutative $A$-polynomial of $(-2,3,n)$ pretzel knots

2011-01-14

arxiv.org/abs/1101.2844v3

Mathematics

Geometric Topology

12 pages, 8 figures

Scientific paper

We study $q$-holonomic sequences that arise as the colored Jones polynomial of knots in 3-space. The minimal-order recurrence for such a sequence is called the (non-commutative) $A$-polynomial of a knot. Using the \emph{method of guessing}, we obtain this polynomial explicitly for the $K_p=(-2,3,3+2p)$ pretzel knots for $p=-5,...,5$. This is a particularly interesting family since the pairs $(K_p,-K_{-p})$ are geometrically similar (in particular, scissors congruent) with similar character varieties. Our computation of the non-commutative $A$-polynomial (a) complements the computation of the $A$-polynomial of the pretzel knots done by the first author and Mattman, (b) supports the AJ Conjecture for knots with reducible $A$-polynomial and (c) numerically computes the Kashaev invariant of pretzel knots in linear time. In a later publication, we will use the numerical computation of the Kashaev invariant to numerically verify the Volume Conjecture for the above mentioned pretzel knots.

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