Stavros Garoufalidis, Christoph Koutschan,
"The non-commutative A-polynomial of (-2,3,n) pretzel knots"
, in Experimental Mathematics, 2011, ISSN: 1058-6458
Original Titel:
The non-commutative A-polynomial of (-2,3,n) pretzel knots
Sprache des Titels:
Englisch
Original Kurzfassung:
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 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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