Cluster algebras and Jones polynomials

-Deformed Rationals and -Continued Fractions

“…Recently, the Jones polynomial was computed using the combinatorics of snake graphs and cluster algebras [22], and in the the Kauffman bracket form it was computed from the combinatorics of Stern-Brocot tree [21]. The combinatorics used in these papers is closely related to that of continued fractions, extensively used in the present paper.…”

“…It can be easily rewritten under the form (1.1). With this observation, we reformulate the result of [22] in terms of q-rationals and q-continuants.…”

“…Proof. Theorem 1.2 in [22] says that the Jones polynomial J r s (q) can be computed as the numerator of a continued fraction. In terms of the continuant the formula is…”

“…Remark B.7. F -polynomials can be used to compute Jones polynomials of rational knots; see [22]. Let us consider weights (x i,j ) on the triangulation T r/s satisfying the Ptolemy relations (B.1) and the same initial assignment (B.2) except for the weight on the side [0, n − 1] that we change to x 0,n−1 = q −1 .…”

“…where as above F 0,k+1 is the F -polynomial corresponding to the longest cluster variable in A(x, y, B r/s ). It was proved in [22] that the above specialization of F 0,k+1 gives precisely the normalized Jones polynomial J r s (q). For instance for r s = 8 3 = [2, 1, 1, 1] = 3, 3 the corresponding graph is 1 → 2 ← 3 → 4, and we compute F 03 = 1 + y 1 + (1 + y 1 + y 1 y 2 )y 3 + ((1 + y 1 + y 1 y 2 )y 3 )y 4…”