The normalized Q1Q^{1} periodicity conjecture for pretzel knots

In the conjectured-properties section, let Q1(c,r)Q^1(c,r) be the first-difference quantity associated with a pretzel knot, and let X(Q1(c,r))X(Q^1(c,r)) denote the factor by which it is normalized. The parameters cc and rr are the knot parameter and symmetric-representation index, respectively.

The normalized Q1Q^1 periodicity conjecture. For the indicated values of cc and rr,

Q1(c,r)X(Q1(c,r))=Q1(c+1,r)X(Q1(c+1,r)).\frac{Q^1(c,r)}{X(Q^1(c,r))}=\frac{Q^1(c+1,r)}{X(Q^1(c+1,r))}.

The paper presents this as an unproved conjectured property, with the surrounding discussion reporting verification only for specified small parameter values.

Sources & referencesView supporting material

Primary source

William Qin, “HOMFLY Polynomials of Pretzel Knots”, arXiv:2101.00695 (2021).

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