Leading-term conjecture for the Jones polynomial of alternating braids

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Let LL be a positive integer, let a1,…,a2La_1,\ldots,a_{2L} satisfy ai≥2a_i\geq 2, and set D=a1+⋯+a2LD=a_1+\cdots+a_{2L}. For the braid word x1a1x2a2…x1a2L−1x2a2Lx_1^{a_1}x_2^{a_2}\dots x_1^{a_{2L-1}}x_2^{a_{2L}}, write V3V_3 for the corresponding Jones polynomial in the variable ss. Leading-term conjecture. The leading term of

V3(x1a1x2a2…x1a2L−1x2a2L)V_3(x_1^{a_1}x_2^{a_2}\dots x_1^{a_{2L-1}}x_2^{a_{2L}})

is s3D−2Ls^{3D-2L}. This predicts the maximal degree and leading coefficient for this family of positive-exponent braid words; the surrounding computations support the formula, while the more general combinatorics of maximal-degree terms is noted to be nontrivial.

References

Primary source

Barbu Berceanu and Abdul Rauf Nizami, “Recurrence relation for Jones polynomials”, arXiv:1002.3735 (2010).

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