Fox Trapezoidal Conjecture for alternating knots

Let KK be an alternating knot with normalized Alexander polynomial

ΔK(T)=i=ggaiTi,\Delta_K(T)=\sum_{i=-g}^g a_iT^i,

where gg is the genus of KK. Fox Trapezoidal Conjecture. The coefficients satisfy

aiai1when 0<ig.|a_i|\le |a_{i-1}| \quad\text{when }0<i\le g.

Moreover, if ai=ai1|a_i|=|a_{i-1}| for some ii, then aj=ai|a_j|=|a_i| whenever 0ji0\le j\le i. This conjecture predicts a trapezoidal pattern for the absolute values of the Alexander polynomial coefficients of alternating knots; the paper proves a special case in which ag=ag1|a_g|=|a_{g-1}|, but the full statement remains open in the supplied source.

Sources & referencesView supporting material

Primary source

Yi Ni, “A characterization of T_2g+1,2 among alternating knots”, arXiv:2007.14629 (2020).

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