Multiplicativity conjecture for the multivariable Alexander polynomial of annulus-summed swatches

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Let L1⊂T2×IL_1\subset T^2\times I and L2⊂T2×IL_2\subset T^2\times I be an m1×nm_1\times n swatch and an m2×nm_2\times n swatch, respectively, where m1,m2,n∈{1,2,…}m_1,m_2,n\in\{1,2,\ldots\}. Let HH be the Hopf link and let m⊂Hm\subset H be the specified component, with associated variable t1t_1. The links H∪f(L1)H\cup f(L_1), H∪f(L2)H\cup f(L_2), and H∪f(L1∗mL2)H\cup f(L_1*_mL_2) each have n+2n+2 components. Multivariable Alexander-polynomial conjecture. The multivariable Alexander polynomial satisfies

ΔH∪f(L1∗mL2)=ΔH∪f(L1) ΔH∪f(L2)(t1−1)n.\Delta_{H\cup f(L_1*_mL_2)}=\frac{\Delta_{H\cup f(L_1)}\,\Delta_{H\cup f(L_2)}}{(t_1-1)^n}.

Here t1t_1 is associated with the component m⊂Hm\subset H. This conjecture is proposed from computed data as a possible algebraic rule for meridional annulus sums and as a tool for classifying weft-knitted stitch patterns; no proof or resolution is supplied.

References

Primary source

Miriam Kuzbary, Shashank G. Markande, Elisabetta A. Matsumoto and Stanley Pritchard, “A study of 2-periodic weft-knitted textiles using the theory of knots and links”, arXiv:2510.08384 (2025).

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