The Alexander polynomial vanishing conjecture for even-component amphicheiral links

Let L=K1KrL=K_1\cup\cdots\cup K_r be an rr-component link, and let

ΔL=ΔL(t1,,tr)Λr:=Z[t1±1,,tr±1],\Delta_L=\Delta_L(t_1,\ldots,t_r)\in \Lambda_r:=\mathbb{Z}[t_1^{\pm1},\ldots,t_r^{\pm1}],

where tit_i corresponds to a meridian of KiK_i. Suppose that LL is algebraically split and component-preservingly amphicheiral, and that rr is even. Alexander polynomial vanishing conjecture. Then

ΔL=0.\Delta_L=0.

The conjecture gives a necessary condition for Alexander polynomials of algebraically split component-preservingly amphicheiral links. The source reports a partial affirmative answer, while the general assertion remains unresolved.

Sources & referencesView supporting material

Primary source

Teruhisa Kadokami, “Amphicheiral links with special properties, I”, arXiv:1107.0377 (2011).

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