Coloring link diagrams by Alexander quandles
Yongju Bae

TL;DR
This paper investigates how link diagrams can be colored using Alexander quandles, revealing conditions under which non-trivial colorings exist based on the Alexander polynomial of the link.
Contribution
It establishes new criteria linking the Alexander polynomial to the existence of non-trivial Alexander quandle colorings of links.
Findings
If the Alexander polynomial is zero, non-trivial colorings exist for any non-trivial Alexander quandle.
If the Alexander polynomial equals one, only trivial colorings are possible.
For non-zero, non-one Alexander polynomials, non-trivial colorings exist in a specific quotient Alexander quandle.
Abstract
In this paper, we study the colorability of link diagrams by the Alexander quandles. We show that if the reduced Alexander polynomial is vanishing, then admits a non-trivial coloring by any non-trivial Alexander quandle , and that if , then admits only the trivial coloring by any Alexander quandle , also show that if , then admits a non-trivial coloring by the Alexander quandle .
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Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Combinatorial Mathematics · Algebraic structures and combinatorial models
