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Prime component-preservingly amphicheiral link with odd minimal crossing number. (English) Zbl 1373.57019

Let \(L\) be an oriented \(r\)-component link in \(S^3\). If there exists an orientation-reversing self-homeomorphism of \(S^3\) that sends each component of \(L\) into another component of \(L\), possibly the same component, with the same or opposite orientation by a permutation, then the link \(L\) is called an amphicheiral link. Especially, if the permutation is the identity, then the amphicheiral link is called a component-preservingly amphicheiral link.
In this paper, the authors state and prove as their main result a theorem which provides a counterexample to A. Stoimenow’s conjecture from [Bull. Am. Math. Soc., New Ser. 45, No. 2, 285–291 (2008; Zbl 1141.57005)], or to what the authors call a generalized version of Tait’s conjecture IV: The minimal crossing number of an amphicheiral link is even.
The main theorem is as follows. Theorem 1.3. For every odd integer \(c > 21\), there exists a prime component-preservingly amphicheiral link with minimal crossing number \(c\).
Actually, A. Kawauchi and the first author [Sci. China, Math. 54, No. 10, 2213–2227 (2011; Zbl 1234.57006)] already found two counterexamples to the conjecture when \(c\leq 13\). However, they are not component-preservingly amphicheiral. So, the new result gives us an answer to the question about the existence of a component-preservingly amphicheiral link with odd crossing number.
The authors construct a prime component-preservingly amphicheiral link which is a 2-component link with linking number \(3\) whose components are a Stoimenow knot and an unknot. The authors use a wonderful technique to construct the example using the Kauffman bracket. In order to show the minimality of the crossing, they refer to A. Stoimenow’s result [J. Reine Angew. Math. 657, 1–55 (2011; Zbl 1257.57010)]. Moreover, the authors show that the Stoimenow knot is not invertible by using the Alexander polynomial.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)

Software:

Knot Atlas
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Full Text: arXiv Euclid

References:

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