Prime component-preservingly amphicheiral link with odd minimal crossing number (Q294169)

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 \textit{amphicheiral} link. Especially, if the permutation is the identity, then the amphicheiral link is called a \textit{component-preservingly} amphicheiral link. NEWLINENEWLINEIn this paper, the authors state and prove as their main result a theorem which provides a counterexample to \textit{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. NEWLINENEWLINEThe 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 \(c\). NEWLINENEWLINEActually, \textit{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. NEWLINENEWLINEThe 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 \textit{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.