Alexander quandle lower bounds for link genera (Q2885407)

Every finite field \(\mathbb{F}_q\), \(q = p^n\), carries several Alexander quandle structures \(\mathbb{X} = (\mathbb{F}_q,\ast)\). The authors denote by \(\mathcal{Q_F}\) the family of these quandles, where \(p\) and \(n\) vary respectively among the odd primes and the positive integers. For every \(k\)-component oriented link \(L\), every partition \(\mathcal{P}\) of \(L\) into \(h := |\mathcal{P}|\) sublinks, and every labeling \(\bar{z} \in \mathbb{N}^h\) of such a partition, the number of \(\mathbb{X}\)-colorings of any diagram of \((L,\bar{z})\) is a well-defined invariant of \((L,\mathcal{P})\), of the form \(q^{a_{\chi}(L,\mathcal{P},\bar{z})+1}\) for some natural number \(a_{\chi}(L,\mathcal{P},\bar{z})\).NEWLINENEWLINELetting \(\mathbb{X}\) and \(\bar{z}\) vary respectively in \(\mathcal{Q_F}\) and among the labelings of \(\mathcal{P}\), define the derived invariant \(\mathcal{A_Q}(L, \mathcal{P}) := \sup\{a_{\chi}(L,\mathcal{P},\bar{z})\}\). If \(\mathcal{P}_M\) is such that \(|\mathcal{P}_M| = k\), the authors show that NEWLINE\[NEWLINE\mathcal{A_Q}(L, \mathcal{P}_M) \leq t(L),NEWLINE\]NEWLINE where \(t(L)\) is the tunnel number of \(L\), generalizing a result by \textit{A. Ishii} [Algebr. Geom. Topol. 8, No. 3, 1403--1418 (2008; Zbl 1151.57007)]. If \(\mathcal{P}\) is a ``boundary partition'' of \(L\) and \(g(L,\mathcal{P})\) denotes the infimum among the sums of the genera of a system of disjoint Seifert surfaces for the \(L_j\)'s, then the following is shown: NEWLINE\[NEWLINE\mathcal{A_Q}(L, \mathcal{P}) \leq 2g(L, \mathcal{P}) + 2k - |\mathcal{P}| - 1.NEWLINE\]NEWLINENEWLINENEWLINEThe authors show that when \(L = K\) is a knot then NEWLINE\[NEWLINE\mathcal{A_Q}(K) \leq \mathcal{A}(K),NEWLINE\]NEWLINE where \(\mathcal{A}(K)\) is the breadth of the Alexander polynomial of \(K\). However, for every \(g \geq 1\) examples of genus-\(g\) knots having the same Alexander polynomial but different quandle invariants \(\mathcal{A_Q}\) are shown. Moreover, in such examples \(\mathcal{A_Q}\) provides sharp lower bounds for the genera of the knots. On the other hand, the authors show that \(\mathcal{A_Q}(L)\) can give better lower bounds on the genus than \(\mathcal{A}(L)\), when \(L\) has \(k \geq 2\) components.