1-dominations between Montesinos knots with at least 6 rational tangles
Let \(K_1\) and \(K_2\) be knots in the three \(3\)-sphere \(S^3\) with exteriors \(E(K_1)\) and \(E(K_2)\), respectively. We say that \(K_1\) dominates \(K_2\) if there exists a proper degree one map \(f : E(K_1) \to E(K_2)\), which is equivalent to saying that there exists a proper degree one map \(f: (S^3, K_1) \to (S^3, K_2)\) such that \(f^{-1}(K_2) = K_1\) by Property P [\textit{P. B. Kronheimer} and \textit{T. S. Mrowka}, Geom. Topol. 8, 295--310 (2004; Zbl 1072.57005)]. It is also known that if \(K_1\) dominates \(K_2\), then we have an epimorphism from the fundamental group of \(E(K_1)\) to that of \(E(K_2)\). In the paper under review, the author studies the existence of proper degree one maps between Montesinos knots. Denote by \(m(\frac{\beta_1}{\alpha_1}, \frac{\beta_2}{\alpha_2}, \cdots, \frac{\beta_s}{\alpha_s} )\) a Montesinos knot which is obtained by joining rational tangles of types \(\frac{\beta_k}{\alpha_k}\) \((\alpha_k \geq 2)\) in a circle, reading clockwise. Although a presentation such as \(m(\frac{\beta_1}{\alpha_1}, \frac{\beta_2}{\alpha_2}, \cdots, \frac{\beta_s}{\alpha_s} )\) is not unique, we can list up all the possible presentations for the given Montesinos knot [\textit{F. Bonahon}, Involutions et fibrés de Seifert dans les variétés de dimension \(3\), These de \(3\)e cycle, Orsay, (1979)]. Note also that \(m(\frac{\beta_1}{\alpha_1}, \frac{\beta_2}{\alpha_2}, \cdots, \frac{\beta_s}{\alpha_s})\) is a knot if and only if either there is exactly one even \(\alpha_k\) or there is no even \(\alpha_k\) and there is an odd number of odd \(\beta_k\). For each presentation of a Montesinos knot \(K = m(\frac{\beta_1}{\alpha_1}, \frac{\beta_2}{\alpha_2}, \cdots, \frac{\beta_s}{\alpha_s})\), the author introduces the associated labeled cycle \(C(K)\), a circle \(S^1\) on which rational numbers \(\frac{\alpha_k}{\beta_k}\) are arranged in the same order as in the presentation of \(K\). Remark that \(C(K)\) depends on a presentation of \(K\). Let \(K_1\) and \(K_2\) be Montesinos knots and assume that \(K_2\) has a presentation \(m(\frac{\beta_1}{\alpha_1}, \frac{\beta_2}{\alpha_2}, \cdots, \frac{\beta_s}{\alpha_s})\). Then the author proves that \(K_1\) dominates \(K_2\) if \(K_1\) has a presentation for which there is a degree one map from the associated labeled cycle \(C(K_1)\) to the associated labeled cycle \(C(K_2)\) satisfying \((a)\) each labeled point in \(C(K_1)\) is sent to a labeled point in \(C(K_2)\), and \((b)\) the pre-image of the point labeled by \(\frac{\beta_k}{\alpha_k}\) \((k = 1, 2, \dots, s)\) in \(C(K_2)\) is the points labeled by \(\pm \frac{\beta_k}{\alpha_k}\) in \(C(K_1)\) and the restriction of this map on a neighborhood of each point in the pre-image labeled by \(\frac{\beta_k}{\alpha_k}\) (resp. \(-\frac{\beta_k}{\alpha_k}\)) is an orientation preserving (resp. reversing) homeomorphism. Furthermore, under the assumption \(s \geq 6\), the author characterizes a proper degree one map between \(K_1\) and \(K_2\) and proves that the converse does hold.
scientific article; zbMATH DE number 5687491
scientific article