Principal congruence link complements

Abstract

International audienceCet article est consacré à un début d’énumération des compléments d’entrelacs dans S 3 provenant des groupes de congruence principaux. Nous utilisons des méthodes théoriques ainsi que des calculs avec MAGMA

All principal congruence link groups

“…Since systole length grows with the norm of the ideal this provides the necessary control. Moreover, when the class number is 1, all ideals are principal and the argument above bounds the absolute value of a generator of the ideal, and when the class number is greater than 1, this bounds the absolute value of some x ∈ I. Summarizing this discussion we obtain (see [BR14,Section 4.1] and [BR17, Lemma 4.1]):…”

“…Concerning the case when h d = 1 (i.e. d ∈ {1, 2, 3, 7, 11, 19}), certain examples already existed in the literature (see [BR14]), and using the 6-Theorem as described in §2 to restrict the possible levels, we subsequently gave 9 new examples of principal congruence link groups in [BR14]. This brought the total known when h d = 1 to 18:…”

“…In this section, we review some earlier work that was used in [BR14], [BR17], and [Goe15] to produce a finite list of potential pairs (d, I).…”

“…To pass from finitely many values of d to finitely many possible pairs (d, I) the norm of the ideal I needs to be bounded. To achieve this, we follow the arguments of [BR14] and [BR17]. We note in passing that a different combinatorial method was used in [Goe15] in his analysis of the cases of d = 1, 3.…”

“…There are only finitely principal congruence manifolds that are link complements in S 3 and the complete list was given in [BGR18]. We refer the reader to [BR18] and [BR14] for an introduction to and further background on congruence manifolds. This report serves as a repository of link diagrams for the cases (d, I) for which such a diagram is known.…”