×

Arithmetic knots in closed 3-manifolds. (English) Zbl 1023.57008

From the introduction: Let \(O_d\) denote the ring of integers in \(\mathbb{Q}(\sqrt{-d})\). An orientable finite volume cusped hyperbolic 3-manifold \(M\) is called arithmetic if the faithful discrete representation of \(\pi_1(M)\) into \(\text{PSL}(2,\mathbb{C})\) is conjugate to a group commensurable with some Bianchi group \(\text{PSL}(2,O_d)\). If \(M\) is a closed orientable 3-manifold, we say a link \(L\subset M\) is arithmetic if \(M\setminus L\) is arithmetic. Since the figure-eight knot complement is well-known to be arithmetic and universal, it follows that every closed orientable 3-manifold contains an arithmetic link. In the case of \(S^3\), the figure-eight knot is the only arithmetic knot. A natural question therefore is: does every closed orientable 3-manifold contain an arithmetic knot? Our main result is the following:
Theorem 1.1. There exist closed orientable 3-manifolds which do not contain an arithmetic knot.
Our methods give much more precise versions of Theorem 1.1, particularly for non-hyperbolic 3-manifolds. We show:
Theorem 1.2. Let \(L\) be a Lens space, and assume that \(L\) contains a knot \(K\) derived from a quaternion algebra then \(L\setminus K\) is homeomorphic to the sister of the figure eight knot complement or the double cover of the figure eight knot complement. Using standard properties about invariant trace fields, we have the following corollary of Theorem 1.2 providing many examples for Theorem 1.1.
Corollary 1.3. Let \(L\) be a Lens space, with \(|\pi_1(L)|= r\) of odd order. If \(r\neq 5\), then \(L\) does not contain an arithmetic knot.
The assumption on odd order is necessary, since \(\mathbb{R}\mathbb{P}^3\) does contain an arithmetic knot.
Little seems to be known about the set of 1-cusped arithmetic hyperbolic 3-manifolds. We suspect that they are very rare. However, at present it is unknown whether there are only finitely many commensurability classes of 1-cusped arithmetic hyperbolic 3-manifolds. Some discussion of what is known is contained in §§5 and 6.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
57M40 Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1016/0040-9383(95)00040-2 · Zbl 0863.57009 · doi:10.1016/0040-9383(95)00040-2
[2] DOI: 10.1007/BF01444486 · Zbl 0830.57008 · doi:10.1007/BF01444486
[3] DOI: 10.1090/S0894-0347-96-00201-9 · Zbl 0936.57010 · doi:10.1090/S0894-0347-96-00201-9
[4] DOI: 10.1090/S0002-9947-1984-0728710-2 · doi:10.1090/S0002-9947-1984-0728710-2
[5] DOI: 10.1080/10586458.2000.10504641 · Zbl 1002.57044 · doi:10.1080/10586458.2000.10504641
[6] DOI: 10.1007/BF01404671 · Zbl 0646.53037 · doi:10.1007/BF01404671
[7] DOI: 10.2307/2154257 · Zbl 0773.20017 · doi:10.2307/2154257
[8] DOI: 10.1016/0040-9383(85)90019-9 · Zbl 0582.57002 · doi:10.1016/0040-9383(85)90019-9
[9] DOI: 10.1007/s002220000047 · Zbl 0947.57016 · doi:10.1007/s002220000047
[10] Reid A. W., J. L. M. S 43 pp 171– (1991)
[11] DOI: 10.1017/S0305004100051094 · Zbl 0309.55002 · doi:10.1017/S0305004100051094
[12] DOI: 10.1016/0001-8708(71)90027-2 · Zbl 0221.20060 · doi:10.1016/0001-8708(71)90027-2
[13] DOI: 10.1007/BF01455567 · Zbl 0553.20023 · doi:10.1007/BF01455567
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.