Arithmeticity of Knot Complements

Reid, Alan W.

doi:10.1112/jlms/s2-43.1.171

Cited by 82 publications

(83 citation statements)

References 19 publications

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“…Moreover in [5] it was shown that for every such d there does exist a link complement. We summarize this in the following result: Although there is a unique arithmetic knot complement in S 3 (the figureeight knot complement, [27]), it is easy to prove that there are infinitely many non-homeomorphic arithmetc link complements in S 3 , even fixing links with 2 components (see for example [27]). …”

Section: 2mentioning

confidence: 99%

Principal congruence link complements

Baker¹,

Reid²

2014

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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Section: 2mentioning

confidence: 99%

Principal congruence link complements

Baker¹,

Reid²

2014

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…This 3-dimensional figure of the link 9 2 24 [Ro76] was made by M. Veve. This link is interesting because A. Hatcher and A. Reid [Re91] showed that its complement has a hyperbolic structure and the group of the knot is an arithmetic group (subgroup of PSL 2 (O 2 ), where O 2 is the ring of integers in the imaginary quadratic number field Q( √ −2)). …”

Section: Corollary 38 (I) ([Bhjs94])mentioning

confidence: 99%

Lissajous knots and billiard knots

Jones¹,

Przytycki²

1998

Banach Center Publ.

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Abstract. We show that Lissajous knots are equivalent to billiard knots in a cube. We consider also knots in general 3-dimensional billiard tables. We analyse symmetry of knots in billiard tables and show in particular that the Alexander polynomial of a Lissajous knot is a square modulo 2. 0. Introduction. A Lissajous knot K is a knot in R 3 given by the parametric equationsfor integers η x , η y , η z . A Lissajous link is a collection of disjoint Lissajous knots. The fundamental question was asked in [BHJS94]: which knots are Lissajous? One defines a billiard knot (or racquetball knot ) as the trajectory inside a cube of a ball which leaves a wall at rational angles with respect to the natural frame, and travels in a straight line except for reflecting perfectly off the walls; generically it will miss the corners and edges, and will form a knot. We will show that these knots are precisely the same as the Lissajous knots. We will also speculate about more general billiard knots, e.g. taking another polyhedron instead of the ball, considering a non-Euclidean metric, or considering the trajectory of a ball in the configuration space of a flat billiard. We will illustrate these by various examples. For instance, the trefoil knot is not a Lissajous knot 1991 Mathematics Subject Classification: 57M25, 58F17. This is an extended version of the talk given in August 1995, at the minisemester on Knot Theory at the Banach Center.We would like to acknowledge the support from USAF grant 1-443964-22502. The paper is in final form and no version of it will be published elsewhere.[145]

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“…For example, given an orbifold with a rigid cusp C, then any horospherical cross-section of a finite covering of C has Euclidean modulus in Q( √ −1) or Q( √ −3) (see [13] or [16]). It is not hard to deduce from this (see [16]) that the invariant trace-field of such a manifold contains Q( √ −1) or Q( √ −3). With this we obtain the following.…”

Section: Theorem 36 Suppose That M Is a Non-arithmetic 1-cusped Hypmentioning

confidence: 99%