scholarly journals Profinite Rigidity, Fibering, and the Figure-Eight Knot

What's Next? ◽  
2020 ◽  
pp. 45-64
Author(s):  
Martin R. Bridson ◽  
Alan W. Reid
Keyword(s):  
Figure Eight Knot

scholarly journals HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations

2012 ◽  
Vol 2012 (7) ◽  
Author(s):  
H. Itoyama ◽  
A. Mironov ◽  
A. Morozov ◽  
And. Morozov
Keyword(s):  
Figure Eight Knot

scholarly journals Constructing 1-cusped isospectral non-isometric hyperbolic 3-manifolds

2017 ◽  
Vol 10 (01) ◽  
pp. 1-25
Author(s):  
Stavros Garoufalidis ◽  
Alan W. Reid
Keyword(s):  
Discrete Spectrum ◽  
Eisenstein Series ◽  
Finite Volume ◽  
Finite Covers ◽  
Figure Eight Knot ◽  
Complex Length

We construct infinitely many examples of pairs of isospectral but non-isometric [Formula: see text]-cusped hyperbolic [Formula: see text]-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an application of Sunada’s method in the cusped setting, and so in addition our pairs are finite covers of the same degree of a 1-cusped hyperbolic 3-orbifold (indeed manifold) and also have the same complex length spectra. Finally we prove that any finite volume hyperbolic 3-manifold isospectral to the figure-eight knot complement is homeomorphic to the figure-eight knot complement.

scholarly journals Volumes and Degeneration of Cone-structures on the Figure-eight Knot

2006 ◽  
Vol 29 (2) ◽  
pp. 445-464 ◽  
Author(s):  
Alexander MEDNYKH ◽  
Alexey RASSKAZOV
Keyword(s):  
Figure Eight Knot

scholarly journals On coverings of figure eight knot surgeries

1991 ◽  
Vol 150 (2) ◽  
pp. 215-228 ◽  
Author(s):  
Mark Baker
Keyword(s):  
Figure Eight Knot

Roll-spun knots

1993 ◽  
Vol 113 (1) ◽  
pp. 91-96 ◽  
Author(s):  
Masakazu Teragaito
Keyword(s):  
Precise Definition ◽  
General Process ◽  
Spinning Process ◽  
Definition Of ◽  
Figure Eight Knot ◽  
Twist Spinning

In this paper we will study 2-knots in the 4-sphere S4 which are obtained by roll-spinning 1-knots in the 3-sphere S3. The process of spinning was introduced by Artin[2] and generalized to twist-spinning by Fox [7] and Zeeman [20]. In [7] Fox introduced another variation of the spinning process, called roll-spinning. He only showed that the roll-spun figure-eight knot cannot be obtained by twist-spinning the figure-eight knot. Finally Litherland [14] described the general process of deform-spinning, and he gave a precise definition of roll-spinning. We remark that Fox's roll-spinning is not the same as Litherland's: Fox's roll-spun figure-eight knot is the symmetry-spun figure-eight knot in terms of [14].

A quadratic parabolic group

1975 ◽  
Vol 77 (2) ◽  
pp. 281-288 ◽  
Author(s):  
Robert Riley
Keyword(s):  
Normal Form ◽  
Half Space ◽  
Transformation Group ◽  
Universal Covering ◽  
Covering Space ◽  
Fundamental Domains ◽  
Universal Covering Space ◽  
Other Information ◽  
Figure Eight Knot

When k is a 2-bridge knot with group πK, there are parabolic representations (p-reps) θ: πK → PSL(): = PSL(2, ). The most obvious problem that this suggests is the determination of a presentation for an image group πKθ. We shall settle the easiest outstanding case in section 2 below, viz. k the figure-eight knot 41, which has the 2-bridge normal form (5, 3). We shall prove that the (two equivalent) p-reps θ for this knot are isomorphisms of πK on πKθ. Furthermore, the universal covering space of S3\k can be realized as Poincaré's upper half space 3, and πKθ is a group of hyperbolic isometries of 3 which is also the deck transformation group of the covering 3 → S3\k. The group πKθ is a subgroup of two closely related groups that we study in section 3. We shall give fundamental domains, presentations, and other information for all these groups.

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