scholarly journals Knot complements and groups

Topology ◽  
1987 ◽  
Vol 26 (1) ◽  
pp. 41-44 ◽  
Author(s):  
Wilbur Whitten
Keyword(s):  
Knot Complements

Incompressible surfaces in 2-bridge knot complements

1985 ◽  
Vol 79 (2) ◽  
pp. 225-246 ◽  
Author(s):  
A. Hatcher ◽  
W. Thurston
Keyword(s):  
Incompressible Surfaces ◽  
Knot Complements

scholarly journals Commensurability classes of 2–bridge knot complements

2008 ◽  
Vol 8 (2) ◽  
pp. 1031-1057 ◽  
Author(s):  
Alan W Reid ◽  
Genevieve S Walsh
Keyword(s):  
Knot Complements

scholarly journals Stable commutator length in amalgamated free products

2015 ◽  
Vol 07 (04) ◽  
pp. 693-717 ◽  
Author(s):  
Tim Susse
Keyword(s):  
Geometric Model ◽  
Free Products ◽  
Cyclic Groups ◽  
Amalgamated Free Products ◽  
Torus Knot ◽  
Knot Complements ◽  
Stable Commutator Length ◽  
Free Abelian Groups ◽  
Commutator Length ◽  
Boundary Mapping

We show that stable commutator length is rational on free products of free abelian groups amalgamated over ℤk, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups and parametrize all surfaces with specified boundary mapping to this space. Using this work we provide a topological algorithm to compute stable commutator length in these groups. Further, we use the methods developed to show that in free products of cyclic groups the stable commutator length of a fixed word varies quasirationally in the orders of the free factors.

scholarly journals Closed incompressible surfaces in knot complements

1999 ◽  
Vol 352 (2) ◽  
pp. 655-677 ◽  
Author(s):  
Elizabeth Finkelstein ◽  
Yoav Moriah
Keyword(s):  
Incompressible Surfaces ◽  
Knot Complements

SEIFERT SURFACES IN KNOT COMPLEMENTS

2007 ◽  
Vol 16 (08) ◽  
pp. 1053-1066 ◽  
Author(s):  
ENSIL KANG
Keyword(s):  
Compact Surface ◽  
Seifert Surface ◽  
Normal Surface ◽  
Incompressible Surface ◽  
Ideal Triangulation ◽  
Seifert Surfaces ◽  
Knot Complements

In the ordinary normal surface for a compact 3-manifold, any incompressible, ∂-incompressible, compact surface can be moved by an isotopy to a normal surface [9]. But in a non-compact 3-manifold with an ideal triangulation, the existence of a normal surface representing an incompressible surface cannot be guaranteed. The figure-8 knot complement is presented in a counterexample in [12]. In this paper, we show the existence of normal Seifert surface under some restriction for a given ideal triangulation of the knot complement.

scholarly journals Knot complements, hidden symmetries and reflection orbifolds

2015 ◽  
Vol 24 (5) ◽  
pp. 1179-1201
Author(s):  
Michel Boileau ◽  
Steven Boyer ◽  
Radu Cebanu ◽  
Genevieve Walsh
Keyword(s):  
Hidden Symmetries ◽  
Knot Complements

scholarly journals Volume function and Mahler measure of exact polynomials

2021 ◽  
Vol 157 (4) ◽  
pp. 809-834
Author(s):  
Antonin Guilloux ◽  
Julien Marché
Keyword(s):  
Mahler Measure ◽  
Two Dimensional ◽  
Closed Formula ◽  
Dimensional Torus ◽  
Volume Function ◽  
Topological Interpretation ◽  
Local Extrema ◽  
Knot Complements ◽  
Extremal Values

We study a class of two-variable polynomials called exact polynomials which contains $A$ -polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the two-dimensional torus and give a closed formula for the Mahler measure in terms of these extremal values. This formula shows that the Mahler measure of an irreducible and exact polynomial divided by $\pi$ is greater than the amplitude of the volume function. We also prove a K-theoretic criterion for a polynomial to be a factor of an $A$ -polynomial and give a topological interpretation of its Mahler measure.

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