ON ARCHIMEDEAN LINK COMPLEMENTS

2002 ◽  
Vol 11 (06) ◽  
pp. 833-868 ◽  
Author(s):  
IAIN R. AITCHISON ◽  
LAWRENCE D. REEVES
Keyword(s):  
Hyperbolic Metric ◽  
Archimedean Solids ◽  
Link Complements ◽  
Hyperbolic Polyhedra ◽  
Figure Eight Knot ◽  
Alternating Links

We study a subclass of alternating links for which the complete hyperbolic metric can be realised directly by pairwise identification of faces of two ideal hyperbolic polyhedra. Our main result is a characterization of these links: essentially, the corresponding polyhedra are exactly the Archimedean solids with trivalent vertices. Furthermore, we show that the only knots which arise are the two dodecahedral knots, and the figure eight knot.

scholarly journals Topological flows for hyperbolic groups

2020 ◽  
pp. 1-47
Author(s):  
RYOKICHI TANAKA
Keyword(s):  
Hausdorff Dimension ◽  
Radon Measure ◽  
Hyperbolic Group ◽  
Hyperbolic Metric ◽  
Hyperbolic Groups ◽  
Group Invariant ◽  
Hyperbolic Metrics ◽  
Measure Class

Abstract Weshow that for every non-elementary hyperbolic group the Bowen–Margulis current associated with a strongly hyperbolic metric forms a unique group-invariant Radon measure class of maximal Hausdorff dimension on the boundary square. Applications include a characterization of roughly similar hyperbolic metrics via mean distortion.

scholarly journals A GEOMETRIC CHARACTERIZATION OF THE UPPER BOUND FOR THE SPAN OF THE JONES POLYNOMIAL

2011 ◽  
Vol 20 (07) ◽  
pp. 1059-1071
Author(s):  
JUAN GONZÁLEZ-MENESES ◽  
PEDRO M. G. MANCHÓN
Keyword(s):  
Upper Bound ◽  
Knot Theory ◽  
Jones Polynomial ◽  
Geometric Proof ◽  
Link Diagram ◽  
Closed Curves ◽  
Number Of Faces ◽  
Alternating Links ◽  
Connected Diagram

Let D be a link diagram with n crossings, sA and sB be its extreme states and |sAD| (respectively, |sBD|) be the number of simple closed curves that appear when smoothing D according to sA (respectively, sB). We give a general formula for the sum |sAD| + |sBD| for a k-almost alternating diagram D, for any k, characterizing this sum as the number of faces in an appropriate triangulation of an appropriate surface with boundary. When D is dealternator connected, the triangulation is especially simple, yielding |sAD| + |sBD| = n + 2 - 2k. This gives a simple geometric proof of the upper bound of the span of the Jones polynomial for dealternator connected diagrams, a result first obtained by Zhu [On Kauffman brackets, J. Knot Theory Ramifications6(1) (1997) 125–148.]. Another upper bound of the span of the Jones polynomial for dealternator connected and dealternator reduced diagrams, discovered historically first by Adams et al. [Almost alternating links, Topology Appl.46(2) (1992) 151–165.], is obtained as a corollary. As a new application, we prove that the Turaev genus is equal to the number k of dealternator crossings for any dealternator connected diagram.

scholarly journals Characterization of quasi-alternating Montesinos links

2015 ◽  
Vol 24 (01) ◽  
pp. 1550002 ◽  
Author(s):  
K. Qazaqzeh ◽  
N. Chbili ◽  
B. Qublan
Keyword(s):  
Knot Theory ◽  
Infinite Family ◽  
Rational Tangles ◽  
Alternating Links ◽  
Alternating Link

Let L be a quasi-alternating link at a crossing c. We construct an infinite family of quasi-alternating links from L by replacing the crossing c by a product of rational tangles, each of which extends c. Consequently, we determine an infinite family of quasi-alternating Montesinos links. This family includes all classes of quasi-alternating Montesinos links that have been detected by Widmer [Quasi-alternating Montesinos links, J. Knot Theory Ramifications18(10) (2009) 1459–1469]. We conjecture that this family contains all non-alternating quasi-alternating Montesinos links.

scholarly journals The Dehn surgery characterization of the trefoil and the figure eight knot

2019 ◽  
Vol 17 (1) ◽  
pp. 251-265
Author(s):  
Peter Ozsváth ◽  
Zoltán Szabó
Keyword(s):  
Dehn Surgery ◽  
Figure Eight Knot

scholarly journals QUASI-ALTERNATING MONTESINOS LINKS

2009 ◽  
Vol 18 (10) ◽  
pp. 1459-1469 ◽  
Author(s):  
TAMARA WIDMER
Keyword(s):  
Open Problem ◽  
Floer Homology ◽  
Natural Generalization ◽  
Khovanov Homology ◽  
Knot Floer Homology ◽  
Interesting Open Problem ◽  
Alternating Links

The aim of this article is to detect new classes of quasi-alternating links. Quasi-alternating links are a natural generalization of alternating links. Their knot Floer and Khovanov homology are particularly easy to compute. Since knot Floer homology detects the genus of a knot as well as whether a knot is fibered, characterization of quasi-alternating links becomes an interesting open problem. We show that there exist classes of non-alternating Montesinos links, which are quasi-alternating.

scholarly journals A characterization of alternating links in thickened surfaces

2021 ◽  
pp. 1-19
Author(s):  
Hans U. Boden ◽  
Homayun Karimi
Keyword(s):  
Opposite Sign ◽  
Bilinear Pairing ◽  
Compact Surface ◽  
Topological Characterization ◽  
Definite Form ◽  
Minimal Genus ◽  
Thickened Surface ◽  
Alternating Links ◽  
Closed Oriented Surface

We use an extension of Gordon–Litherland pairing to thickened surfaces to give a topological characterization of alternating links in thickened surfaces. If $\Sigma$ is a closed oriented surface and $F$ is a compact unoriented surface in $\Sigma \times I$ , then the Gordon–Litherland pairing defines a symmetric bilinear pairing on the first homology of $F$ . A compact surface in $\Sigma \times I$ is called definite if its Gordon–Litherland pairing is a definite form. We prove that a link $L$ in a thickened surface is non-split, alternating, and of minimal genus if and only if it bounds two definite surfaces of opposite sign.

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