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On Essential and Inessential Polygons in Embedded Graphs

R.Bruce Richter, R.P. Vitray
2002 Journal of combinatorial theory. Series B (Print)  
In this article, we present a number of results of the following type: A given subgraph of an embedded graph either is embedded in a disc or it has a face chain containing a non-contractible closed path  ...  Our main application is to prove that any two faces of a 4-representative embedding are simultaneously contained in a disc bounded by a polygon.  ...  ACKNOWLEDGMENTS This research was conducted in part while the first author visited the second at Rollins College in the fall of 1992.  ... 
doi:10.1006/jctb.2001.2065 fatcat:ewjqlokgkfdlrbezr4zroli3ze

Knotted lattice-like space fillers

W. Kuperberg
1995 Discrete & Computational Geometry  
The aim of this note is to construct a solid torus, knotted in an arbitrarily given knot type, which tiles E3 by a lattice.  ...  More generally, a lattice-like space filler is constructed that is a handlebody of arbitrary genus, whose handles are arbitrarily knotted and arbitrarily linked with each other, and, furthermore, through  ...  Every embedding of a (finite) graph in ~ 3 is topologically equivalent to an embedding of the graph in I • T.  ... 
doi:10.1007/bf02574062 fatcat:con7ts6plzgibak2ukay5tfkiu

Regular projective polyhedra with planar faces I

J. L. Arocha, J. Bracho, L. Montejano
2000 Aequationes Mathematicae  
polyhedra, and conclude with the case in which the polyhedron is an embedded surface.  ...  Here, we develop the basic notions; we introduce a new diophantine trigonometric equation, which plays a key role in the classification theorem, relating the combinatorial and geometric parameters of such  ...  Javier Bracho thanks the funding of DGAPA-UNAM and for hospitality of UMASS-Amherst.  ... 
doi:10.1007/pl00000128 fatcat:vbvprn3tbneqxjntbaqivq5b4e

Alternating Quadrisecants of Knots [article]

E. Denne
2005 arXiv   pre-print
Comparing the order of the four points along the line and knot we can distinguish three types of quadrisecants; the alternating ones have the most relevance for the geometry of a knot.  ...  In this paper we prove that every (nontrivial tame) knot has an alternating quadrisecant. This result had applications to the total curvature, second hull and ropelength of knots.  ...  Sullivan for suggesting this problem and for his endless support and advice. Thanks also to S.B. Alexander, R.L. Bishop and J. Canteralla for their corrections and many helpful suggestions.  ... 
arXiv:math/0510561v1 fatcat:nynrcafpijg5jfwg2kdujzo6gq

Page 7041 of Mathematical Reviews Vol. , Issue 2002J [page]

2002 Mathematical Reviews  
Bruce (3-CARL-SMS; Ottawa, ON); Vitray, R. P. On essential and inessential polygons in embedded graphs. (English summary) J. Combin. Theory Ser. B 84 (2002), no. 1, 190-117.  ...  Then G contains a polygon which is noncontractible and separates ~ into two nontrivial parts. (ii) Let G be an r-representative embedding in an orientable surface of genus at least 2.  ... 

Quadrisecants and essential secants of knots with applications to the geometry of knots [chapter]

Elizabeth Denne, Philipp Reiter, Simon Blatt, Armin Schikorra
2017 New Directions in Geometric and Applied Knot Theory  
A quadrisecant line is one which intersects a curve in at least four points, while an essential secant captures something about the knottedness of a knot.  ...  Quadrisecants of types abcd and abdc are called simple and flipped respectively.  ...  Acknowledgment: I first worked on quadrisecants as part of my PhD thesis, and I am forever grateful to my advisor John M. Sullivan for introducing me to them.  ... 
doi:10.1515/9783110571493-007 fatcat:3cshle6pb5hnpkk5dxkxu3tet4

Cantor sets in 3-manifolds

David C. Wright
1979 Rocky Mountain Journal of Mathematics  
On the other hand if y is inessential in M -C, then y is inessential in the complement of the 1skeleton of V and hence in M -V which is contained in M -K. Punctured cells.  ...  On the other hand, if y is a loop of M -U which is inessential in M -P, y is inessential in M -N and hence in M -C. I am indebted to Ric Ancel for pointing out the following result.  ... 
doi:10.1216/rmj-1979-9-2-377 fatcat:wv26i544vfbmtjkrezhrwtdkve

Quadrisecants and essential secants of knots: with applications to the geometry of knots [article]

Elizabeth Denne
2016 arXiv   pre-print
A quadrisecant line is one which intersects a curve in at least four points, while an essential secant captures something about the knottedness of a knot.  ...  This survey article gives a brief history of these ideas, and shows how they may be applied to questions about the geometry of a knot via the total curvature, ropelength and distortion of a knot.  ...  In the knotted Θ-graph α ∪ β ∪ γ, the ordered triple (α, β, γ) is essential.  ... 
arXiv:1608.02608v1 fatcat:mtpoii6fxbbenmdaqp5tixupty

Re-embedding structures of 5-connected projective-planar graphs

Shigeru Kitakubo, Seiya Negami
2002 Discrete Mathematics  
We shall identify the structures of 5-connected projective-planar graphs which generate their inequivalent embeddings on the projective plane and show a short proof of the result, due to Kitakubo [3],  ...  that every 5-connected projective-planar nonplanar graph has exactly 1, 2, 3, 4, 6, 9 or 12 inequivalent embeddings on the projective plane.  ...  For example, each spoke of ML n in the standard embedding is essential in M and an inessential are clips o a diagonal 2-cell from M . Each bridge B for C is placed in either D or M .  ... 
doi:10.1016/s0012-365x(01)00084-x fatcat:l3crzexqfjg4fmthqcw7dufvuq

Quadrisecants give new lower bounds for the ropelength of a knot

Elizabeth Denne, Yuanan Diao, John M Sullivan
2006 Geometry and Topology  
Our work was partially supported by the National Science Foundation through grants DMS-03-10562 (Diao) and DMS-00-71520 (Sullivan and Denne), and by the UIUC Campus Research Board.  ...  Acknowledgements We extend our thanks to Stephanie Alexander, Dick Bishop, Jason Cantarella, Rob Kusner and Nancy Wrinkle for helpful conversations and suggestions.  ...  Since K i is essential, by definition, we can find an "-perturbation S 0 i of S i such that ‚ i WD K i [ S 0 i is an embedded essential ‚-graph.  ... 
doi:10.2140/gt.2006.10.1 fatcat:bb2dtg65wbh7fbhe3jsfzhp5hy

A table of n-component handlebody links of genus n+1 up to six crossings [article]

Giovanni Bellettini, Giovanni Paolini, Maurizio Paolini, Yi-Sheng Wang
2020 arXiv   pre-print
In this paper, we consider handlebody links with one genus 2 handlebody and n-1 solid tori, n>1.  ...  A handlebody link is a union of handlebodies of positive genus embedded in 3-space, which generalizes the notion of links in classical knot theory.  ...  Note also it cannot be the case that one component of ∂ is essential in ∂ HL 1 and the other inessential by the irreducibility of HL. Suppose both components are inessential in ∂ HL 1 .  ... 
arXiv:2003.03748v2 fatcat:lyqqkyzdebeb7bm6z4bgzuxzeu

Convergence and isotopy type for graphs of finite total curvature [article]

Elizabeth Denne
2007 arXiv   pre-print
Generalizing Milnor's result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy.  ...  We apply our main theorem to prove a limiting result for essential subarcs of a knot.  ...  Acknowledgments We extend our thanks to Stephanie Alexander, Dick Bishop and Dylan Thurston for their interest and helpful conversations.  ... 
arXiv:math/0606008v2 fatcat:n7l7whljvzf4pa6rdmwvzjzczm

Surface bundles versus Heegaard splittings

David Bachman, Saul Schleimer
2005 Communications in analysis and geometry  
As we shall see in the proofs of Theorems 3.1 and 6.1, essential surfaces and strongly irreducible splittings in the mapping torus M (ϕ) yield ϕ-invariant lines in the curve complex of the fibre.  ...  Here the Farey graph takes the place of the curve complex. An analysis of ϕ-invariant lines in the Farey graph allows a complete classification of incompressible surfaces in such bundles.  ...  When (θ, t) lies on such an edge the singular component Σ is a four-valent graph with one vertex, embedded in both the fibre F (θ) and the level H(t).  ... 
doi:10.4310/cag.2005.v13.n5.a3 fatcat:2l7bwqmanbbcxc6vawrrisupju

On Spaces of Inscribed Triangles [article]

Richard Evan Schwartz
2020 arXiv   pre-print
Our result comes from taking the limit of a structural result about spaces of triangles inscribed in polygons.  ...  On the left, a sequence of inessential loops converges to a proper arc of some loop. On the right, a sequence of inessential loops converges to an essential loop.  ...  The one remaining component is the one coming from Condition 1 above. It is an embedded polygon disjoint from all these.  ... 
arXiv:1908.08174v2 fatcat:oa2um74e65e2pcklzz5uuggt5a

Surface bundles versus Heegaard splittings [article]

David Bachman, Saul Schleimer
2002 arXiv   pre-print
We prove that the translation distance is bounded above in terms of the genus of any strongly irreducible Heegaard splitting.  ...  When (θ, t) lies on such an edge the singular component Σ is a four valent graph with one vertex, embedded in both the fibre F (θ) and the level H(t).  ...  As p is not an essential saddle at least one boundary component of P is inessential in its fibre. If all three are inessential then H ′′ is a twosphere.  ... 
arXiv:math/0212104v1 fatcat:zm4osux24jaepenvfbcq7rp6xu
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