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On Essential and Inessential Polygons in Embedded Graphs

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. ...

##
###
Knotted lattice-like space fillers

1995
*
Discrete & Computational Geometry
*

The aim of this note is to construct a solid torus, knotted

doi:10.1007/bf02574062
fatcat:con7ts6plzgibak2ukay5tfkiu
*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. ...##
###
Regular projective polyhedra with planar faces I

2000
*
Aequationes Mathematicae
*

polyhedra,

doi:10.1007/pl00000128
fatcat:vbvprn3tbneqxjntbaqivq5b4e
*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. ...##
###
Alternating Quadrisecants of Knots
[article]

2005
*
arXiv
*
pre-print

Comparing the order of the four points along the line

arXiv:math/0510561v1
fatcat:nynrcafpijg5jfwg2kdujzo6gq
*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. ...##
###
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]

2017
*
New Directions in Geometric and Applied Knot Theory
*

A quadrisecant line is

doi:10.1515/9783110571493-007
fatcat:3cshle6pb5hnpkk5dxkxu3tet4
*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. ...##
###
Cantor sets in 3-manifolds

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. ...

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

2016
*
arXiv
*
pre-print

A quadrisecant line is

arXiv:1608.02608v1
fatcat:mtpoii6fxbbenmdaqp5tixupty
*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*. ...##
###
Re-embedding structures of 5-connected projective-planar graphs

2002
*
Discrete Mathematics
*

We shall identify the structures of 5-connected projective-planar

doi:10.1016/s0012-365x(01)00084-x
fatcat:l3crzexqfjg4fmthqcw7dufvuq
*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 . ...##
###
Quadrisecants give new lower bounds for the ropelength of a knot

2006
*
Geometry and Topology
*

Our work was partially supported by the National Science Foundation through grants DMS-03-10562 (Diao)

doi:10.2140/gt.2006.10.1
fatcat:bb2dtg65wbh7fbhe3jsfzhp5hy
*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*. ...##
###
A table of n-component handlebody links of genus n+1 up to six crossings
[article]

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 ∂Â is

*essential*

*in*∂ HL 1

*and*the other

*inessential*by the irreducibility of HL. Suppose both components are

*inessential*

*in*∂ HL 1 . ...

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

2007
*
arXiv
*
pre-print

Generalizing Milnor's result that an FTC (finite total curvature) knot has an isotopic inscribed

arXiv:math/0606008v2
fatcat:n7l7whljvzf4pa6rdmwvzjzczm
*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. ...##
###
Surface bundles versus Heegaard splittings

2005
*
Communications in analysis and geometry
*

As we shall see

doi:10.4310/cag.2005.v13.n5.a3
fatcat:2l7bwqmanbbcxc6vawrrisupju
*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). ...##
###
On Spaces of Inscribed Triangles
[article]

2020
*
arXiv
*
pre-print

Our result comes from taking the limit of a structural result about spaces of triangles inscribed

arXiv:1908.08174v2
fatcat:oa2um74e65e2pcklzz5uuggt5a
*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. ...##
###
Surface bundles versus Heegaard splittings
[article]

2002
*
arXiv
*
pre-print

We prove that the translation distance is bounded above

arXiv:math/0212104v1
fatcat:zm4osux24jaepenvfbcq7rp6xu
*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. ...
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