COMBINATORICS AND FOUR BRIDGED KNOTS

2001 ◽  
Vol 10 (04) ◽  
pp. 493-527 ◽  
Author(s):  
DANIEL MATIGNON
Keyword(s):  
Projective Plane ◽  
Dehn Surgery ◽  
Knot Space

The ℝ P 3-Conjecture states a non-trivial knot in S 3 cannot yield ℝ P 3 by a Dehn surgery. Generically, in the knot-space S3-N(K), the intersection of a projective plane ℝP2 in ℝ P 3, and any 2-sphere S2 in S3 pierced by K, is a 1-complex which can be viewed as a graph in either the projective plane or the 2-sphere. Gordon and Luecke have used similar graphs arising as the intersection of two 2-spheres, to prove that a knot in S3 is determined by its complement. A part of this paper concerns some new combinatorial results on these graphs. They are considered as an unavoidable step towards showing that the ℝ P 3-Conjecture is true. Moreover, we use these results to prove that any non-trivial knot that could yield ℝ P 3 has at least five bridges.

On Non-Integral Dehn Surgeries Creating Non-Orientable Surfaces

2006 ◽  
Vol 49 (4) ◽  
pp. 624-627
Author(s):  
Masakazu Teragaito
Keyword(s):  
Projective Plane ◽  
Klein Bottle ◽  
Orientable Surface ◽  
Dehn Surgery ◽  
Orientable Surfaces

AbstractFor a non-trivial knot in the 3-sphere, only integral Dehn surgery can create a closed 3-manifold containing a projective plane. If we restrict ourselves to hyperbolic knots, the corresponding claim for a Klein bottle is still true. In contrast to these, we show that non-integral surgery on a hyperbolic knot can create a closed non-orientable surface of any genus greater than two.

THERE ARE NO STRICT GREAT x-CYCLES AFTER A REDUCING OR P2 SURGERY ON A KNOT

1998 ◽  
Vol 07 (05) ◽  
pp. 549-569 ◽  
Author(s):  
JAMES A. HOFFMAN
Keyword(s):  
Projective Plane ◽  
Planar Graphs ◽  
Combinatorial Analysis ◽  
Regular Neighborhood ◽  
Dehn Surgery ◽  
Dehn Filling ◽  
The Core ◽  
Essential Step ◽  
Spherical Boundary ◽  
Cabling Conjecture

The Cabling conjecture states that Dehn surgery on a nontrivial knot K in S3 yields a reducible manifold only when K is a nontrivial cabled knot. One idea is to attack this problem with the techniques used by Gordon and Luecke in the Knot Complement Problem. This involves a combinatorial analysis of two intersecting planar graphs. In the context of reducible surgery, one of the two planar graphs will necessarily contain a Scharlemann cycle. So, we define a strict x-cycle to be any x-cycle which is not a Scharlemann cycle; likewise, for strict great x-cycles. We show that if the reducing sphere meets the core of the Dehn filling minimally, then strict great x-cycles are not permitted. Thus, strict great x-cycles can play a role similar to that of the Scharlemann cycle in the Knot Complement Problem. The obstruction of finding strict great x-cycles is considered an essential step in the program. A second conjecture states that Dehn surgery on a nontrivial knot K in S3 yields a manifold containing an embedded projective plane only when K is a nontrivial cabled knot. We show how the Gordon and Luecke technique can be applied towards this conjecture by considering the spherical boundary of a regular neighborhood of the projective plane. And if the projective plane is chosen to meet the core of the Dehn filling minimally, we show that strict great x-cycles are not permitted.

Dehn surgery on crosscap number two knots and projective planes

2002 ◽  
Vol 11 (06) ◽  
pp. 869-886
Author(s):  
Masakazu Teragaito
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Projective Planes ◽  
Dehn Surgery ◽  
The Creation

In the present paper, we will study the creation of projective planes by Dehn surgery on knots in the 3-sphere. It is shown that a projective plane cannot be created by Dehn surgery on a crosscap number two knot. As a corollary, we will prove that crosscap number two knots satisfy the projective space conjecture, which asserts that the projective 3-space cannot be obtained by Dehn surgery on a non-trivial knot in the 3-sphere.

MINIMISING THE BOUNDARIES OF PUNCTURED PROJECTIVE PLANES IN S3

2001 ◽  
Vol 10 (03) ◽  
pp. 415-430 ◽  
Author(s):  
MICHEL DOMERGUE ◽  
DANIEL MATIGNON
Keyword(s):  
Projective Plane ◽  
Solid Torus ◽  
Projective Planes ◽  
Dehn Surgery ◽  
The Core

This paper concerns 3-manifolds X obtained by Dehn surgery on a knot in S 3, in particular those which contain embedded projective planes. Either, they are homeomorphic to the 3-real projeclive space ℝ P 3, or they are reducible. Let p be the number of intersections of a projective plane in X with the core of the solid torus added during surgery. We prove here that either X is reducible or p is bigger than or equal to five. Consequently, if X is homeomorphic to ℝ P 3 then all its projective planes are pierced at least in five points by the core of the surgery. This result is considered as a step towards showing that ℝ P 3 cannot be obtained by a Dehn surgery along a knot in S 3.

Gilles Deleuze's Luminous Philosophy

2020 ◽  
Author(s):  
Hanjo Berressem
Keyword(s):  
Projective Plane ◽  
Visual Arts ◽  
Whole Body ◽  
Literary Studies ◽  
Chronological Order ◽  
Real Projective Plane ◽  
Cinema Studies ◽  
The Real

Providing a comprehensive reading of Deleuzian philosophy, Gilles Deleuze’s Luminous Philosophy argues that this philosophy’s most consistent conceptual spine and figure of thought is its inherent luminism. When Deleuze notes in Cinema 1 that ‘the plane of immanence is entirely made up of light’, he ties this philosophical luminism directly to the notion of the complementarity of the photon in its aspects of both particle and wave. Engaging, in chronological order, the whole body and range of Deleuze’s and Deleuze and Guattari’s writing, the book traces the ‘line of light’ that runs through Deleuze’s work, and it considers the implications of Deleuze’s luminism for the fields of literary studies, historical studies, the visual arts and cinema studies. It contours Deleuze’s luminism both against recent studies that promote a ‘dark Deleuze’ and against the prevalent view that Deleuzian philosophy is a philosophy of difference. Instead, it argues, it is a philosophy of the complementarity of difference and diversity, considered as two reciprocally determining fields that are, in Deleuze’s view, formally distinct but ontologically one. The book, which is the companion volume toFélix Guattari’s Schizoanalytic Ecology, argues that the ‘real projective plane’ is the ‘surface of thought’ of Deleuze’s philosophical luminism.

scholarly journals Group-labeled light dual multinets in the projective plane

2018 ◽  
Vol 341 (8) ◽  
pp. 2121-2130 ◽  
Author(s):  
Gábor Korchmáros ◽  
Gábor P. Nagy
Keyword(s):  
Projective Plane

scholarly journals On 3-syzygy and unexpected plane curves

2021 ◽  
Author(s):  
Grzegorz Malara ◽  
Piotr Pokora ◽  
Halszka Tutaj-Gasińska
Keyword(s):  
Projective Plane ◽  
Plane Curves ◽  
Complex Projective Plane ◽  
Geometric Properties ◽  
Complex Projective

AbstractIn this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties. More generally, we study 3-syzygy curve arrangements and we present examples that admit unexpected curves.

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