264 research outputs found
Symplectic homology and the Eilenberg-Steenrod axioms
We give a definition of symplectic homology for pairs of filled Liouville
cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod
axioms except for the dimension axiom. The resulting long exact sequence of a
pair generalizes various earlier long exact sequences such as the handle
attaching sequence, the Legendrian duality sequence, and the exact sequence
relating symplectic homology and Rabinowitz Floer homology. New consequences of
this framework include a Mayer-Vietoris exact sequence for symplectic homology,
invariance of Rabinowitz Floer homology under subcritical handle attachment,
and a new product on Rabinowitz Floer homology unifying the pair-of-pants
product on symplectic homology with a secondary coproduct on positive
symplectic homology.
In the appendix, joint with Peter Albers, we discuss obstructions to the
existence of certain Liouville cobordisms.Comment: v3: corrected Lemma 7.11. Various other minor modifications and
reformatting. Final version to be published in Algebraic and Geometric
Topolog
Maxwell's Theory of Solid Angle and the Construction of Knotted Fields
We provide a systematic description of the solid angle function as a means of
constructing a knotted field for any curve or link in . This is a
purely geometric construction in which all of the properties of the entire
knotted field derive from the geometry of the curve, and from projective and
spherical geometry. We emphasise a fundamental homotopy formula as unifying
different formulae for computing the solid angle. The solid angle induces a
natural framing of the curve, which we show is related to its writhe and use to
characterise the local structure in a neighborhood of the knot. Finally, we
discuss computational implementation of the formulae derived, with C code
provided, and give illustrations for how the solid angle may be used to give
explicit constructions of knotted scroll waves in excitable media and knotted
director fields around disclination lines in nematic liquid crystals.Comment: 20 pages, 9 figure
- …