83 research outputs found
Bridge trisections in rational surfaces
We study smooth isotopy classes of complex curves in complex surfaces from
the perspective of the theory of bridge trisections, with a special focus on
curves in and . We are
especially interested in bridge trisections and trisections that are as simple
as possible, which we call "efficient". We show that any curve in
or admits an efficient
bridge trisection. Because bridge trisections and trisections are nicely
related via branched covering operations, we are able to give many examples of
complex surfaces that admit efficient trisections. Among these are
hypersurfaces in , the elliptic surfaces , the Horikawa
surfaces , and complete intersections of hypersurfaces in
. As a corollary, we observe that, in many cases, manifolds that
are homeomorphic but not diffeomorphic have the same trisection genus, which is
consistent with the conjecture that trisection genus is additive under
connected sum. We give many trisection diagrams to illustrate our examples.Comment: 46 pages, 28 color figure
The Dihedral Genus of a Knot
Let be a Fox -colored knot and assume bounds a locally
flat surface over which the given -coloring extends. This
coloring of induces a dihedral branched cover . Its branching set
is a closed surface embedded in locally flatly away from one singularity
whose link is . When is homotopy ribbon and a definite
four-manifold, a condition relating the signature of and the Murasugi
signature of guarantees that in fact realizes the four-genus of . We
exhibit an infinite family of knots with this property, each with a {Fox
3-}colored surface of minimal genus . As a consequence, we classify the
signatures of manifolds which arise as dihedral covers of in the
above sense.Comment: 19 pages, 10 figures, 3 footnotes. Final versio
Trisections of 4-manifolds via Lefschetz fibrations
We develop a technique for gluing relative trisection diagrams of
-manifolds with nonempty connected boundary to obtain trisection diagrams
for closed -manifolds. As an application, we describe a trisection of any
closed -manifold which admits a Lefschetz fibration over equipped with
a section of square , by an explicit diagram determined by the vanishing
cycles of the Lefschetz fibration. In particular, we obtain a trisection
diagram for some simply connected minimal complex surface of general type. As a
consequence, we obtain explicit trisection diagrams for a pair of closed
-manifolds which are homeomorphic but not diffeomorphic. Moreover, we
describe a trisection for any oriented -bundle over any closed surface and
in particular we draw the corresponding diagrams for and using our gluing technique. Furthermore, we provide an
alternate proof of a recent result of Gay and Kirby which says that every
closed -manifold admits a trisection. The key feature of our proof is that
Cerf theory takes a back seat to contact geometry.Comment: 34 pages, 21 figure
Bridge trisections of knotted surfaces in 4--manifolds
We prove that every smoothly embedded surface in a 4--manifold can be
isotoped to be in bridge position with respect to a given trisection of the
ambient 4--manifold; that is, after isotopy, the surface meets components of
the trisection in trivial disks or arcs. Such a decomposition, which we call a
\emph{generalized bridge trisection}, extends the authors' definition of bridge
trisections for surfaces in . Using this new construction, we give
diagrammatic representations called \emph{shadow diagrams} for knotted surfaces
in 4--manifolds. We also provide a low-complexity classification for these
structures and describe several examples, including the important case of
complex curves inside . Using these examples, we prove that
there exist exotic 4--manifolds with --trisections for certain values of
. We conclude by sketching a conjectural uniqueness result that would
provide a complete diagrammatic calculus for studying knotted surfaces through
their shadow diagrams.Comment: 17 pages, 5 figures. Comments welcom
Trisections and spun 4-manifolds
We study trisections of 4-manifolds obtained by spinning and twist-spinning
3-manifolds, and we show that, given a (suitable) Heegaard diagram for the
3-manifold, one can perform simple local modifications to obtain a trisection
diagram for the 4-manifold. We also show that this local modification can be
used to convert a (suitable) doubly-pointed Heegaard diagram for a
3-manifold/knot pair into a doubly-pointed trisection diagram for the
4-manifold/2-knot pair resulting from the twist-spinning operation.
This technique offers a rich list of new manifolds that admit trisection
diagrams that are amenable to study. We formulate a conjecture about
4-manifolds with trisection genus three and provide some supporting evidence.Comment: 16 pages, 12 figures. Comments welcome
Algorithms for l-sections on genus two curves over finite fields and applications
We study \ell-section algorithms for Jacobian of genus two over finite fields. We provide trisection (division by
\ell=3) algorithms for Jacobians of genus 2 curves over finite fields \F_q of odd and even characteristic. In odd
characteristic we obtain a symbolic trisection polynomial whose roots correspond (bijectively) to the set of
trisections of the given divisor. We also construct a polynomial whose roots allow us to calculate the 3-torsion
divisors. We show the relation between the rank of the 3-torsion subgroup and the factorization of this 3-torsion
polynomial, and describe the factorization of the trisection polynomials in terms of the galois structure of the 3-
torsion subgroup. We generalize these ideas and we determine the field of definition of an \ell-section with \ell \in {3,
5, 7}. In characteristic two for non-supersingular hyperelliptic curves we characterize the 3-torsion divisors and
provide a polynomial whose roots correspond to the set of trisections of the given divisor. We also present a
generalization of the known algorithms for the computation of the 2-Sylow subgroup to the case of the \ell-Sylow
subgroup in general and we present explicit algorithms for the computation of the 3-Sylow subgroup. Finally we
show some examples where we can obtain the central coefficients of the characteristic polynomial of the Frobenius
endomorphism reduced modulo 3 using the generators obtained with the 3-Sylow algorithm.En esta tesis se estudian algoritmos de \ell-división para Jacobianas de curvas de género 2. Se presentan algoritmos
de trisección (división por \ell=3) para Jacobianas de curvas de género 2 definidas sobre cuerpos finitos \F_q de
característica par o impar indistintamente. En característica impar se obtiene explícitamente un polinomio de
trisección, cuyas raíces se corresponden biyectivamente con el conjunto de trisecciones de un divisor cualquiera de la
Jacobiana. Asimismo se proporciona otro polinomio a partir de cuyas raíces se calcula el conjunto de los divisores de
orden 3. Se muestra la relación entre el rango del subgrupo de 3-torsión y la factorización del polinomio de la 3-
torsión, y se describe la factorización del polinomio de trisección en términos de las órbitas galoisianas de la 3-
torsión. Se generalizan estas ideas para otros valores de \ell y se determina el cuerpo de definición de una \ell-sección
para \ell=3,5,7. Para curvas no-supersingulares en característica par también se da una caracterización de la 3-torsión
y se proporciona un polinomio de trisección para un divisor cualquiera. Se da una generalización, para \ell arbitraria,
de los algoritmos conocidos para el cómputo explícito del subgrupo de 2-Sylow, y se detalla explícitamente el
algoritmo para el cómputo del subgrupo de 3-Sylow. Finalmente, se dan ejemplos de cómo obtener los valores de la
reducción módulo 3 de los coeficientes centrales del polinomio característico del endomorfismo de Frobenius
mediante los generadores proporcionados por el algoritmo de cálculo del 3-Sylow.En aquesta tesi s'estudien algoritmes de \ell-divisió per a grups de punts de Jacobianes de corbes de gènere 2. Es
presenten algoritmes de trisecció (divisió per \ell=3) per a Jacobianes de corbes de gènere 2 definides sobre cossos
finits \F_q de característica parell o senar indistintament. En característica parell s'obté explícitament un polinomi de
trisecció, les arrels del qual estan en bijecció amb el conjunt de triseccions d'un divisor de la Jacobiana qualsevol. De
manera semblant, es proporciona un altre polinomi amb les arrels del qual es calcula el conjunt dels divisors d'ordre
3. Es mostra la relació entre el rang del subgrup de 3-torsió i la factorització del polinomi de la 3-torsió, i es descriu
la factorització del polinomi de trisecció en termes de les òrbites galoisianes de la 3-torsió. Es generalitzen aquestes
idees a altres valors de \ell i es determina el cos de definició d'una \ell-secció per a \ell=3,5,7. Per a corbes nosupersingulars
en característica 2 també es proporciona una caracterització de la 3-torsió i un polinomi de trisecció
per a un divisor qualsevol. Es dóna una generalització, per a \ell arbitrària, dels algoritmes coneguts per al càlcul
explícit del subgrup de 2-Sylow, i es detalla explícitament en el cas del 3-Sylow. Finalment es mostren exemples de
com obtenir els valors de la reducció mòdul 3 dels coeficients centrals del polinomi característic de l'endomorfisme
de Frobenius fent servir els generadors proporcionats per l'algoritme de càlcul del 3-Sylow
Compact hyperbolic manifolds without spin structures
We exhibit the first examples of compact orientable hyperbolic manifolds that
do not have any spin structure. We show that such manifolds exist in all
dimensions . The core of the argument is the construction of a
compact orientable hyperbolic -manifold that contains a surface of
genus with self intersection . The -manifold has an odd
intersection form and is hence not spin. It is built by carefully assembling
some right angled -cells along a pattern inspired by the minimum
trisection of . The manifold is also the first
example of a compact orientable hyperbolic -manifold satisfying any of these
conditions: 1) is not generated by geodesically immersed
surfaces. 2) There is a covering that is a non-trivial bundle over
a compact surface.Comment: 23 pages, 16 figure
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