Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGeometric triangulations of a family of hyperbolic 3-braids
145
0
0.0
(
0
)
Authors
Barbara Nimershiem
Ask ChatGPT about the research
No Arabic abstract
We construct topological triangulations for complements of $(-2,3,n)$-pretzel knots and links with $nge7$. Following a procedure outlined by Futer and Gueritaud, we use a theorem of Casson and Rivin to prove the constructed triangulations are geometric. Futer, Kalfagianni, and Purcell have shown (indirectly) that such braids are hyperbolic. The new result here is a direct proof.
rate research
Read More
We classify 3-braids arising from collision-free choreographic motions of 3 bodies on Lissajous plane curves, and present a parametrization in terms of levels and (Christoffel) slopes. Each of these Lissajous 3-braids represents a pseudo-Anosov mapping class whose dilatation increases when the level ascends in the natural numbers or when the slope descends in the Stern-Brocot tree. We also discuss 4-symbol frieze patterns that encode cutting sequences of geodesics along the Farey tessellation in relation to odd continued fractions of quadratic surds for the Lissajous 3-braids.
In this paper, we explore minimal contact triangulations on contact 3-manifolds. We give many explicit examples of contact triangulations that are close to minimal ones. The main results of this article say that on any closed oriented 3-manifold the number of vertices for minimal contact triangulations for overtwisted contact structures grows at most linearly with respect to the relative $d^3$ invariant. We conjecture that this bound is optimal. We also discuss, in great details, contact triangulations for a certain family of overtwisted contact structures on 3-torus.
It is well known that a triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is neighbourly and orientable. No such characterization of tightness was previously known for higher dimensional manifolds. In this paper, we prove that a triangulation of a closed 3-manifold is tight with respect to a field of odd characteristic if and only if it is neighbourly, orientable and stacked. In consequence, the K{u}hnel-Lutz conjecture is valid in dimension three for fields of odd characteristic. Next let $mathbb{F}$ be a field of characteristic two. It is known that, in this case, any neighbourly and stacked triangulation of a closed 3-manifold is $mathbb{F}$-tight. For triangulated closed 3-manifolds with at most 71 vertices or with first Betti number at most 188, we show that the converse is true. But the possibility of an $mathbb{F}$-tight non-stacked triangulation on a larger number of vertices remains open. We prove the following upper bound theorem on such triangulations. If an $mathbb{F}$-tight triangulation of a closed 3-manifold has $n$ vertices and first Betti number $beta_1$, then $(n-4)(617n- 3861) leq 15444beta_1$. Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra.
It is conjectured that every cusped hyperbolic 3-manifold admits a geometric triangulation, i.e. it is decomposed into positive volume ideal hyperbolic tetrahedra. Here, we show that sufficiently highly twisted knots admit a geometric triangulation. In addition, by extending work of Gueritaud and Schleimer, we also give quantifi
A key result in computational 3-manifold topology is that any two triangulations of the same 3-manifold are connected by a finite sequence of bistellar flips, also known as Pachner moves. One limitation of this result is that little is known about the structure of this sequences; knowing more about the structure could help both proofs and algorithms. Motivated by this, we show that there must be a sequence that satisfies a rigid property that we call semi-monotonicity. We also study this result empirically: we implement an algorithm to find such semi-monotonic sequences, and compare their characteristics to less structured sequences, in order to better understand the practical and theoretical utility of this result.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
