Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA note on the existence of essential tribranched surfaces
154
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The second author and Hara introduced the notion of an essential tribranched surface that is a generalisation of the notion of an essential embedded surface in a 3-manifold. We show that any 3-manifold for which the fundamental group has at least rank four admits an essential tribranched surface.
rate research
Read More
Extending Culler-Shalen theory, Hara and the second author presented a way to construct certain kinds of branched surfaces in a $3$-manifold from an ideal point of a curve in the $operatorname{SL}_n$-character variety. There exists an essential surface in some $3$-manifold known to be not detected in the classical $operatorname{SL}_2$-theory. We prove that every connected essential surface in a $3$-manifold is given by an ideal point of a rational curve in the $operatorname{SL}_n$-character variety for some $n$.
We present a practical algorithm to test whether a 3-manifold given by a triangulation or an ideal triangulation contains a closed essential surface. This property has important theoretical and algorithmic consequences. As a testament to its practicality, we run the algorithm over a comprehensive body of closed 3-manifolds and knot exteriors, yielding results that were not previously known. The algorithm derives from the original Jaco-Oertel framework, involves both enumeration and optimisation procedures, and combines several techniques from normal surface theory. Our methods are relevant for other difficult computational problems in 3-manifold theory, such as the recognition problem for knots, links and 3-manifolds.
Checkerboard surfaces in alternating link complements are used frequently to determine information about the link. However, when many crossings are added to a single twist region of a link diagram, the geometry of the link complement stabilizes (approaches a geometric limit), but a corresponding checkerboard surface increases in complexity with crossing number. In this paper, we generalize checkerboard surfaces to certain immersed surfaces, called twisted checkerboard surfaces, whose geometry better reflects that of the alternating link in many cases. We describe the surfaces, show that they are essential in the complement of an alternating link, and discuss their properties, including an analysis of homotopy classes of arcs on the surfaces in the link complement.
We show that if K is a non-trivial knot inside a homology sphere Y, then the rank of knot Floer homology associated with K is strictly bigger than the rank of Heegaard Floer homology of Y.
In this note, we revisit the $Theta$-invariant as defined by R. Bott and the first author. The $Theta$-invariant is an invariant of rational homology 3-spheres with acyclic orthogonal local systems, which is a generalization of the 2-loop term of the Chern-Simons perturbation theory. The $Theta$-invariant can be defined when a cohomology group is vanishing. In this note, we give a slightly modified version of the $Theta$-invariant that we can define even if the cohomology group is not vanishing.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
