Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userState graphs and fibered state surfaces
137
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such planar graph is a checkerboard surface, and each state surface is a fiber if and only if all of its associated checkerboard surfaces are fibers. We give an algebraic condition that characterizes which checkerboard surfaces are fibers directly from their state graphs. We use this to classify fibering of checkerboard surfaces for several families of planar graphs, including those associated with 2-bridge links. This characterizes fibering for many families of state surfaces.
rate research
Read More
By work of Uhlenbeck, the largest principal curvature of any least area fiber of a hyperbolic $3$-manifold fibering over the circle is bounded below by one. We give a short argument to show that, along certain families of fibered hyperbolic $3$-manifolds, there is a uniform lower bound for the maximum principal curvatures of a least area minimal surface which is greater than one.
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface (we call these admissible). There is a natural necessary condition on such graphs, which we call combinatorial admissibility. Our first main result is that this condition is also sufficient. It follows that a sub-graph of an admissible graph is admissible. Our second major result is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).
We associate to triangulations of infinite type surface a type of flip graph where simultaneous flips are allowed. Our main focus is on understanding exactly when two triangulations can be related by a sequence of flips. A consequence of our results is that flip graphs for infinite type surfaces have uncountably many connected components.
An embedding of a metric graph $(G, d)$ on a closed hyperbolic surface is emph{essential}, if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_e(G)$ of $(G, d)$ is the lowest genus of a surface on which such an embedding is possible. In the next result, we establish a formula to compute $g_e(G)$. Furthermore, we show that for every integer $ggeq g_e(G)$, $(G, d)$ admits such an embedding (possibly after a rescaling of $d$) on a surface of genus $g$. Next, we study minimal embeddings where each complementary region has Euler characteristic $-1$. The maximum essential genus $g_e^{max}(G)$ of $(G, d)$ is the largest genus of a surface on which the graph is minimally embedded. Finally, we describe a method explicitly for an essential embedding of $(G, d)$, where $g_e(G)$ and $g_e^{max}(G)$ are realized.
We study finite type invariants of nullhomologous knots in a closed 3-manifold $M$ defined in terms of certain descending filtration ${mathscr{K}_n(M)}_{ngeq 0}$ of the vector space $mathscr{K}(M)$ spanned by isotopy classes of nullhomologous knots in $M$. The filtration ${mathscr{K}_n(M)}_{n geq 0}$ is defined by surgeries on special kinds of claspers in $M$ having one special leaf. More precisely, when $M$ is fibered over $S^1$ and $H_1(M)=mathbb{Z}$, we study how far the natural surgery map from the space of $mathbb{Q}[t^{pm 1}]$-colored Jacobi diagrams on $S^1$ of degree $n$ to the graded quotient $mathscr{K}_n(M)/mathscr{K}_{n+1}(M)$ can be injective for $nleq 2$. To do this, we construct a finite type invariant of nullhomologous knots in $M$ up to degree 2 that is an analogue of the invariant given in our previous paper arXiv:1503.08735, which is based on Lescops construction of $mathbb{Z}$-equivariant perturbative invariant of 3-manifolds.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
