arXiv subjects

Xifeng Jin

Publications and source records attributed to Xifeng Jin.

5 recordsLinked to original sources

arXiv2023

Super efficiency of efficient geodesics in the complex of curves

We show that efficient geodesics have the strong property of "super efficiency". For any two vertices, $v , w \in \mathcal{C}(S_g)$, in the complex of curves of a closed oriented surface of genus $g \geq 2 $, and any efficient geodesic, $v = v_1 , \cdots , v_{\text d}=w$, it was previously established by Birman, Margalit and the second author (see arXiv:1408.4133) that there is an explicitly computable list of at most ${\text d}^{(6g-6)}$ candidates for the $v_1$ vertex. In this note we establish a bound for this computable list that is independent of ${\text d}$-distance and only dependent on genus -- the super efficiency property. The proof relies on a new intersection growth inequality between intersection number of curves and their distance in the complex of curves, together with a thorough analysis of the dot graph associated with the intersection sequence.

math.GT↗

arXiv2021

Origami edge-paths in the curve graph

An "origami" (or flat structure) on a closed oriented surface, $S_g$, of genus $g \geq 2$ is obtained from a finite collection of unit Euclidean squares by gluing each right edge to a left one and each top edge to a bottom one. The main objects of study in this note are "origami pairs of curves" -- filling pairs of simple closed curves, $ (α,β)$, in $S_g$ such that their minimal intersection is equal to their algebraic intersection -- they are "coherent". An origami pair of curves is naturally associated with an origami on $S_g$. Our main result establishes that for any origami pair of curves there exists an "origami edge-path", a sequence of curves, $α=α_0, α_1, α_2, \cdots, α_n = β$, such that: $α_i$ intersects $α_{i+1}$ at exactly once; any pair $(α_i, α_j)$ is coherent; and thus, any filling pair, $(α_i, α_j)$, is also an origami. With their existence established, we offer shortest origami edge-paths as an area of investigation.

math.GT↗

arXiv2021

Heegaard distance of the link complements in $S^3$

We show that, for any integers, $g \geq 3$ and $n \geq 2$, there exists a link in $S^3$ such that its complement has a genus $g$ Heegaard splitting with distance $n$.

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arXiv2020

Bounded geodesic image theorem via bicorn curves

We give a uniform bound of the bounded geodesic image theorem for the closed oriented surfaces. The proof utilizes the bicorn curves introduced by Przytycki and Sisto (see arXiv:1502.02176). With the uniformly bounded Hausdorff distance of the bicorn paths and 1-slimness of the bicorn curve triangles, we are able to show the bound is 44 for both non-annular and annular subsurfaces. In a particular case when the distance between a geodesic and an essential boundary component of subsurface (or core if it is annular) is $\geq 18$, then the bound can be as small as 3, which is comparable to the bound 4 in the motivating examples by Masur and Minsky (see arXiv:9807150), and is same as the bound given by Webb for non-annular subsurfaces.

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arXiv2013

On Homology Roses and the D(2)-problem

For a commutative ring $R$ with a unit, an $R$-homology rose is a topological space whose homology groups with $R$-coefficients agree with those of a bouquet of cirlces. In this paper, we study some special properties of covering spaces and fundamental groups of $R$-homology roses, from which we obtain some result supporting the Carlsson conjecture on free $(Z_p)^r$ actions. In addition, for a group $G$ and a field $F$, we define an integer called the $F$-gap of $G$, which is an obstruction for $G$ to be realized as the fundamental group of a 2-dimensional $F$-homology rose. Furthermore, we discuss how to search candidates of the counterexamples of Wall's D(2)-problem among $F$-homology roses and $F$-acyclic spaces.

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