The cyclic cycle complex of a surface

Digraphs and cycle polynomials for free-by-cyclic groups

Let $ϕ\in \mbox{Out}(F_n)$ be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism $ϕ$ determines a free-by-cyclic group $Γ=F_n \rtimes_ϕ\mathbb Z,$ and a homomorphism $α\in H^1(Γ; \mathbb Z)$. By work of Neumann, Bieri-Neumann-Strebel and Dowdall-Kapovich-Leininger, $α$ has an open cone neighborhood $\mathcal A$ in $H^1(Γ;\mathbb R)$ whose integral points correspond to other fibrations of $Γ$ whose associated outer automorphisms are themselves representable by expanding irreducible train-track maps. In this paper, we define an …

Cyclic branched coverings of knots and quandle homology

We give a construction of quandle cocycles from group cocycles, especially, for any integer p \geq 3, quandle cocycles of the dihedral quandle R_p from group cocycles of the cyclic group Z/p. We will show that a group 3-cocycle of Z/p gives rise to a non-trivial quandle 3-cocycle of R_p. When p is an odd prime, since dim_{F_p} H_Q^3(R_p; F_p) = 1, our 3-cocycle is a constant multiple of the Mochizuki 3-cocycle up to coboundary. Dually, we construct a group cycle represented by a cyclic branched covering branched along a knot K from the quandle cycle associated with a colored diagram of K.

Equivelar and d-Covered Triangulations of Surfaces. II. Cyclic Triangulations and Tessellations

With the $[0,1,2]$-family of cyclic triangulations we introduce a rich class of vertex-transitive triangulations of surfaces. In particular, there are infinite series of cyclic $q$-equivelar triangulations of orientable and non-orientable surfaces for every $q=3k$, $k\geq 2$, and every $q=3k+1$, $k\geq 3$. Series of cyclic tessellations of surfaces are derived from these triangulated series.

Systoles of hyperbolic surfaces with big cyclic symmetry

We obtain the exact values of the systoles of these hyperbolic surfaces of genus $g$ with cyclic symmetries of the maximum order and the next maximum order. Precisely: for genus $g$ hyperbolic surface with order $4g+2$ cyclic symmetry, the systole is $2\mathrm{arccosh} (1+\cos \fracπ{2g+1}+\cos \frac{2π}{2g+1})$ when $g\ge 7$, and for genus $g$ hyperbolic surface with order $4g$ cyclic symmetry, the systole is $2\mathrm{arccosh} (1+2\cos \fracπ{2g})$ when $g\ge4$.

来自曲面的循环对象

Cyclic Objects from Surfaces

该研究为亏格g≥1的闭曲面族赋予三维配边范畴中的（余）循环对象结构，推导得出三维TQFT可诱导（余）循环模，并完成Reshetikhin-Turaev TQFT对应模的代数计算。