arXiv subjects

Aoi Wakuda

Publications and source records attributed to Aoi Wakuda.

3 recordsLinked to original sources

arXiv2025

Separability criteria for loops via the Goldman bracket

We provide some explicit algebraic criteria in terms of the Goldman bracket to decide whether two free homotopy classes of loops on an oriented surface admit disjoint representatives. We extend Kabiraj's method using the hyperbolic geometry of surfaces to prove these criteria. As an application, we show that the center of the Goldman Lie algebra of a pair of pants is generated by the class of the constant loop together with the classes of loops that wind multiple times around a single puncture or boundary component. This case was not covered by Kabiraj, since a pair of pants is not filled by simple closed curves.

math.GT↗

arXiv2025

Puncture loops on a non-orientable surface

On a connected surface $N$ with negative Euler characteristic, the free homotopy class of a loop obtained by smoothing an intersection of two closed geodesics may wind around a puncture. Chas and Kabiraj showed that this phenomenon does not occur when the surface $N$ is orientable. In this paper, we prove that it occurs when $N$ is non-orientable and both geodesics involved in the smoothing are actually one-sided. In particular, we study a loop obtained by traversing a one-sided closed geodesic and the $m$-th power of another one-sided closed geodesic for odd $m$. Then we show that its free homotopy class may wind aroud a puncture at most two values of $m$. Furthermore, if two such $m$'s exist, they are consecutive odd integers.

math.GT↗

arXiv2024

A generalization of the Center Theorem of the Thurston-Wolpert-Goldman Lie algebra

The Goldman Lie algebra of an oriented surface was defined by Goldman. By the natural involution that opposes the orientation of curves, the Goldman Lie algebra becomes a $\mathbb{Z}_{2}$-graded Lie algebra. Its even part is isomorphic to the Thurston-Wolpert-Goldman Lie algebra or, briefly, the TWG Lie algebra. Chas and Kabiraj proved the center of the TWG Lie algebra is generated by the class of the unoriented trivial loop and the classes of unoriented loops parallel to boundary components or punctures. The center of the even part can be rephrased as the set of elements of the even part annihilated by all the elements of the even part. We also prove some similar statements for the remaining 3 cases involving the odd part. Moreover, we compute the elements of the symmetric algebra and the universal enveloping algebra of the Goldman Lie algebra annihilated by all the even elements of the Goldman Lie algebra, and those annilated by all the odd elements.

math.GT↗