论文标题
转向三角形的多项式不变
A polynomial invariant for veering triangulations
论文作者
论文摘要
我们在\ mathbb {z}中引入了多项式不变$v_τ\ [h_1（m）/\ text {torsion}] $与$ 3 $ -manifold $ m $的veering三角剖分$τ$相关。在三角剖分分层的特殊情况下，即来自纤维化，$v_τ$恢复了纤维面的Teichmüller多项式与$τ$相关。通过Dehn Filling，这给出了Teichmüller多项式的组合描述。 对于一般的旋转三角剖分$τ$，我们表明$τ$所携带的表面确定同源性的圆锥，该锥与其正闭合横向的锥双重。此外，我们证明这是thurston norm Ball（通常是非纤维）面上（通常是无纤维的）面向锥体的$ \ textit {等} $，并且$τ$从精确的意义上计算出该锥体的标准。我们还对$ \ textit {flow Graph} $ for $τ$及其perron多项式提供了$v_τ$的组合描述。这种观点使我们能够表征何时旋转三角剖分，而更普遍地计算出由$τ$确定的瑟斯顿标准的面。
We introduce a polynomial invariant $V_τ\in \mathbb{Z}[H_1(M)/\text{torsion}]$ associated to a veering triangulation $τ$ of a $3$-manifold $M$. In the special case where the triangulation is layered, i.e. comes from a fibration, $V_τ$ recovers the Teichmüller polynomial of the fibered faces canonically associated to $τ$. Via Dehn filling, this gives a combinatorial description of the Teichmüller polynomial for any hyperbolic fibered $3$-manifold. For a general veering triangulation $τ$, we show that the surfaces carried by $τ$ determine a cone in homology that is dual to its cone of positive closed transversals. Moreover, we prove that this is $\textit{equal}$ to the cone over a (generally non-fibered) face of the Thurston norm ball, and that $τ$ computes the norm on this cone in a precise sense. We also give a combinatorial description of $V_τ$ in terms of the $\textit{flow graph}$ for $τ$ and its Perron polynomial. This perspective allows us to characterize when a veering triangulation comes from a fibration, and more generally to compute the face of the Thurston norm determined by $τ$.
