论文标题
基于理想的三角形的封闭框架$ 3 $ manifolds的量子不变
Quantum invariants of closed framed $3$-manifolds based on ideal triangulations
论文作者
论文摘要
我们构建了一种封闭框架$ 3 $ manifolds的新型量子不变，而第一个betti数字消失了。不变的定义是针对任何有限的尺寸HOPF代数，例如小量子组，并且基于理想的三角剖分。我们使用Heisenberg Double的规范元素，该元素满足五角大楼方程，以及R. Benedetti和C. Petronio引入的$ 3 $ manifolds的图形表示。构造简单易用，很容易直观地理解。五角大楼方程反映了理想三角形的Pachner $（2,3）$的移动，而HOPF代数的无效性则反映了框架。对于一个不变的Hopf代数，不变式将封闭的梳子$ 3 $ manifolds的不变式减少。对于一个不可测量的圆形HOPF代数，不变的降低到了封闭的$ 3 $ manifolds的拓扑不变，这是我们上一篇论文中引入的。在本文中，我们在对称关键类别中使用更多的HOPF MONOID对结构进行形式化，并使用张量网络进行计算。
We construct a new type of quantum invariant of closed framed $3$-manifolds with the vanishing first Betti number. The invariant is defined for any finite dimensional Hopf algebra, such as small quantum groups, and is based on ideal triangulations. We use the canonical element of the Heisenberg double, which satisfies a pentagon equation, and graphical representations of $3$-manifolds introduced by R. Benedetti and C. Petronio. The construction is simple and easy to be understood intuitively; the pentagon equation reflects the Pachner $(2,3)$ move of ideal triangulations and the non-involutiveness of the Hopf algebra reflects framings. For an involutory Hopf algebra, the invariant reduces to an invariant of closed combed $3$-manifolds. For an involutory unimodular counimodular Hopf algebra, the invariant reduces to the topological invariant of closed $3$-manifolds which is introduced in our previous paper. In this paper we formalize the construction using more generally a Hopf monoid in a symmetric pivotal category and use tensor networks for calculations.
