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Wigner symbols and combinatorial invariants of three-manifolds with boundary

机译：带边界的三流形的维格纳符号和组合不变量

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In this paper we generalize the partition function proposed by Ponzano and Regge in 1968 to the case of a compact 3-dimensional simplicial pair (M, partial derivative M). The resulting state sum Z[(M, partial derivative M)] contains both Wigner 6 j symbols associated with tetrahedra and Wigner 3 jm symbols associated with triangular faces lying in partial derivative M. In order to show the invariance of Z[(M, partial derivative M)] under PL-homeomorphisms we exploit some results due to Pachner on the equivalence of n-dimensional PL-pairs both under bistellar moves on n-simplices in the interior of M and under elementary boundary operations (shellings and inverse shellings) acting on n-simplices which have some component in partial derivative M. We find, in particular, the algebraic identities - involving a suitable number of Wigner symbols - which realize the complete set of Pachner's boundary operations in n = 3. The results established for the classical SU (2)-invariant Z[(M, partial derivative M)] are further extended to the case of the quantum enveloping algebra U-q (sl (2, (C)) (q a root of unity). The corresponding quantum invariant, Z(q)[(M, partial derivative M)], turns out to be the counterpart of the Turaev-Viro invariant for a closed 3-dimensional PL-manifold. [References: 25]
机译：在本文中，我们将Ponzano和Regge于1968年提出的分区函数推广为一个紧凑的3维单纯形对（M，偏导数M）的情况。结果状态总和Z [（M，偏导数M）]包含与四面体相关的Wigner 6 j符号和与位于偏导数M中的三角形面相关的Wigner 3 jm符号。为了显示Z [（M， PL同胚同态下的偏导数M）]，由于Pachner在M内部的n单纯形上的双星运动和基本边界运算（脱壳和反脱壳）下的n维PL对的等价性，我们利用了一些结果作用于在偏导数M中具有某些分量的n个单形。我们特别发现代数恒等式-包括适当数量的Wigner符号-在n = 3时实现了完整的Pachner边界运算集。经典的SU（2）不变Z [（M，偏导数M）]进一步扩展到量子包络代数Uq（sl（2，（C））（单位为qa的根）的情况。 ，Z（q）[（M，部分阶导数M）]，证明它是闭合3维PL流形的Turaev-Viro不变量的对应物。 [参考：25]

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