On the Tutte-Krushkal-Renardy polynomial for cell complexes
On the Tutte-Krushkal-Renardy polynomial for cell complexes
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关于细胞复合体的 Tutte-Krushkal-Renardy 多项式
DOI：
10.1016/j.jcta.2013.12.006
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
S. Chmutov
中科院分区：
文献类型：
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作者：
Carlos Bajo;B. Burdick;S. Chmutov
最近，V. Krushkal和D. Renardy将Tutte多项式从图推广到细胞复合体。我们证明了在原点对这个多项式求值可以得到A. Duval， C. Klivans和J. Martin意义上的细胞生成树的个数。此外，在稍作修改后，Tutte-Krushkal-Renardy多项式在原点处求值给出了一个加权的元胞生成树计数，因此它的自由项可以由Duval等人的元胞矩阵-树定理计算。在球的元分解情况下，这个修正多项式满足与原多项式相同的对偶恒等式。我们发现沿某条直线对Tutte-Krushkal-Renardy求值可以得到Bott多项式。最后证明了Tutte-Krushkal-Renardy多项式的绞结关系。
Recently V. Krushkal and D. Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A. Duval, C. Klivans, and J. Martin. Moreover, after a slight modification, the Tutte–Krushkal–Renardy polynomial evaluated at the origin gives a weighted count of cellular spanning trees, and therefore its free term can be calculated by the cellular matrix-tree theorem of Duval et al. In the case of cell decompositions of a sphere, this modified polynomial satisfies the same duality identity as the original polynomial. We find that evaluating the Tutte–Krushkal–Renardy along a certain line gives the Bott polynomial. Finally we prove skein relations for the Tutte–Krushkal–Renardy polynomial.