矩阵积分与虚缠结和虚链环的产生和计数
Matrix Integrals and the Generation and Counting of Virtual Tangles and Links
摘要：
虚链路是经典链路的推广，可以用嵌入“加厚”曲面$Σ\times I$中的链路来表示，该曲面是亏格$h$的黎曼曲面与区间的乘积。在本文中，我们表明，虚拟交替链接和缠结自然与$1/N^2$展开的积分在$N\times N$复杂的矩阵。我们建议，这是足够的计数这些图模普通（平面）苍蝇的等价类。为了验证这一假设，我们使用的算法编码相应的费曼图的排列，生成虚拟图多达6个交叉点，并计算各种不变量。在此假设下，我们利用已知的矩阵积分结果，得到亏格为1到5直到10阶的虚交错缠结的生成函数（即\ 10真正的十字路口）。对于$n$的渐近行为，大的数目的链接和缠结的属$h$和与$n$的交叉也计算了$h=1,2,3$和一般$h$。
英文摘要：
Virtual links are generalizations of classical links that can be represented by links embedded in a ``thickened'' surface $Σ\times I$, product of a Riemann surface of genus $h$ with an interval. In this paper, we show that virtual alternating links and tangles are naturally associated with the $1/N^2$ expansion of an integral over $N\times N$ complex matrices. We suggest that it is sufficient to count the equivalence classes of these diagrams modulo ordinary (planar) flypes. To test this hypothesis, we use an algorithm coding the corresponding Feynman diagrams by means of permutations that generates virtual diagrams up to 6 crossings and computes various invariants. Under this hypothesis, we use known results on matrix integrals to get the generating functions of virtual alternating tangles of genus 1 to 5 up to order 10 (i.e.\ 10 real crossings). The asymptotic behavior for $n$ large of the numbers of links and tangles of genus $h$ and with $n$ crossings is also computed for $h=1,2,3$ and conjectured for general $h$.
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