论文标题
条形码posets：组合属性和连接
Barcode Posets: Combinatorial Properties and Connections
论文作者
论文摘要
条形码是真实行上间隔的有限多组，$ b = \ {（b_i，d_i）\} _ {i = 1}^n $。条形码是拓扑数据分析中的重要对象，它们可以作为过滤的持续同源组的摘要。也已经研究了条形码的组合特性，主要是在间隔顺序和间隔图的背景下进行研究。在本文中，我们定义了一个新的地图家族，从带有$ n $ bars的条形码的空间中定义了各种多种多样的置换集，称为多极。这些多晶线在条形码的空间上提供了新的组合不变性。然后，我们定义了对这些多晶线的订单关系，我们显示的可以解释为条形码的交叉数，让人联想到图兰的跨越数字。接下来，我们表明所产生的posets是订单形态的，对众所周知的poset的主要理想，称为多项式纽曼晶格。因此，这些poset形成了多面体的分级面晶格，我们称之为条形码晶格或条形码多面体。最后，我们表明，对于大型的条形码，这些不变性可以在一对条形码之间在Wasserstein和瓶颈距离上提供界限，从而将这些离散的不变性链接到条形码上的连续指标。
A barcode is a finite multiset of intervals on the real line, $B = \{ (b_i, d_i)\}_{i=1}^n$. Barcodes are important objects in topological data analysis, where they serve as summaries of the persistent homology groups of a filtration. The combinatorial properties of barcodes have also been studied, mainly in the context of interval orders and interval graphs. In this paper, we define a new family of maps from the space of barcodes with $n$ bars to the permutation sets of various multisets, known as multipermutations. These multipermutations provide new combinatorial invariants on the space of barcodes. We then define an order relation on these multipermutations, which we show can be interpreted as a crossing number for barcodes, reminiscent of Túran's crossing number for graphs. Next, we show that the resulting posets are order-isomorphic to principal ideals of a well known poset known as the multinomial Newman lattice. Consequently, these posets form the graded face-lattices of polytopes, which we refer to as barcode lattices or barcode polytopes. Finally, we show that for a large class of barcodes, these invariants can provide bounds on the Wasserstein and bottleneck distances between a pair of barcodes, linking these discrete invariants to continuous metrics on barcodes.
