Braiding vineyards

Abstract: Vineyards are a common way to study persistence diagrams of a data set which is changing, as strong stability means that it is possible to pair points in ``nearby'' persistence diagrams, yielding a family of point sets which connect into curves when stacked. Recent work has also studied monodromy in the persistent homology transform, demonstrating some interesting connections between an input shape and monodromy in the persistent homology transform for 0-dimensional homology embedded in $\mathbb{R}<sup>2$. In this work, we re-characterize monodromy in terms of periodicity of the associated vineyard of persistence diagrams. We construct a family of objects in any dimension which have non-trivial monodromy for $l$-persistence of any periodicity and for any $l$. More generally we prove that any knot or link can appear as a vineyard for a shape in $\mathbb{R}<sup>d$, with $d\geq 3$. This shows an intriguing and, to the best of our knowledge, previously unknown connection between knots and persistence vineyards. In particular this shows that vineyards are topologically as rich as one could possibly hope.