Models of random knots. (English) Zbl 1404.57007
The first half of the article contains a review of various models of random knots, such as self avoiding closed walks on the cubic lattice \(\mathbb{Z}^3\), random polygons in \(\mathbb{R}^3\), smoothed Brownian motion in \(\mathbb{R}^3\), random grid diagrams in \(\mathbb{R}^2\), random planar diagrams in \(\mathbb{R}^2\), random braids and random planar Lissajous curves. For each model the main known results on the knot distribution, the nature of these models and the properties of the knots they produce are discussed. This part of the paper can serve as an excellent source of the current literature on these models. Of particular interest to the author is a Petaluma model where a knot diagram has only a single crossing and several projected strands smoothly traverse this crossing at different heights. C. Adams et al. [J. Knot Theory Ramifications 24, No. 3, Article ID 1550011 (2015; Zbl 1322.57004) and ibid. 24, No. 2, Article ID 1550012, 16 p. (2015; Zbl 1316.57011)] have shown that every knot or link has a planar projection with a single crossing – the so called multi-crossing or über-crossing. Moreover, every knot has a petal projection, where the loops that emanate from the multi-crossing point have disjoint interiors. Consequently, petal projections are represented by a rose-shaped curve with an odd number of petals. In order to reconstruct the original knot one needs only the relative ordering of the heights of the strands above the multi-crossing point and this information can be encoded in a single permutation \(\sigma\). Random knots are constructed by picking a permutation \(\sigma\in S_{2n+1}\) with uniform probability. The author reports on rigorous results and numerical experiments concerning the asymptotic distribution of finite type invariants of random knots in Petaluma model, see also the following [C. Even-Zohar et al., Discrete Comput. Geom. 56, No. 2, 274–314 (2016; Zbl 1354.57012)]. The paper concludes with questions about the universality and classification of the various random knot models.
Reviewer: Claus Ernst (Bowling Green)
MathOverflow Questions:
Complexity of random knot with vertices on sphereComplexity of random knot with vertices on sphere
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 60B05 | Probability measures on topological spaces |
Keywords:
random knot; models of random knots; Petaluma model; petal numbers; über-crossing; finite type invariantsSoftware:
KnotTheory; KnotInfo; plantri; SnapPy; Knotscape; GitHub; Knot Atlas; ABCDEFG; MathOverflowReferences:
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