TY - JOUR

T1 - The distribution of knots in the petaluma model

AU - Even-Zohar, C.

AU - Hass, J.

AU - Linial, N.

AU - Nowik, T.

N1 - Cited By :1 Export Date: 21 February 2022

PY - 2018/10/18

Y1 - 2018/10/18

N2 - The representation of knots by petal diagrams (Adams et al 2012) naturally defines a sequence of distributions on the set of knots. We establish some basic properties of this randomized knot model. We prove that in the random n–petal model the probability of obtaining every specific knot type decays to zero as n, the number of petals, grows. In addition we improve the bounds relating the crossing number and the petal number of a knot. This implies that the n–petal model represents at least exponentially many distinct knots. Past approaches to showing, in some random models, that individual knot types occur with vanishing probability rely on the prevalence of localized connect summands as the complexity of the knot increases. However, this phenomenon is not clear in other models, including petal diagrams, random grid diagrams and uniform random polygons. Thus we provide a new approach to investigate this question.

AB - The representation of knots by petal diagrams (Adams et al 2012) naturally defines a sequence of distributions on the set of knots. We establish some basic properties of this randomized knot model. We prove that in the random n–petal model the probability of obtaining every specific knot type decays to zero as n, the number of petals, grows. In addition we improve the bounds relating the crossing number and the petal number of a knot. This implies that the n–petal model represents at least exponentially many distinct knots. Past approaches to showing, in some random models, that individual knot types occur with vanishing probability rely on the prevalence of localized connect summands as the complexity of the knot increases. However, this phenomenon is not clear in other models, including petal diagrams, random grid diagrams and uniform random polygons. Thus we provide a new approach to investigate this question.

UR - http://www.scopus.com/inward/record.url?scp=85055632271&partnerID=8YFLogxK

U2 - https://doi.org/10.2140/agt.2018.18.3647

DO - https://doi.org/10.2140/agt.2018.18.3647

M3 - Article

SN - 1472-2747

VL - 18

SP - 3647

EP - 3667

JO - Algebraic and Geometric Topology

JF - Algebraic and Geometric Topology

IS - 6

ER -