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
–petal
model the probability of obtaining every specific knot type decays to zero as
,
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
–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.
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