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Knots, Slipknots, and Ephemeral Knots in Random Walks and Equilateral Polygons
 

Summary: Knots, Slipknots, and Ephemeral Knots in
Random Walks and Equilateral Polygons
Kenneth C. Millett
December 1, 2008
Abstract
The probability that a random walk or polygon in the 3-space or in
the simple cubic lattice contains a knot goes to one at the length goes
to infinity. Here, we prove that this is also true for slipknots consisting
of unknotted portions, called the slipknot, that contain a smaller knotted
portion, called the ephemeral knot. As is the case with knots, we prove
that any topological knot type occurs as the ephemeral knotted portion
of a slipknot.
1 Introduction
Self-avoiding random walks and polygons in 3-space or in the cubic lattice pro-
vide a popular model for linear polymers under certain physical conditions.
With increasing length, the probability that a random walk or polygon con-
tains a knot goes to one [SW88, Pip89, DPS94, Dia95] proving a conjecture
of Frisch and Wasserman [FW61] and of Delbruck [Del62]. In addition, this
knotted portion can be of any desired topological type. As a consequence, the
influence of knotting in the statistical mechanical and physical properties of

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics