Random knots using Chebyshev billiard table diagrams

详细信息    查看全文

• 作者：Moshe Cohen¹ ; ^{mcohen@tx.technion.ac.il" class="auth_mail" title="E-mail the corresponding author} ; Sunder Ram Krishnan ; ² ; ^{eeksunderram@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：57M25 ; 57M27 ; 05C80 ; 60C05 ; 60B99
• 刊名：Topology and its Applications
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：194
• 期：Complete
• 页码：4-21
• 全文大小：523 K

文摘

We use the Chebyshev knot diagram model of Koseleff and Pecker in order to introduce a random knot diagram model by assigning the crossings to be positive or negative uniformly at random. We give a formula for the probability of choosing a knot at random among all knots with bridge index at most 2. Restricted to this class, we define internal and external reduction moves that decrease the number of crossings of the diagram. We make calculations based on our formula, showing the numerics in graphs and providing evidence for our conjecture that the probability of any knot appearing in this model decays to zero as the number of crossings goes to infinity.