ENUMERATING THE PRIME ALTERNATING KNOTS, PART I

Enumerating the Prime Alternating Knots, Part Ii

Alternating Knots

Enumerating Alternating Trees

In this paper we examine the enumeration of alternating trees. We give a bijective proof of the fact that the number of alternating unrooted trees with n vertices is given by 1 n2 n?1 P n k=1 ? n k k n?1 , a problem rst posed by Postnikov in 4]. We also prove, using formal arguments, that the number of alternating plane trees with n vertices is 2(n ? 1) n?1 .

Enumerating alternating tree families

We study two enumeration problems for up-down alternating trees, i.e., rooted labelled trees T , where the labels v1, v2, v3, . . . on every path starting at the root of T satisfy v1 < v2 > v3 < v4 > · · · . First we consider various tree families of interest in combinatorics (such as unordered, ordered, d-ary and Motzkin trees) and study the number Tn of different up-down alternating labelled ...

Alternating Quadrisecants of Knots

It is known [Pann, Kup] that for every knotted curve in space, there is a line intersecting it in four places, a quadrisecant. Comparing the order of the four points along the line and the knot we can distinguish three types of quadrisecants; the alternating ones have the most relevance for the geometry of a knot. I show that every (nontrivial, tame) knot in R has an alternating quadrisecant. T...