Satellite constructions and geometric classification of Brunnian links

We study satellite operations on Brunnian links. First, we find two special satellite operations, both of which can construct infinitely many distinct Brunnian links from almost every Brunnian link. Second, we give a geometric classification theorem for Brunnian links, characterize the companionship graph defined by Budney in [JSJ-decompositions of knot and link complements in [Formula: see text], Enseign. Math. 3 (2005) 319–359], and develop a canonical geometric decomposition, which is simpler than JSJ-decomposition, for Brunnian links. The building blocks of Brunnian links then turn out to be Hopf [Formula: see text]-links, hyperbolic Brunnian links, and hyperbolic Brunnian links in unlink-complements. Third, we define an operation to reduce a Brunnian link in an unlink-complement into a new Brunnian link in [Formula: see text] and point out some phenomena concerning this operation.

New criteria and constructions of brunnian links

We present two practical and widely applicable methods, including some criteria and a general procedure, for detecting Brunnian property of a link, if each component is known to be unknot. The methods are based on observation and handwork. They are used successfully for all Brunnian links known so far. Typical examples and extensive experiments illustrate their efficiency. As an application, infinite families of Brunnian links are created and we establish a general way to construct new ones in bulk.

All for One and One for All

Chapter 23 tells the story of the Borromeo family and the origin of the motif on their coat of arms. It is a link known to mathematicians as the Borromean rings. The link is formed of three rings, which cannot be separated although no two of the rings are linked. This motif has a long history. It was known to the Vikings as the Walknot, and is inscribed on the Stora Hammars 1 picture-stone. Some of John Robinson’s abstract geometrical sculptures take the form of the Borromean rings. The mathematician Hermann Brun investigated how the structure of the Borromean rings could be extended to form other links, and these are known as Brunnian links. The emblem of the Principia Discordia—a satirical counter-culture text written in 1963—is a pentagonal Brunnian link formed of five nonagons known as the Mandala Discordia.

On two constructions of Brunnian links

Brunnian links and braids are those that become trivial upon removing any of the components. It is well known that any link is the closure of some braid. However, a Brunnian link might not be the closure of any Brunnian braid. In this paper, we present two methods of constructing Brunnian links from Brunnian braids and show that our methods do result in Brunnian links that cannot be obtained as the closure of a Brunnian braid.

On the universal sl2 invariant of Brunnian bottom tangles

The universal sl2 invariant is an invariant of bottom tangles from which one can recover the colored Jones polynomial of links. We are interested in the relationship between topological properties of bottom tangles and algebraic properties of the universal sl2 invariant. A bottom tangle T is called Brunnian if every proper subtangle of T is trivial. In this paper, we prove that the universal sl2 invariant of n-component Brunnian bottom tangles takes values in a small subalgebra of the n-fold completed tensor power of the quantized enveloping algebra Uh(sl2). As an application, we give a divisibility property of the colored Jones polynomial of Brunnian links.

FINITE TYPE INVARIANTS AND MILNOR INVARIANTS FOR BRUNNIAN LINKS

A link L in the 3-sphere is called Brunnian if every proper sublink of L is trivial. In a previous paper, Habiro proved that the restriction to Brunnian links of any Goussarov–Vassiliev finite type invariant of (n + 1)-component links of degree < 2n is trivial. The purpose of this paper is to study the first nontrivial case. We show that the restriction of an invariant of degree 2n to (n + 1)-component Brunnian links can be expressed as a quadratic form on the Milnor link-homotopy invariants of length n + 1.