knot theory In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be und ...

, a branch of

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

, a Brunnian link is a nontrivial link that becomes a set of trivial unlinked circles if any one component is removed. In other words, cutting any loop frees all the other loops (so that no two loops can be directly linked). The name ''Brunnian'' is after

Hermann Brunn Karl Hermann Brunn (1 August 1862 – 20 September 1939) was a German mathematician, known for his work in convex geometry (see Brunn–Minkowski theorem, Brunn–Minkowski inequality) and in knot theory. Brunnian links are named after him, as h ...

. Brunn's 1892 article ''Über Verkettung'' included examples of such links.

Examples

The best-known and simplest possible Brunnian link is the

Borromean rings In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are link (knot theory), topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops wh ...

, a link of three

unknot In the knot theory, mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a Knot (mathematics), knot tied into it, unknotted. To a knot ...

s. However for every number three or above, there are an infinite number of links with the Brunnian property containing that number of loops. Here are some relatively simple three-component Brunnian links which are not the same as the Borromean rings: Image:Brunnian-3-not-Borromean.svg, 12-crossing link. Image:Three-triang-18crossings-Brunnian.svg, 18-crossing link. Image:Three-interlaced-squares-Brunnian-24crossings.svg, 24-crossing link. The simplest Brunnian link other than the 6-crossing Borromean rings is presumably the 10-crossing L10a140 link. An example of an ''n''-component Brunnian link is given by the "rubberband" Brunnian Links, where each component is looped around the next as ''aba''⁻¹''b''⁻¹, with the last looping around the first, forming a circle. In 2020, new and much more complicated Brunnian links were discovered in using highly flexible geometric-topology methods. See Section 6.

Non-circularity

It is impossible for a Brunnian link to be constructed from geometric circles. Somewhat more generally, if a link has the property that each component is a circle and no two components are linked, then it is trivial. The proof, by

Michael Freedman Michael Hartley Freedman (born April 21, 1951) is an American mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the 4-dimensional gen ...

and Richard Skora, embeds the three-dimensional space containing the link as the boundary of a Poincaré ball model of four-dimensional

hyperbolic space In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to −1. It is homogeneous, and satisfies the stronger property of being a symme ...

, and considers the hyperbolic

convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

s of the circles. These are two-dimensional subspaces of the hyperbolic space, and their intersection patterns reflect the pairwise linking of the circles: if two circles are linked, then their hulls have a point of intersection, but with the assumption that pairs of circles are unlinked, the hulls are disjoint. Taking cross-sections of the Poincaré ball by concentric three-dimensional spheres, the intersection of each sphere with the hulls of the circles is again a link made out of circles, and this family of cross-sections provides a continuous motion of all of the circles that shrinks each of them to a point without crossing any of the others.; see in particular Lemma 3.2, p. 89

Classification

Brunnian links were classified up to link-homotopy by

John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...

in , and the invariants he introduced are now called Milnor invariants. An (''n'' + 1)-component Brunnian link can be thought of as an element of the

link group In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Ph.D. thesis, . Notably, the link group is not in general the fundamental group of the link comp ...

– which in this case (but not in general) is the

fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...

of the link complement – of the ''n''-component unlink, since by Brunnianness removing the last link unlinks the others. The link group of the ''n''-component unlink is the

free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''− ...

on ''n'' generators, ''F''_''n'', as the link group of a single link is the knot group of the

, which is the integers, and the link group of an unlinked union is the

free product In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, an ...

of the link groups of the components. Not every element of the link group gives a Brunnian link, as removing any ''other'' component must also unlink the remaining ''n'' elements. Milnor showed that the group elements that do correspond to Brunnian links are related to the

graded Lie algebra In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operati ...

of the

lower central series In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a centr ...

of the free group, which can be interpreted as "relations" in the

free Lie algebra In mathematics, a free Lie algebra over a field ''K'' is a Lie algebra generated by a set ''X'', without any imposed relations other than the defining relations of alternating ''K''-bilinearity and the Jacobi identity. Definition The definition ...

. In 2021, two special satellite operations were investigated for Brunnian links in 3-sphere, called "satellite-sum" and "satellite-tie", both of which can be used to construct infinitely many distinct Brunnian links from almost every Brunnian link. A geometric classification theorem for Brunnian links was given. More interestingly, a canonical geometric decomposition in terms of satellite-sum and satellite-tie, which is simpler than JSJ-decomposition, for Brunnian links, was developed. The building blocks of Brunnian links therein turn out to be Hopf -links, hyperbolic Brunnian links, and hyperbolic Brunnian links in unlink-complements, the last of which can be further reduced into a Brunnian link in 3-sphere.

Massey products

Brunnian links can be understood in

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

via

Massey product In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product. The Massey product was created by William S. Massey, an American algebraic topologist. Massey triple product L ...

s: a Massey product is an ''n''-fold product which is only defined if all (''n'' − 1)-fold products of its terms vanish. This corresponds to the Brunnian property of all (''n'' − 1)-component sublinks being unlinked, but the overall ''n''-component link being non-trivially linked.

Brunnian braids

A Brunnian

braid A braid (also referred to as a plait; ) is a complex structure or pattern formed by interlacing three or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strand ...

is a braid that becomes trivial upon removal of any one of its strings. Brunnian braids form a

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

of the

braid group In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of Braid theory, -braids (e.g. under ambient isotopy), and whose group operation is composition of ...

. Brunnian braids over the 2-

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

that are not Brunnian over the 2- disk give rise to non-trivial elements in the homotopy groups of the 2-sphere. For example, the "standard" braid corresponding to the Borromean rings gives rise to the

Hopf fibration In differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an infl ...

''S''³ → ''S''², and iteration of this (as in everyday braiding) is likewise Brunnian.

Real-world examples

Many

disentanglement puzzle Disentanglement puzzles (also called entanglement puzzles, tanglement puzzles, tavern puzzles or topological puzzles) are a type or group of mechanical puzzle that involves disentangling one piece or set of pieces from another piece or set of pie ...

s and some

mechanical puzzles A mechanical puzzle is a puzzle presented as a set of mechanically interlinked pieces in which the solution is to manipulate the whole object or parts of it. While puzzles of this type have been in use by humanity as early as the 3rd century BC ...

are variants of Brunnian Links, with the goal being to free a single piece only partially linked to the rest, thus dismantling the structure. Brunnian chains are also used to create wearable and decorative items out of elastic bands using devices such as the

Rainbow Loom Rainbow Loom is a plastic tool used to weave colourful rubber band, rubber and plastic bands (called loom bands) into decorative items such as bracelets and charms. It was invented in 2010 by Cheong Choon Ng in Novi, Michigan. Description The R ...

or Wonder Loom.

References

External links

"Are Borromean Links so Rare?", by Slavik Jablan
(also available in its original form as published in the journal ''Forma'
here (PDF file)
). * {{Braiding Links (knot theory)