# nLab weight systems are cohomology of loop space of configuration space

Contents

### Context

#### Knot theory

knot theory

Examples/classes:

Types

knot invariants

Related concepts:

category: knot theory

# Contents

## Statement

### For horizontal chord diagrams

###### Remark

(Hopf algebra-structures)

Both the ordinary homology of a based loop space as well as the universal enveloping algebra of a Lie algebra are canonically Hopf algebras, the former via the Pontrjagin ring-structure (see at homology of loop spaces).

###### Proposition

(ordinary homology of based loop space of ordered configuration space of points is universal enveloping algebra of infinitesimal braid Lie algebra)

For $D, n \in \mathbb{N}$ natural numbers and for any ground field $\mathbb{F}$ (in fact over every commutative ring) the ordinary homology of the based loop space of the ordered configuration space of points in the Cartesian space/Euclidean space $\mathbb{R}^D$ is isomorphic, as a Hopf algebra (Remark ), to the universal enveloping algebra of the infinitesimal braid Lie algebra:

$H_\bullet \big( \Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}(\mathbb{R}^D) \big) \;\simeq\; \mathcal{U} \big( \mathcal{L}_n(D) \big) \,.$

This is due to Fadell-Husseini 01, Theorem 2.2, re-stated as Cohen-Gitler 01, Theorem 4.1, Cohen-Gitler 02, Theorem 2.3.

Notice also that:

###### Proposition

(universal enveloping of infinitesimal braids is horizontal chord diagrams)

$\Big( \mathcal{A}_n^{pb} \;\coloneqq\; Span \big( \mathcal{D}_n^{pb} \big)/(2T, 4T) , \circ \Big)$

of horizontal chord diagrams on $n$ strands with product given by concatenation of strands (this Def.), modulo the 2T relations and 4T relations (this Def.) is isomorphic to the universal enveloping algebra of the infinitesimal braid Lie algebra (Def. ):

$\big(\mathcal{A}_n^{pb}, \circ\big) \;\simeq\; \mathcal{U}(\mathcal{L}_n(D)) \,.$

The combination of Prop. and Prop. yields:

###### Corollary

For $D, n \in \mathbb{N}$ and for any ground field $\mathbb{F}$ (in fact over every commutative ring) the ordinary homology of the based loop space of the ordered configuration space of points in the Cartesian space/Euclidean space $\mathbb{R}^D$ is isomorphic, as a Hopf algebra, to the associative algebra of horizontal chord diagrams on $n$ strands with product given by concatenation of strands (this Def.), modulo the 2T relations and 4T relations (this Def.):

$H_\bullet \big( \Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}(\mathbb{R}^D) \big) \;\simeq\; \big(\mathcal{A}_n^{pb}, \circ\big) \,.$

### For horizontal weight systems

###### Proposition

(integral horizontal weight systems are integral cohomology of based loop space of ordered configuration space of points in Euclidean space)

Given any ground field $\mathbb{F}$ (in fact any ground ring, notably the integers) there is, for each natural number $n$, a canonical isomorphism of graded abelian groups between

1. $\mathcal{W}_n^{pb} \;\coloneqq\; Hom_{\mathbb{F} Mod} \big( \underset{ \mathcal{A}_n^{pb} }{ \underbrace{ Span \big( \mathcal{D}_n^{pb} \big) /(2T,4T) } } , \mathbb{F} \big)$

on horizontal chord diagrams of $n$ strands (elements of the set $\mathcal{D}^{pb}$)

$H^\bullet \big( \Omega \underset{ {}^{\{1,\cdots,n\}} }{Conf} (\mathbb{R}^D) \big) \;\simeq\; (\mathcal{W}_n^{pb})^\bullet \;\simeq\; Gr^\bullet( \mathcal{V}_{pb} ) \,.$

(the second equivalence on the right is the fact that weight systems are associated graded of Vassiliev invariants, for $D =3$).

This appears stated as Kohno 02, Theorem 4.1; it follows immediately by Corollary of Prop. .

### For round weight systems

###### Proposition

(weight systems are inside real cohomology of based loop space of ordered configuration space of points in Euclidean space)

For ground field $k = \mathbb{R}$ the real numbers, there is a canonical injection of the real vector space $\mathcal{W}$ of framed weight systems (here) into the real cohomology of the based loop spaces of the ordered configuration spaces of points in 3-dimensional Euclidean space:

$\mathcal{W} \;\overset{\;\;\;\;}{\hookrightarrow}\; H\mathbb{R}^\bullet \Big( \underset{n \in \mathbb{N}}{\sqcup} \Omega \underset{{}^{\{1,\cdots,n\}}}{Conf} \big( \mathbb{R}^3 \big) \Big)$

This is stated as Kohno 02, Theorem 4.2

The statement relating the ordinary homology of the based loop space of the ordered configuration space of points to the universal enveloping algebra of the infinitesimal braid Lie algebra:

The dual statement identifying the ordinary cohomology of the based loop space of the ordered configuration space of points with the space of weight systems on horizontal chord diagrams:
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