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The exponent sum of a braid

Definition

Let $n \geq 1$ and let $B_{n} = ⟨ σ_{1}, \dots, σ_{n - 1} ∣ Artin relators ⟩$ be the braid group of the Artin presentation (The braid group by Artin presentation); recall that $B_{0}$ and $B_{1}$ are trivial. The exponent sum is the unique homomorphism

$e_{n} : B_{n} ⟶ Z, e_{n} (σ_{i}) = 1 (1 \leq i \leq n - 1),$

where $Z$ is the additive group of integers. It is well defined because every defining relator of the Artin presentation has exponent sum $0$ : the braid relator $σ_{i} σ_{i + 1} σ_{i} = σ_{i + 1} σ_{i} σ_{i + 1}$ has both sides of exponent sum $3$ , and each far-commutation relator has both sides of exponent sum $2$ , so the assignment $σ_{i} \mapsto 1$ kills all relators, and Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group applies. It is surjective for $n \geq 2$ and it is the trivial map on the trivial group $B_{1}$ . For every Artin word $β = σ_{i_{1}}^{ε_{1}} \dots σ_{i_{k}}^{ε_{k}}$ one has $e_{n} (β) = \sum_{r = 1}^{k} ε_{r}$ independently of the word, and $e_{n} (β^{- 1}) = - e_{n} (β)$ . Caveat: $e_{n}$ is the composite of the abelianisation $B_{n} \to B_{n}^{a b}$ with a surjection onto the free cyclic group $Z$ (for $n \geq 2$ ): since the target is abelian, every commutator is sent to $0$ , so $e_{n}$ factors through $B_{n} / [B_{n}, B_{n}]$ (The abelianisation $G^{a b} : = G / [G, G]$ and its canonical map). No normal form, Garside structure or faithfulness statement is used anywhere.

Facts & Assumptions

Given: An integer $n \geq 1$ and the Artin presentation of $B_{n}$ ; no choice principle is used.

[F1]

$B_{n} = ⟨ σ_{1}, \dots, σ_{n - 1} ∣ σ_{i} σ_{i + 1} σ_{i} = σ_{i + 1} σ_{i} σ_{i + 1} (1 \leq i \leq n - 2), σ_{i} σ_{j} = σ_{j} σ_{i} (∣ i - j ∣ > 1) ⟩$ , with $B_{0} = B_{1}$ trivial and $B_{n}$ defined by the empty presentation for $n \leq 1$ (The braid group by Artin presentation).

[F2]

Von Dyck: if a function from the generators of a presented group to a group $H$ sends every defining relator to the identity of $H$ , then it extends to a unique homomorphism from the presented group to $H$ (Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group).

[F3]

A group homomorphism $f : G \to G^{'}$ satisfies $f (x y) = f (x) f (y)$ for all $x, y$ , hence $f (x^{k}) = f (x)^{k}$ for all $k \in Z$ (Monoid homomorphism and group homomorphism).

Proof

1.1F1F2given

Well-definedness. Write the target additively, so the identity of $Z$ is $0$ . Define $u$ on generators by $u (σ_{i}) : = 1$ . At the braid relator both sides receive $1 + 1 + 1 = 3$ ; at each far-commutation relator both sides receive $1 + 1 = 2$ . Hence every defining relator of [F1] is sent to $0$ , and [F2] produces a unique homomorphism $e_{n} : B_{n} \to Z$ with $e_{n} (σ_{i}) = 1$ . For $n \leq 1$ the group $B_{n}$ is trivial by [F1] and the unique homomorphism to $Z$ is the trivial one; this is the case $n = 1$ of the statement.

2.1F3step 1.1algebra

Values on words and inverses. Let $β = σ_{i_{1}}^{ε_{1}} \dots σ_{i_{k}}^{ε_{k}}$ be an Artin word. By [F3] applied successively, $e_{n} (β) = \sum_{r = 1}^{k} ε_{r} e_{n} (σ_{i_{r}}) = \sum_{r = 1}^{k} ε_{r}$ ; in particular the value does not depend on the word chosen to represent the element $β$ , because it equals the value of the well-defined map $e_{n}$ at $β$ . Taking $k = 1$ , $e_{n} (σ_{i}^{- 1}) = - 1 = - e_{n} (σ_{i})$ , so the same computation gives $e_{n} (β^{- 1}) = - e_{n} (β)$ for every Artin word by [F3] and the multiplicativity of group homomorphisms.

3.1F1step 1.1step 2.1given∎

Surjectivity. For $n \geq 2$ the element $e_{n} (σ_{1}) = 1$ generates the additive group $Z$ , so $e_{n}$ is surjective; for $n = 1$ the domain $B_{1}$ is trivial, and the exponent sum is the (trivial, hence not surjective) map into $Z$ . This proves all claims of the definition and completes the construction of the unique homomorphism with the prescribed values.

Depends on

Used by

An unnormalized Hecke trace is not Markov invariant Counterexample
The HOMFLYPT polynomial from the Hecke Markov trace Definition
The Hecke trace skein calculation for a three-crossing braid Example
The Jones specialization of a two-strand closure Example
The Hecke trace construction is an oriented link invariant Theorem
The HOMFLYPT skein relation Theorem

Dependency tree · two levels

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Sources

Birman and Brendle, Braids: A Survey, Handbook of Knot Theory chapter, author manuscript; section 2.2 (the exponent sum of a braid word), printed pp. 14-16 (standard reference, not scraped)