Colored Alexander invariants

A homological representation formula of colored Alexander invariants

Abstract.

We give a formula of the colored Alexander invariant in terms of the homological representation of the braid groups which we call truncated Lawrence’s representation. This formula generalizes the famous Burau representation formula of the Alexander polynomial.

Key words and phrases:
Alexander polynomial, Colored Alexander invariant, (truncated) Lawrence’s representation
2010 Mathematics Subject Classification:
Primary 57M27 , Secondary 57M25,20F36

1. Introduction

The Alexander polynomial is one of the most important and fundamental knot invariant having various definitions and interpretations. Each definition brings a different prospect and often leads to different generalizations.

In this paper we explore one of the generalizations of the Alexander polynomial, the colored Alexander invariant introduced in [ADO]. This is a family of invariants of a link indexed by positive integers , and coincides with the (multivariable) Alexander polynomial [Mu1]. The first definition of the colored Alexander invariant in [ADO] uses a state-sum inspired from solvable vertex model in physics. As is already noted in [ADO] and made it clarified in [Mu2], the -th colored Alexander invariant is a version of a quantum invariant at -th root of unity.

Throughout the paper we treat the case that is a knot. Then the colored Alexander invariant is a function of one variable . We give a new formulation of the colored Alexander invariant, in the same spirit as [It2] where we gave a topological formulation of the loop expansion of the colored Jones polynomials.

In Theorem 5.3 we show that the colored Alexander invariant is written as a sum of the traces of homological representations which we call truncated Lawrence’s representation. These are variants of Lawrence’s representation studied in [Law, It1, It2], derived from an action on the configuration space. Our formula can be seen as a generalization of the Burau representation formula of the Alexander polynomial and has more topological flavor.

A key result is Theorem 4.2, where we show that truncated Lawrence’s representation is identified with a certain quantum braid group representation, defined for the case the quantum parameter is put as -th root of unity. This generalizes a relation between Lawrence’s representation and generic quantum -representation [Koh, It1], and is interesting in its own right.

Unfortunately, unlike the Burau representation formula, our formula is not completely topological since it heavily depends on a particular presentation (closed braid representative) of knots, and we cannot see its topological invariance directly, although it suggests a relationship to the topology of abelian coverings of the configuration space.

We remark that the colored Alexander invariant at is equal to the -colored Jones polynomial at -th root of unity [MuMu], which in turn, is equal to Kashaev’s invariant derived from the quantum dilogarithm function [Kas]. Thus, our formula also brings a new prospect for Kashaev’s invariant and the volume conjecture.

Acknowledgements

The author gratefully thanks Jun Murakami who suggests a generalization of the author’s previous work on colored Jones polynomial to the colored Alexander invariants. He also wish to thank Tomotada Ohtsuki for stimulating discussion. The author was supported by JSPS KAKENHI Grant Numbers 25887030 and 15K17540.

2. Quantum representation and colored Alexander invariant

2.1. Generic quantum representation

For a complex parameter , we define

Let be the quantum , defined by

For , let be a -vector space spanned by , equipped with an -module structure by

Here . We define the -operator by

where is the transposition and is a universal -matrix of . When , gives rise to a braid group representation

Let be a subspace of spanned by , and let . Then and are finite-dimensional braid group representations with dimension and , respectively.

To relate with irreducible finite dimensional -modules, let be subspace of spanned by , where we define . Then is a sub -module with much familiar action

(2.1)

If is equal to a positive integer , then for , and is nothing but the -dimensional irreducible -module.

As in the case , leads to the braid group representation , and and also give rise to braid group representations.

In the definition of , the term corresponds to a part of the framing correction (see also Remark 2.3). Thanks to this modification, the action of is written by

(2.2)

A remarkable feature is that each coefficient in (2.2) is a Laurent polynomial of and . Thus by regarding and as abstract variables, one can define the braid group representation over the Laurent polynomial ring , which we call a generic quantum -representation (see [JK, It1] for details).

We will consider the following two kind of genericity conditions.

Definition 2.1.

We say that is generic with respect to if . We also say that and are fully generic if the subset is algebraically independent.

Lemma 2.2.

  1. If is generic with respect to , then and are isomorphic as -modules. In particular, the braid group representations and , ( and , and ) are isomorphic.

  2. If and are fully generic, then the braid group representation splits as

Proof.

(i) follows from the definition of . (ii) is [JK, Lemma 13]. (We remark that the modules and in this paper correspond to and in [JK]). ∎

From now on, we will always assume that is generic with respect to so that we do not need to distinguish with . We will mainly work on .

2.2. The case is a root of unity

Let us put as the -th root of unity . As we remarked in the previous section, we assume that is generic with respect to .

By (2.1), spans an irreducible -dimensional irreducible -module, a central deformation of the -dimensional irreducible module [Mu2]. We denote this -dimensional irreducible -module by , and the corresponding braid group representation by .

As an -module, the tensor product decomposes as

(2.3)

so iterated use of (2.3) gives a decomposition as an module

(2.4)

Here is the intertwiner space , the set of endomorphisms equivariant with respect to the -actions. The dimension of is the multiplicity of the direct summands.

The -module is generated by a highest weight vector , so we identify with the subspace consisting of highest weight vectors

(2.5)

In particular, we may view the decomposition (2.4) as

(2.6)

Next we look at the braid group action. As in the previous section, let be the subspace of spanned by , and let . Both and are braid group representations, and as a braid group representation, splits as

By definition, for . Hence by (2.5) the intertwiner space in (2.4) is identified with the direct sums of .

(2.7)

Summarizing, the decomposition of as -representation (2.4) is written in terms of as

(2.8)

2.3. Colored Alexander invariant

The colored Alexander invariant is the quantum invariant coming from . However, the quantum trace of the quantum representation is trivial so we need a trick (see [GPMT] for a general framework).

For a finite dimensional free -vector space , , where is the dual of , and is identified with the contraction. Let be the tensor products of copies of . The partial trace

is the map defined by the composition of the following natural maps,

Let be an oriented knot in represented as the closure of an -braid . We cut the first strand of to get an -tangle . Let be an operator invariant of the tangle (see [Oht, Chapter 3]), defined by

(2.9)

where is given by , and is the exponent sum homomorphism, defined by .

Remark 2.3.

By [Mu2] the framing correction is given by , but as we noted in the previous section, the part of the framing correction is already included in the definition of -operator. This explains the framing correction term in (2.9).

Since is irreducible, is a scalar multiple of the identity

(2.10)

It turns out is independent of a choice of a closed braid representative.

Definition 2.4 (Colored Alexander invariant [Ado],[Mu2]).

The colored Alexander invariant of color is the scalar in (2.10), viewed as a function of the variable .

It is convenient to put and write , a rational function of the variable (Note that by (2.2) and (2.9), is indeed a rational function of .) We call the colored Alexander polynomial, whose name is justified by the fact , where denotes the Conway polynomial (See [Mu1] and Example 5.4).

3. Homological representations

3.1. Lawrence’s representation

We review a homological braid group representation which we call Lawrence’s representation. See [Law], [It1, Section 3], [It2, Appendix]. In the case , it is known as Lawrence-Krammer-Bigelow representation studied in [Kra, Big], and in the case it is nothing but the reduced Burau representation.

We identify the braid group with the mapping class group of the -punctured disc

so that the standard generator of corresponds to the right-handed half Dehn twist along the real line segment .

Let be the unordered configuration space of -points in ,

Here denotes the symmetric group acting as a permutation of the indices. Fix a sufficiently small , and let . We use as a base point of . It is known that , where the first components are spanned by the meridians of the hyperplanes , and the last component is spanned by the meridian of the discriminant .

Let be the homomorphism

where the first map is the Hurewicz homomorphism and the second map is defined by . Let be the covering that corresponds to . Take a point and we use as a base point of . By identifying and as deck translations, is a free -module of rank . Moreover, since is invariant under the action, acts on as -module automorphisms.

Lawrence’s representation is a variant of . It is a sub-representation of , the homology of locally finite chains.

Let be the -shaped graph with four vertices and oriented edges as shown in Figure 1(1). A fork with the base point is an embedded image of into such that:

  • All points of are mapped to the interior of .

  • The vertex is mapped to .

  • The other two external vertices and are mapped to the puncture points.

The image of the edge , and the image of viewed as a single oriented arc, are denoted by and respectively. We call and the handle and the tine of .

Figure 1. Forks and multiforks: to distinguish tines and handle, we often write tine of forks by a bold gray line.

A multifork of dimension is an ordered tuples of forks such that

  • is a fork based on .

  • .

  • .

Figure 1 (2) illustrates an example of a multifork of dimension . We represent a multifork consisting of parallel forks by drawing a single fork labelled by , as shown in Figure 1 (3).

Let . For each , we assign a multifork in Figure 2, which we call a standard multifork.

Figure 2. Standard multifork for

A multifork of dimension gives rise to a homology class in a following manner. Each handle of is seen as a path , so the handles of defines a path . Let be a lift of , taken so that . Let and let be the connected component of containing . We equip an orientation of so that a canonical homeomorphism is orientation preserving. Then represents an element of . By abuse of notation, we use to represent both multifork and the homology class .

We graphically express relations among homology classes represented by multiforks by Figure 3, which we call fork rules.

Figure 3. Geometric rewriting formula for multiforks. Here is a different version of -integer, and is the version of -binomial coefficient.

Let be the subspace of spanned by all multiforks. The set of standard multiforks forms a free basis of . is invariant under the action, so we have the braid group representation

which we call Lawrence’s representation.

When and are put as fully generic complex numbers, all representations , and are the same, but at non-generic parameters they might be different.

Example 3.1.

Here we give a sample graphical calculation using the fork rules.

3.2. Truncated Lawrence’s representation

We introduce a new braid group representation, defined in the case that the variable is put as the -th root of unity .

Let be the subspace of spanned by , where

We define . By abuse of notation, we use the same symbol to represent the image of the projection . By definition, the set of standard multiforks without more than parallel tines , where

forms a basis of . We denote the dimension of , the cardinality of by .

Proposition-Definition 3.2.

At , is a -invariant subspace of , hence we have a linear representation

We call truncated Lawrence’s representation.

Proof.

For and a standard multifork with , is a multifork more than parallel tines. A crucial point is that for , unless . This observation and an iterated use of the fork rule (F4) show , as desired. ∎

4. Truncated Lawrence’s representation and quantum representation

In this section we explore the connection among the braid group representations we introduced in Section 2 and 3. We continue to assume that is generic with respect to .

Put . There is an isomorphism as -vector spaces (for the precise definition of , which is not important here, see [It2]). Thus, by sending a natural basis of we get a basis of , indexed by the same set as the standard multiforks. Using this basis, we express the braid group representation as

The next result says that and are completely the same.

Theorem 4.1.

[Koh, Theorem 6.1],[It1, Theorem 4.5] If is generic with respect to , then for an -braid we have an identity of matrices

(4.1)

We extend the identity (4.1) for the quantum representation and truncated Lawrence’s representation . For , we define . Then and forms a basis of . We express the braid group representation by using this basis as

Theorem 4.2.

If is generic with respect to , for an -braid we have an identity of matrices

Proof.

We view as a quotient rather as a submodule of , and denote the quotient map by . Then is also seen as a quotient, where is the quotient map from . Since is an -module homomorphism, is a surjection as a braid group representation.

By definition for the inclusion map , . Thus, for . This observation, together with Theorem 4.1, shows that the basis of corresponds to the standard multifork basis of , with substitution . ∎

5. Homological representation formula of the colored Alexander invariant

Theorem 4.2 philosophically provides a formula of the colored Alexander invariant, but rewriting the definition of colored Alexander invariant (2.10) in terms of requires several non-trivial computations. Our main task is to express the partial trace as a linear combination of the traces on intertwiner space.

To begin with, we compute the partial trace for . Recall that , where is the intertwiner space which is one-dimensional. Thus acts on each as a scalar multiple by .

Lemma 5.1.

The partial trace of is given by

(5.1)

Here satisfies the symmetry

(5.2)

and is given by the formula

(5.3)
Proof.

Throughout the proof we adopt the convention that all indices are considered modulo , unless otherwise specified.

First we prove the symmetry (5.2). It is sufficient to check (5.2) for the case . A key observation is that

This implies an isomorphism of intertwiner spaces , hence . Also by definition of ,

That is, the actions of on and are the same. Therefore we conclude