Twisted Alexander Invariants of Twisted Links
Let be an oriented link in , and let be the -component link regarded in the homology 3-sphere that results from performing -surgery on . Results about the Alexander polynomial and twisted Alexander polynomials of corresponding to finite-image representations are obtained. The behavior of the invariants as increases without bound is described.
Keywords: Knot, link, twisted Alexander polynomial, Mahler measure.
MSC 2010: Primary 57M25; secondary 37B10, 37B40.
Let be an oriented -component link in the 3-sphere . For any positive integer , let be the -component link regarded in the homology 3-sphere that results from -surgery on , removing a closed tubular neighborhood of and then replacing it in such a way that its meridian wraps once about the meridian of and times around the longitude. If is trivial and bounds a disk , then is a link in , and it can be obtained from by giving full twists to those strands that pass through .
In previous work , we considered the multivariable Alexander polynomial and the limiting behavior of its Mahler measure as increases without bound. The case in which has zero linking number with each of the other components of was treated separately and combinatorially. The first goal here is to provide a more topological perspective of this case. The second goal is to generalize our twisting results for Alexander polynomials twisted by representations of the link group in a finite group.
We are grateful to Stefan Friedl and Jonathan Hillman for suggestions and helpful remarks.
2 Twisting and Mahler measure.
We survey some results about Mahler measure and Alexander polynomials, and offer motivation for the results that follow.
The Mahler measure of a nonzero integral polynomial in variables is
The Mahler measure of the zero polynomial is .
(1) In the case of a polynomial in a single variable, Jensen’s formula implies that is equal to
where is the leading coefficient of and the are the zeros (with possible multiplicity) of . A proof of this and the following fundamental facts about Mahler measure can be found in .
(2) for any nonzero polynomials . We define the Mahler measure of a rational function , , to be .
(3) if and only if is the product of a unit and generalized cyclotomic polynomials. A generalized cyclotomic polynomial is a cyclotomic polynomial evaluated at a monomial; e.g.,
(4) The Mahler measure of a nonzero Laurent polynomial can be defined either directly from the definition or by normalizing, multiplying by a monomial, so that all exponents are nonnegative.
The Mahler measure of the Alexander polynomial of a link provides a measure of growth of homology torsion of its finite abelian branched covers. We identify with the free abelian multiplicative group generated by .
 Let be an oriented -component link with nonzero Alexander polynomial . Then
where is a finite-index subgroup of , , is the order of the torsion subgroup of and is the associated abelian cover of branched over . In the case that is a knot (that is, ), limit superior can be replaced by an ordinary limit.
We recall that the Alexander polynomial is the first term in a sequence of polynomial invariants that are successive divisors (see below). The authors conjectured  that when vanishes, the conclusion of Theorem 2.3 holds with replaced by the first nonvanishing higher Alexander polynomial. The conjecture was recently proved by T. Le .
Much of the interest in Mahler measures of Alexander polynomials is motivated by an open question posed by D.H. Lehmer in 1933.
Lehmer’s Question. Is 1 a limit point of the set of Mahler measures of integral polynomials in a single variable?
There are no known values in this set between 1 and the Mahler measure of a polynomial found by Lehmer,
Lehmer’s polynomial has a single root outside the unit circle. When is replaced by , the result is the Alexander polynomial of a fibered, hyperbolic knot, the -pretzel knot.
A method for constructing many polynomials with small Mahler measure greater than 1 is to begin with a product of cyclotomic polynomials and perturb the middle coefficient. Lehmer’s polynomial arises as , where is the th cyclotomic polynomial. In fact, all known Mahler measures less than 1.23 have been found in this way .
Our principal motivation for studying the twisting effects on Alexander polynomials comes from a topological analogue of the above procedure: Starting with a periodic -braid (the closure of which has an Alexander polynomial with Mahler measure equal to 1), one perturbs it by twisting consecutive strands, where , and then forms the closure. Knots obtained this way are known as twisted torus knots , and the Mahler measure of is often small but greater than 1. In particular, the closure of the 3-braid is the -pretzel knot.
Observe that is the product of the 3-braid and two full twists on all strands. Let be the 2-component link , where is the closure of while is an encircling unknot. With orientations chosen appropriately, the -pretzel knot is , the result of -surgery on .
Since is a perturbation of the closure of the periodic braid , we might expect that replacing with higher powers would produce twisted torus knots with small Mahler measure. Indeed, calculations suggest that converges to a relatively low value of . The following theorem shows that this is in fact so, and moreover the limit is the Mahler measure of .
 Let be an oriented link in . If has non-zero linking number with some other component, then
In the case that has zero linking number with all other components of , the Mahler measures of can increase without bound. However, converges. In fact, the polynomials converge in a strong sense, as we argued combinatorially in . In the next section, we examine this case more closely, from a topological perspective.
3 Twisting about an unlinked component.
Alexander polynomials are defined for any finitely presented group and epimorphism , . We regard as a multiplicative free abelian group with generators . One considers the kernel of . Its abelianization is a finitely generated -module, where is the ring of Laurent polynomials in . As is a Noetherian unique factorization domain, a sequence of elementary ideals is defined for . The greatest common divisor of the elements of is the th Alexander polynomial of , denoted here by . These polynomials, which are defined up to multiplication by units of , form a sequence of successive divisors. The 0th polynomial is called the Alexander polynomial of and we denote it more simply by .
For purposes of computation, one considers a presentation of :
With no loss of generality, we assume that and .
The epimorphism pulls back to the free group generated by , inducing a unique extension to a map of group rings that, by abuse of notation, we denote also by . One forms an Alexander matrix of the presentation:
where are Fox partial derivatives (see, for example, ). Then is equal to the greatest common divisor of the -minors of , provided that ; when , the result is divisible by , and we must divide by it.
Pairs as above are objects of a category, morphisms being homomorphims such that . If is a morphism, then divides . In particular, is equal to up a unit factor of (denoted ) whenever is an isomorphism.
When is the group of an oriented link of -components in a homology 3-sphere, there is a natural augmentation that maps the meridianal generators of the th component to , for . The Alexander polynomial , an invariant of the link, is denoted here by .
Assume that is an oriented -component non-split link in the 3-sphere such that has zero linking number with . Let
be a Wirtinger presentation for . We will assume throughout, without loss of generality, that corresponds to a meridian of .
Note that we have omitted a Wirtinger relation in the presentation (3.3). It is well known that any single Wirtinger relation is a consequence of the remaining ones. We assume throughout, again without loss of generality, that the omitted relation does not involve meridianal generators of .
The longitude of any component of is a simple closed curve in the torus boundary of a neighborhood of the component that is null-homologous in the link complement. Up to conjugacy, the longtitude represents an element of , and a representative word in the Wirtinger presentation associated to a diagram can be read by tracing around the component of the link, recording generators or their inverses as we pass under arcs, and finally multiplying by an appropriate power of the generator corresponding to the arc where we began so that the element represented has trivial abelianization.
Let be the homotopy class of the longitude of . For any integer , one of the relators in the quotient group presentation
For the purpose of the proof, relabel the meridianal generators of by such that follows as we travel along in its preferred direction (subscripts regarded modulo ). Some arc of the link diagram passes between and . The Wirtinger relations at such crossings allow us to express as a conjugate of , and then as a conjugate of , and so forth. The final relator, which expresses as a conjugate of itself, is . Substituting trivializes the relator, and hence it is redundant in the presence of the other relators. ∎
We denote by the oriented -component sublink . We denote by the exterior of . Performing -surgery on yields a homology sphere . We regard as an oriented link . When is unknotted, , and is the link obtained from by giving full twists to those strands passing through a disk with boundary . It is clear that (3.4) is a presentation of . An Alexander matrix for results from the Alexander matrix for by adjoining a row consisting of the Fox partial derivatives of the added relation . By adding nugatory crossings to , we can arrange that the word representing the longitude does not contain or . Then in the new row we see in the column corresponding to . In other columns we see an element of multiplied by the integer . The reason is the following. Assume that represents , where each or . For any , the generator occurs times, and it contributes , if ; it is equal to , if . Here we use the fact that since has zero linking number with .
Lemma 3.1 allows us to discard some row of other than the last without affecting the module presented. Let be the resulting square matrix. The Alexander polynomial is the determinant of , which we can write as a Laurent polynomial . Note that appears linearly in .
The limit is the same as the determinant of the matrix obtained from by modifying the last row: setting equal to 1 and replacing with the coefficient in the column corresponding to . The modified row agrees with the Fox partial derivatives of the relation =1, the relation that arises from 0-framed surgery on .
The determinant of the modified matrix is the Alexander polynomial , where is obtained from the exterior via 0-framed surgery on , and is induced by the augmentation of that is standard on but maps the class of trivially.
We have proved:
Assume that is an oriented -component link in the 3-sphere such that has zero linking number with each component of the sublink . Then there exists a polynomial with the following properties:
(1) with ;
(2) For any positive integer , ;
where is the 3-manifold obtained from the exterior of by 0-framed surgery on , and is induced by the augmentation of that is standard on but maps a meridian of trivially.
The polynomial provides a uniform normalization of the Alexander polynomials . Using it, we can speak of the coefficients of .
Consider the Borromean rings, the oriented 3-component link in Figure 1 with Wirtinger generators indicated. The link appears in Figure 2.
The homotopy class of the longitude of is . Using Wirtinger relations, it can be written as . The Alexander matrix for , with the first 5 rows corresponding to relations at numbered crossings, is
The Alexander polynomial is equal to .
The matrix (*) computed in Example 3.4 can be computed for any link for which Theorem 3.2 applies. The limit in statement (3) will vanish if and only if the matrix is singular. Equivalently, the limit vanishes if and only if , that is, the polynomials do not depend on .
Assume the hypotheses of Theorem 3.2. The limit in statement (3) vanishes if and only if the sequence of polynomials is constant.
In the remainder of the section, we characterize homologically the case in which the limit of Theorem 3.2 (3) vanishes.
Let denote the -cover of the exterior of associated to the augmentation in Theorem 3.2. The longitude of lifts to a closed oriented curve in ; we fix a lift and regard it as an element of the -module . By abuse of notation, we let denote this class.
The Alexander polynomial is equal to if ; it is equal to if (see Proposition 7.3.10 of , for example). If has zero linking number with , then the Torres conditions  imply that . Hence . We recall that the rank of an -module , denoted by , is the dimension of regarded as a -vector space, where is the field of fractions of .
Assume that is an oriented -component link in the 3-sphere such that has zero linking number with each component of . The following statements are equivalent.
(1) the sequence of polynomials is constant;
(2) or is a torsion element of .
Recall that is the 3-manifold obtained from by 0-framed surgery on . Let be the -cover induced by . The homology is the quotient of by the submodule generated by . By Theorem 3.2 (3) and Corollary 3.5, the sequence of polynomials is constant if and only if . Hence statement (1) is equivalent to the assertion that is nontrivial. Since the latter module is isomorphic to , statements (1) and (2) are equivalent. ∎
The restriction of to , which we also denote by , is the standard abelianization. When the Alexander polynomial of is nontrivial, the conclusion of Theorem 3.6 simplifies.
Assume in addition to the hypotheses of Theorem 3.6 that . Then the sequence of polynomials is constant if and only if is a torsion element of .
The hypothesis implies that . The long exact sequence of the pair together with Excision yields a short exact sequence:
Since , tensoring with shows that . Theorem 3.6 completes the proof.
A torsion element of Corollary 3.7 need not be trivial in , as we demonstrate. Consider the 2-component link with labeled Wirtinger generators in Figure 3. A straightforward calculation shows that has module presentation
The element is conjugate in to and it represents in . Since is isomorphic to the direct sum , it is clear that is a nontrivial torsion element in .
The link in Example 3 is a homology boundary link (see , p. 23). A link is a homology boundary link if there exist mutually disjoint properly embedded orientable surfaces in the link exterior , corresponding to the components , such that the boundary of is homologous to the longitude of the th component. Since the linking number of any curve with the th longitude is given by intersection number with , each inclusion map induces a trivial homomorphism on first homology.
The following proposition implies that performing -surgery on any component of a homology boundary -component link produces a sequence of -component links having the same Alexander polynomial.
Let be an oriented -component link as in Theorem 3.6 . Assume that there exists a properly embedded orientable surface in with boundary homologous to the longitude of and such that the inclusion map induces a trivial homomorphism on first homology. Then the sequence of polynomials is constant.
We conclude the section with two examples and a conjecture.
Consider the oriented 2-component link in Figure 4. A straightforward calculation shows that . Here .
By replacing the 2 full-twists in by full-twists, becomes and .
Consider the oriented 3-component link in Figure 5. The sublink is a Hopf link, and has linking number zero with each of its components. A straightforward calculation shows that , a product of generalized cyclotomic polynomials.
For any oriented link as in Theorem 3.2, is an integer multiple of a product of generalized cyclotomic polynomials.
4 Twisted Alexander polynomials.
As in the previous section, let be a pair consisting of group with presentation 3.1 and an epimorphism to a nontrivial free abelian group generated by . We assume, as we did above, that .
Let be a Noetherian unique factorization domain, and a linear representation. Below we will consider only the case that and the image of is a finite group of permutation matrices; in other words, is a representation of in the group of permutations of . Define
by mapping to , and extending linearly.
Twisted Alexander invariants generalize the (classical) Alexander polynomial by incorporating information from the representation . They were introduced by X.-S. Lin in  and later extended by many authors. Particularly relevant here are publications of Wada  and Kirk and Livingston . The reader is referred to  for a comprehensive survey of twisted Alexander invariants.
Wada’s approach considers the twisted Alexander matrix
We regard as an matrix over by removing inner parentheses. The Alexander-Lin polynomial is the greatest common divisor of the maximal minors, well defined up to multiplication by units in . It is an invariant of the triple (equivalence defined as for pairs but respecting the distinguished group element ) and the conjugacy class of the representation (see ).
Dividing by the determinant of eliminates the dependence on the distinguished element. The resulting rational function, well defined up to unit multiplication, is often called the Wada invariant of and , denoted here by .
The Wada invariant has a homological interpretation with theoretical advantages over the combinatorial approach. Let be a finite CW complex with having a single 0-cell as well as 1- and 2-cells corresponding to the generators and relations in (3.1). (When is the group of an oriented link, working with a Wirtinger presentation will ensure that is homotopy equivalent to the link exterior.) Let denote the universal cover of , with the structure of a CW complex that is lifted from . Consider the chain complex
where is a free module on which acts via , while denotes the cellular chain complex of with coefficients in . The group acts on the left by deck transformations. The tensor product has the structure of a right -module via
The homology groups of the chain complex are finitely generated -modules. As above, elementary ideals are defined. The module order of – that is, the greatest common divisor of the elements of its 0th elementary ideal – is an invariant of and the conjugacy class of . In  it is shown that is equal to the Wada invariant .
When is the group of an oriented link in a homology 3-sphere, is called the twisted Alexander polynomial of with respect to the augmentation and representation , which we denote by . Similarly, we denote the Wada invariant by . When the augmentation is standard, we omit it from our notation.
Henceforth we assume that is a finite-image permutation representation. It is easy to see that the denominator of Wada’s invariant is a product of cyclotomic polynomials. In this case, the Mahler measures of , and are equal. Moreover, Shapiro’s lemma  implies that is isomorphic to , where is the -fold cover of the link exterior that is induced by .
An analogue of Theorem 2.3 was proven in . For this we replace by unbranched abelian cover of corresponding to the finite-index subgroup . Since is a subgroup of , the representation restricts. Let be the -fold induced cover. We replace by the order of the torsion subgroup , denoted by . Then Theorem 3.10  implies that
Formulating a theorem analogous to Theorem 2.4 is more problematic, since the groups are quotients rather than subgroups of . Our approach passes to an appropriate arithmetic subsequence of .
Assume as in Section 3 that is an oriented link in such that has nonzero linking number with some other component. Let be a finite-image permutation representation, and let denote the order of , where is the class of the longitude of . For any positive integer , the group is a quotient of by the relation , where is a merdianal generator of . Consequently, the standard augmentation extends to the standard augmentation of , mapping the class of a meridian of to , where is the linking number of and . Moreover, induces a representation that maps a meridianal generator of trivially.
Let be an oriented link in such that has nonzero linking number with some other component. Let be the class of the longitude of . Let be a finite-image permutation representation and the extension to mapping trivially. Then
where is the order of .
By the excision isomorphism, is trivial unless . One checks that
and hence its module order is a product of generalized cyclotomic polynomials. Consequently, there exists a short exact sequence
where is a factor of . It follows that , the module order of , is the product of and , the module order of (see , for example). Hence vanishes if and only if vanishes. In such a case, the conclusion of the theorem is trivial. Therefore we assume that and are nonzero.
The condition that is nonzero implies that (see Proposition 2 (5) of ). The exact sequence above becomes short exact:
Hence is the product of and the module order of .
Recall that the Alexander-Lin polynomial has the same Mahler measure as the twisted Alexander polynomial . Working with the former is relatively easy: it is the determinant of the twisted Alexander matrix (4.1) with replaced by . We conclude that the Mahler measure of is equal to that of . Corollary 3.2 of , a consequence of a lemma of D. Boyd and W. Lawton  (see also ), implies that the limit of the Mahler measure of as increases without bound is , which is equal to the Mahler measure of . ∎
Consider the oriented 2-component link in Figure 6. The knot is the torus knot , drawn as a pretzel knot. The knot is the -pretzel knot. Let be the permutation representation mapping , and to matrices corresponding to the 5-cycles and , respectively. The class of the longitude of maps to a 3-cycle. The twisted Alexander polynomial is the product of Lehmer’s polynomial evaluated at and a second, irreducible polynomial of degree 34. The Mahler measure is The fact that the (classical) Alexander polynomial is a factor of any twisted Alexander polynomial, whenever a finite-image permutation representation is employed, is a well-known consequence of the fact that the representation fixes a 1-dimensional subspace. (See discussion preceding Corollary 4.4.)
The limit is approximately 4.18.
Theorem 3.2 also generalizes.