# Alexander invariants

Invariants connected with the module structure of the one-dimensional homology of a manifold $\widetilde{M}$, freely acted upon by a free Abelian group $J ^ {a}$ of rank $a$ with a fixed system of generators $t _ {1} \dots t _ {a}$.

The projection of the manifold $\widetilde{M}$ onto the space $M$ of orbits (cf. Orbit) is a covering which corresponds to the kernel $K _ {a}$ of the homomorphism $\gamma : G \rightarrow J _ {a}$ of the fundamental group $\pi _ {1} (M) = G$ of the manifold $M$. Since $K _ {a} = \pi _ {1} ( \widetilde{M} )$, the group $B _ {a} = K _ {a} / K _ {a} ^ \prime$, where $K _ {a} ^ \prime$ is the commutator subgroup of the kernel $K _ {a}$, is isomorphic to the one-dimensional homology group $H _ {1} ( \widetilde{M} , \mathbf Z )$. The extension $1 \rightarrow K _ {a} \rightarrow G \rightarrow J ^ {a} \rightarrow 1$ generates the extension $(*) : 1 \rightarrow B _ {a} \rightarrow G/ K _ {a} ^ \prime \rightarrow J ^ {a} \rightarrow 1$, which determines on $B _ {a}$ the structure of a module over the integer group ring $\mathbf Z (J ^ {a} )$ of the group $J ^ {a}$( cf. Group algebra). The same structure is induced on $B _ {a}$ by the given action of $J ^ {a}$ on $\widetilde{M}$. Fixation of the generators $t _ {i}$ in $J ^ {a}$ identifies $\mathbf Z ( J ^ {a} )$ with the ring $L _ {a} = L _ {a} (t _ {1} \dots t _ {a} ) = \mathbf Z [ t _ {1} , t ^ {-1} \dots t _ {a} , t _ {a} ^ {-1} ]$ of Laurent polynomials in the variables $t _ {i}$. Purely algebraically the extension

defines and is defined by the extension of modules $(**): 0 \rightarrow B _ {a} \rightarrow A _ {a} \rightarrow I _ {a} \rightarrow 0$[5]. Here $I _ {a}$ is the kernel of the homomorphism $\epsilon : L _ {a} \rightarrow \mathbf Z$ $( \epsilon t _ {i} = 1 )$. The module $A _ {a}$ is called the Alexander module of the covering $\widetilde{M} \rightarrow M$. In the case first studied by J.W. Alexander [1] when $M = M (k)$ is the complementary space of some link $k$ of multiplicity $\mu$ in the three-dimensional sphere $S ^ {3}$, while the covering corresponds to the commutation homomorphism $\gamma _ \mu : G(k) \rightarrow J ^ \mu$ of the link group, $A _ \mu$ is the Alexander module of the link $k$. The principal properties of $G$ which are relevant to what follows are: $G/ G ^ \prime$ is a free Abelian group, the defect of the group $G$ is 1, $G$ has the presentation $\{ x _ {1} \dots x _ {m+1} ; r _ {1} \dots r _ {m} \}$ for which $\gamma _ \mu (x _ {i} ) = t _ {i}$, $1 \leq i \leq \mu$; $\gamma _ \nu (x _ {i} ) = 1$, $i > \mu$( cf. Knot and link diagrams). In the case of links the generators $t _ {i} \in J ^ \mu$ correspond to the meridians of the components $k _ {i} \subset k$ and are fixed by the orientations of these components and of the sphere.

As a rule, $M$ is the complementary space $M(k)$ of $k$, consisting of $\mu$ $(n - 2)$- dimensional spheres $k _ {i}$ in $S ^ {n}$. In addition to the homomorphism $\gamma _ {m}$, one also considers the homomorphism $\gamma _ \sigma : G(k) \rightarrow J$, where $\gamma (x)$ is equal to the sum of the link coefficients of the loop representing $x$ with all $k _ {i}$.

The matrix $\mathfrak M _ {a}$ of the module relations of a module $A _ {a}$ is called the Alexander covering matrix and, in the case of links, the Alexander link matrix. It may be obtained as the matrix

$$\left ( \frac{\partial r _ {i} }{\partial x _ {j} } \right ) ^ {\gamma _ {a} \phi } ,$$

where $\{ x _ {i} ; r _ {i} \}$ is a presentation of the group $G$. If $\mu = 1$, the matrix $\mathfrak N _ {a}$ of module relations for $B _ {a}$ is obtained from $\mathfrak M _ {a}$ by discarding the zero column. The matrices $\mathfrak M _ {a}$ and $\mathfrak N _ {a}$ are defined by the modules $A _ {a}$ and $B _ {a}$ up to transformations corresponding to transitions to other presentations of the module. However, they can be used to calculate a number of module invariants. Alexander ideals are ideals of the module $A _ {a}$, i.e. series of ideals $E _ {i} ( A _ {a} )$ of the ring $L _ {a} : (0) \subseteq E _ {0} \subseteq \dots \subseteq E _ {i-1} \subseteq E _ {i} \subseteq \dots \subseteq (1)$, where $E _ {i}$ is generated by the minors of $\mathfrak M _ {a}$ of order $(m - i) \times (m - i)$ and $E _ {i} = L _ {a}$ for $m - i < 1$. The opposite numbering sequence may also be employed. Since $L _ {a}$ is both a Gaussian ring and a Noetherian ring, each ideal $E _ {i}$ lies in a minimal principal ideal $( \Delta _ {i} )$; its generator $\Delta _ {i}$ is defined up to unit divisors $t _ {i} ^ {k}$. The Laurent polynomial $\Delta _ {i} (t _ {1} \dots t _ \mu )$ is simply called the Alexander polynomial of $k$( or of the covering $\widetilde{M} \rightarrow M$). If $\Delta _ {i} \neq 0$, it is multiplied by $t _ {1} ^ {k _ {1} } \dots t _ \mu ^ {k _ \mu }$ so that $\Delta _ {i} (0 \dots 0) \neq 0$ and $\neq \infty$. To the homomorphism $\gamma _ \sigma$ there correspond a module $\overline{A}\;$, ideals $\overline{E}\; _ {i}$ and polynomials $\overline \Delta \; _ {i}$, designated, respectively, as Alexander's reduced module, Alexander's reduced ideals and Alexander's reduced polynomials of $k$( or of the covering ${\widetilde{M} } _ \sigma \rightarrow M$). If $\mu = 1$, then $A = \overline{A}\;$. $\mathfrak M ( \overline{A}\; )$ is obtained from $\mathfrak M$ by replacing all $t _ {i}$ by $t$. If $\mu \geq 2$, $\overline \Delta \; _ {1}$ is divisible by ${(t - 1) } ^ {\mu - 2 }$. The polynomial $\nabla (t) = {\overline \Delta \; } _ {1} (t) / {(t - 1) } ^ {\mu - 2 }$ is known as the Hosokawa polynomial. The module properties of $A (k)$ have been studied [4], [8], [10]. The case of links has not yet been thoroughly investigated. For $\mu = 1$, the group $H _ {1} ( \widetilde{M} ; R)$ is finitely generated over any ring $R$ containing $\mathbf Z$ in which $\Delta (0)$ is invertible [7], in particular over the field of rational numbers, and, if $\Delta (0) = +1$, then also over $\mathbf Z$. In such a case $\Delta (t)$ is the characteristic polynomial of the transformation $t : H _ {1} ( \widetilde{M} ; R) \rightarrow H _ {1} ( \widetilde{M} ; R)$. The degree of $\Delta _ {1} (t)$ is equal to the rank of $H _ {1} ( \widetilde{M} ; R)$; in particular, $\Delta _ {1} (t) = 1$ if and only if $H _ {1} ( \widetilde{M} ; \mathbf Z) = 0$. If $n = 3$, the link ideals have the following symmetry property: $E _ {i} = {\overline{E}\; } _ {i}$, where the bar denotes that the image is taken under the automorphism generated by replacing all $t _ {i}$ by $t _ {i} ^ {-1}$. It follows that $\Delta _ {i} ( t _ {1} ^ {-1} \dots t _ \mu ^ {-1} ) = t _ {1} ^ {N _ {1} } \dots t _ \mu ^ {N _ \mu } \Delta _ {i} ( t _ {1} \dots t _ \mu )$ for certain integers $N _ {i}$. This symmetry is the result of the Fox–Trotter duality for knot and link groups. It may also be deduced from the Poincaré duality for the manifold $\widetilde{M}$, taking into account the free action of $J ^ {a}$[3]. If $\Delta _ {1} (t _ {1} \dots t _ \mu ) \neq 0$, then the chain complex $C _ {*} ( \widetilde{M} )$ over the field of fractions $P _ \mu$ of the ring $L _ \mu$ is acyclic ( $n = 3$), and the Reidemeister torsion $\tau \in P _ \mu / \Pi$ corresponding to the imbedding $L _ \mu \subset P _ \mu$, where $\Pi$ is the group of units of $L _ \mu$, is defined accordingly. If $\mu = 2$, then $\tau = \Delta _ {1}$; if $\mu = 1$, then $\tau = \Delta _ {1} / t - 1$( up to units of $L _ \mu$). The symmetry of $\Delta _ {1}$ for $n = 3$ is a consequence of the symmetry of $\tau$. If $\mu = 1$, it follows from the symmetry of $\Delta _ {i} (t)$ and from the property $\Delta _ {i} (1) = \pm 1$ that the degree of $\Delta _ {i} (t)$ is even. The degree of $\nabla (t)$ is also even [4]. The following properties of the knot polynomials $\Delta _ {i} (t)$ are characteristic: $\Delta _ {i} (1) = \pm 1$; $\Delta _ {i} (t) = t ^ {2k} \Delta _ {i} ( t ^ {-1} )$; $\Delta _ {i+1}$ divides $\Delta _ {i}$; and $\Delta _ {i} = 1$ for all $i$ greater than a certain value $N$, i.e. for each selection $\Delta _ {i} (t)$ with these properties there exists a knot $k$ for which they serve as the Alexander polynomials. The Hosokawa polynomials [4] are characterized by the property $\nabla (t) = t ^ {2k} \nabla (t ^ {-1} )$ for any $\mu \geq 2$; the polynomials $\Delta _ {1}$ of two-dimensional knots by the property $\Delta _ {1} (1) = 1$.

Alexander invariants, and in the first place the polynomials, are powerful tools for distinguishing knots and links. Thus, $\Delta _ {1}$ fails to distinguish between only three pairs out of the knots in a table containing fewer than 9 double points (cf. Knot table). See also Knot theory; Alternating knots and links.

#### References

 [1] J.W. Alexander, "Topological invariants of knots and links" Trans. Amer. Math. Soc. , 30 (1928) pp. 275–306 [2] K. Reidemeister, "Knotentheorie" , Chelsea, reprint (1948) [3] R.C. Blanchfield, "Intersection theory of manifolds with operators with applications to knot theory" Ann. of Math. (2) , 65 : 2 (1957) pp. 340–356 [4] F. Hosokawa, "On -polynomials of links" Osaka J. Math. , 10 (1958) pp. 273–282 [5] R.H. Crowell, "Corresponding groups and module sequences" Nagoya Math. J. , 19 (1961) pp. 27–40 [6] R.H. Crowell, R.H. Fox, "Introduction to knot theory" , Ginn (1963) [7] L.P. Neuwirth, "Knot groups" , Princeton Univ. Press (1965) [8] R.H. Crowell, "Torsion in link modules" J. Math. Mech. , 14 : 2 (1965) pp. 289–298 [9] J. Levine, "A method for generating link polynomials" Amer. J. Math. , 89 (1967) pp. 69–84 [10] J.W. Milnor, "Multidimensional knots" , Conference on the topology of manifolds , 13 , Boston (1968) pp. 115–133
How to Cite This Entry:
Alexander invariants. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Alexander_invariants&oldid=45055
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article