I nternat. J. Mah. & 4ath Si.
Vol. 2 No. 2 (1979) 229-237
229
ON THE ALEXANDER POLYNOMIALS OF ALTERNATING
TWO-COMPONENT LINKS
MARK E. KIDWELL
Department of Mathematics
Amherst College
Amherst, Massachusetts 01002 U.S.A.
(Received September 5, 1978)
ABSTRACT.
A(x,y).
Let
L
be an alternating two-component link with Alexander polynomial
Then the polynomials (i- x) A (x,y)
alternating.
a way that
That is, (i
(-l)i+J
y) A (x,y)
and (i- y) A (x,y)
can be written as
7
i,j
c..
13
are
x
i
yj
in such
c.. > O.
KEY WORDS AND PHRASES.
Alternating link projection, Alexander 3ix and
polynomial, Adjacency matrix, rooted tree.
AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.
i.
Pimary 55A25.
INTRODUCTION.
This paper extends the graphical methods of Crowell [I] from the study of
the reduced Alexander polynomial A(t) to the study of the unreduced Alexander
polynomial
A(x,y).
This extension requires extra care in labelling the edges
of our graph and in comparing its adjacency matrix to the Alexander matrix of the
alternating link.
230
M. E. KIDWELL
Crowell also used his method to prove that for alternating links,
A(t)
deg
2h,
h
where
equalities relating the
is the genus.
In a future paper, we hope to prove
x- and y-degrees of
A(x,y)
to geometric properties
of an alternating link.
The author would like to thank James Bailey and Joan Hutchinson for their
helpful comments.
2.
ALTERNATING LINK PROJECTIONS.
Let
P be an alternating, regular, planar projection of a two-component
K U
L
link
K2,
1
6 [4, p.16]
such as the projection of the link
The components of
L
are oriented, and
P
in Fig. 1.
inherits this link-orientation.
The two thick arrows off the projection in Fig. I indicate link orientation.
We call K
1
the x-component and
K
2
the y-component.
The crossings of a projection are either positive, as shown in Fig.
negative, as shown in Fig.
ent segments.
2b),
We also distinguish four types of crossings depending upon how
x-component overcrosses
y-component.
crossings
y-component overcrosses
x-component.
r
Y
s -crossings
x-component overcrosses itself.
s -crossings
y-component overcrosses itself.
x
Y
The capital letters
then
These are:
r -crossings
x
type.
Rx
YR--’ Sx,
and S
y
denote the number of crossings of each
We follow Crowell’s convention that if
P
R
x
R
y
0
for a projection
is not alternating.
We shall need the following not-quite-obvious fact:
LEMMA 2.1.
or
depending upon the orientations of the constitu-
x- and y-components enter in.
the
2a),
In an alternating link projection
P,
R
x
R
y
P,
231
POLYNOMIALS OF ALTERNATING LINKS
Let
PROOF.
0
be the union of the
f
r
(J) e Ux,
(i)
R
x
x
(i)
Let
U
s -crossings and let
0
be a crossing in
x
must terminate (go under) at an adjacent
since the projection is alternating.
0
and
x
Ux,
Define
R
so
x
(J).
f((i))
+ S
x
R + S
x
y
R
y
x
If a projection has
that
and
x
x
is a one-to-one correspondence between
and
r
s -crossings.
and
y
x-component at
The overcrossing
vertex
be the union of the
x
(1)
be an
(2Rx+l),...,(d)
crossings labelled
d
r -crossing,
x
(2),...,(2Rx)
s
s -crossings.
be
or
x
be
(1),(2),... ,(d),
r
x
or
we stipulate
r -crossings, and
y
y
A Wirtinger presentation [2, p.86] for the link group
HI(S3- L)
has one
generator for each overcrossing segment in a regular projection of the link.
If the projection is alternating, we can label the segment which overcrosses
at vertex (i)
"x i
oP
(Thus we
y-component of the link.
"x."
I
Yi
depending on whether it belongs to the
"Yi"
The relators
define only one of the two symbols
1,...,d,
r i, i
or
x-
of the Wirtinger presentation are
of the form
r
for the crossing of Fig.
2a)
xj Yi Xk
i
3.
-1
Yi
and of the form
ri
for the crossing of Fig.
-1
xi
YJ
-i
xi
-i
Yk
Pb).
ASSOCIATED GRAPH THEORY.
We now regard the projection
vertices and the segments of
P
P
as a graph with the crossings of
Joining vertices as edges.
P
as
Thus each over-
crossing segment in an alternating projection contributes two edges to the
graph.
We will use the word
"vertex"
when we are thinking of
P
strictly as
M.E. KIDWELL
232
a graph, and the word "crossing" when we are thinking of
P
We next orient and label the edges of the graph of P.
toward (i).
incident
"i".
P.
the edges of
"-y"
edges which start at
0.) We
bij
.
J
det
B’
H(T),
J,
the sum of the labels on the
(If there are no such edges, then
i.
J$i
b..11
#
i
b
THEOREM 3.I (Matrix-tree theorem).
adjacency matrix
Fig. I shows the alternating
of an oriented and labelled graph is the
entry,
and end at
also define
if the over-
"-t".
(bij)
B
(i,J)
matrix having for
dd
"-y"
We differ from Crowell [I, pp.260-61] only in using
where he uses simply
The adjacency matrix
"-x"
We indicate the alternating orientation by
orientation and labelling of
and
(i)
The edge on the right receives the label
crossing segment belongs to the y-component.
"-x"
we
The edge which is on the left with respect to the overcrossing seg-
if the overcrossing segment belongs to the x-component and
o__n
P,
edges which belong to undercrossing segments at
ment receives the label
arrows
This alternating
At each vertex (i) of
orientation differs from the link orientation.
orient the two
as a link projection.
i
J"
Let
Thus each row-sum of
B’
B
is zero.
be the matrix obtained from an
B by crossing out the first row and column.
Then
where the summation is over all maximal rooted trees
T
with
T
origin
(I)
in the graph
[i, footnote 2, p. 262] and H(T)
of the labels on the edges of
PROOF.
See [3, p.?].
T.
Our matrix
regarding the difference in labelling).
while Moon
[3]
denotes the product
makes his row-sums zero.
B
is the transpose of Crowell’s
Crowell makes his column-sums zero
This discrepancy arises because Crowell
is counting trees oriented away from the origin
trees oriented toward the origin.
(dis-
(as we are) while Moon
is counting
POLYNOMIALS OF ALTERNATING LINKS
233
COMPARISON WITH THE ALEXANDER MATRIX.
det B’
Our next task is to prove that the polynomial
given in the last
theorem is a link-type invariant.
A [2, p.100] of
The Alexander matrix
and its Wirtinger presentation is the
Fd/El(S3-L)
map
A
.
(aij)
matrix
T\-V
(The
displays the link group as a homomorphic image of a free
group, and the Abelianizing map
and columns of
dd
P
with respect to the projection
L
(
sends all
yi/y.)
and all
x.1 /x
(1),...,(d)
correspond naturally to the vertices
The rows
of
the relators and generators of the Wirtinger presentation so correspond.
P
since
Define
A to be the matrix obtained from A by multiplying any row which corresponds
to a negative crossing by
-1.
A
From now on, we will call
the Alexander
matrix.
LEMMA 4.1.
+-X
y-+q
in
[x,
x
-1
A by deleting the first
be the matrix obtained from
,
Then there is a polynomial
row and column.
+p
A’
Let
y,y
-1
],
det
A(x,y)
such that, up to units
(1-x)A(x,y)
This is a special case of a theorem of Torres
A(x,y)
is defined to be the Alexander polynomial of the link.
link-type invariant
[2, p.120].
to a segment from the
THEOREM .2.
PROOF.
b..
’
corresponding
(1-y)A(x,y).
r
x
x-1
A
are identical, except for the diagonal entries in
or
r -crossings.
y
(resp. y-l)
(i)
If
while
aii
is an
y-1
r
x
(resp.
ry-)
(resp. x-l).
This is a routine application of the free differential calculus, and
exactly follows [i
(2 10)
p.261]
Notice that
b
ii
sum of the labels on all edges coming i__n to vertex (i).
or
det
It is a
For a given alternating link projection, the adjacency matrix B
rows corresponding to
crossing, then
A
If we had deleted a column of
y-component, we would have gotten
and the Alexander matrix
x-I
[5, p.61].
PROOF.
y-l.
is defined to be minus the
This sum must be either
D
234
M. E. KIDWELL
We give below the matrices
link
6
A
B
and
for the alternating projection of the
shown in Fig. 1.
B
(z,)
()(3),()()(6)
(i),
x-i
-x
1
(3)
-y
(6)
0
0
y-i
0
0
y-I
0
0
0
0
0
0
x-1
0
0
x-1
0
0
-x
-y
0
1
0
y-1
o
o
y-1
-x
0
1
0
-y
1
y-i
-y
1
0
-y
0
1
0
-x
x-I
0
i
0
0
0
1
0
1
0
-y
0
0
1
0
0
y-1
0
1
0
y-1
-y
o
-y
y- /
A
d
Recall that
det
B’
oelT" (-l)i=2H bio(i)
Z
where
is the group of
d
permutations of
{2,3,...,d}.
written as a product of
O(O(J)),...)
Let
b
O
H
(algebraic) cycles
Any permutation
bio(i ).
i=2
a
can be
(i,a(i), o((i)),...)(J,o(J),
A (geometric) cycle in a graph is a collection of oriented
edges which forms a closed curve.
LEMMA
2.3. Let
b
be a non-zero entry in
(i,(i), ...)
trivial cycle
in
c,
det B’.
Then for every non-
P
there is a cycle of edges in
Joining
vertex (i) to vertex c(i), etc.
PROOF.
This is clear from the definition of the adjacency matrix.
In Fig. l, hashmrks indicate the (geometric) cycles corresponding to
(3 h)(5 6).
LEMMA h.h.
to vertex o(i)crossing.
PROOF.
If
For this example,
(y l) (1) (-x) (-y) (-’y)
Suppose there is an edge of the graph of
P
Joining vertex (i)
(i) is an r x-crossing, o(i) is an ry or s x
or s -crossing.
is an r -crossing, then o(i) is an r
y
x
y
Then if
(I)
(i)
The edge in question is part of an overcrossing segment at
which belongs to the
x-
segment must terminate.
crossing.
b
(resp.
Thus
y-)
a(i)
component.
is an
r
y
o(1),
At
or
s
x
this overcrossing
(resp.
r
x
or
s
-)
y
POLYNOMIALS OF ALTERNATING LINKS
COR0Y
from
4.5.
If
b
0,
e
the number of off-diagonal factors
If
negative integers
b
b
(x
e
l)Pl-l(y
I)
det A’
there are
diagonal factors of
b
1
e
pl
R -i
x
PROOF.
By Lemma h.1,
a
O
(x-l)ao’.
(-l)ebO
nating.
That is,
(-1) i+J
PROOF.
A’
By Corollary
(l-y)A(x,y),
THEOREM 4.8.
x
(l-y)A(x,y),
det
R
r-rows and
.
reversed.
det B’
and non-
and
x
ao,
In
b
e
ao
Since the off-
y
r -rows, there must be
y
this imbalance is
up to units in
(-1)
4.6,
r
r -rows.
x
2[x,x -1 ,y,y-1 ].
(1-A (x,y),
(y-l)ae’.
up to units.
Thus
det
B’
up to units.
The polynomials
(l-y)A(x,y)
(l-y)A(x,y)
and
can be written as
.
(l-x)A(x,y)
c
i,J
i
jxiy
j
are alter-
in such a way
0.
cij
The argument is identical to
we can compute
Pl
[D
PI-I( -x) P2 (-y)PB(x-I)P&(y-I) P 5
in the diagonal elements.
h.7.
Y
(-x)P2(-y)P3(x -I) P& (y-l) p5
must be paired between
THEOREM
r -rows.
is
(x-l) Pl (y-l)
In B’,
an imbalance of
then there is a positive integer
coming
such that
P2’P3 ’pI’p5
ae
PROOF.
0,
e
The corresponding term of
that
bie
r -rows equals the number of off-diagonal elements coming from
x
COROLLARY 4.6.
Let
235
det
B’
(1-y)A(x,y)
maximal rooted trees in the graph of
[i, (2.13), p.262].
by summing the products of the labels on
P.
One of these products is positive if
and only if it has an even number of edges labelled
results we have developed for
by making vertex (i) an
(l-y)A(x,y)
r-crossing.
Y
By Theorem 3.I,
"-x"
or
"-y".
can be duplicated for
All the
(l-x)A(x,y)
236
M. E. KIDWELL
It would be interesting to know whether the polynomial A (x,y)
alternating for alternating links.
from our results.
must be
This does not follow by elementary algebra
For example, the polynomial
3- 2x + 2x
2
2x2y
+2y2 2xy2 + 3x2y2
-2y- xy-
becomes alternating upon multiplication by
the Torres conditions
.
[5, P.57],
(i-x)
or
(i-y).
According to
ths polynomial could be the Alexander polynomial
of a linking-number-one link each of whose components has Alexander polynomial
3- 5t + 3t
REFERENCES
i.
Crowell, R. H.
Cnus of alternating link types, Annals of Math.
258-275.
69 (1959)
2.
Crowell, E. H. and R. H. Fox.
New York, 1963.
3.
Moon, J. W.
h.
Rolfsen, D. Knots and Links, Mathematical Lecture Series No. 7, Publish or
Perish, Berkeley, 1976.
5.
Torres, G.
Introduction to Knot Theory, Ginn & Company,
Count.ing Labelled Trees, Canadian Mathematical Monographs No. l,
William Clowes and Sons, London, 1970.
On the Alexander polynomial, Annals of Math. 57 (1953)
57-89.
POLYNOMIALS OF ALTERNATING LINKS
K
K
2
I
(5
6)
(2)
(t)
FIGURE 1
a) + crossing
b)
FIGURE 2
crossing