NAKANISHI, YASUTAKA

Osaka City University Repository

Nakanishi, Y.
Osaka J. Math.
27 (1990), 207-215
ON FOX'S CONGRUENCE CLASSES OF KNOTS, II
Dedicated to Professor Akio Hattori on his sixtieth birthday
YASUTAKA NAKANISHI*
(Received April 7, 1989)
R.H. Fox introduced the notion of congruence class of knots in [3], and he
gave a necessary condition for congruence in terms of Alexander matrices and
polynomials. S. Suzuki and the author [9] improved his condition and showed
that there exist infinitely many congruence classes of knots modulo n, q if nφl
and (n, #)Φ(2, 1), (2, 2). Further, they conjectured that all knots are con-
gruent modulo 2, 1 and 2, 2. In this note we will generalize the notion of
congruence class for links and give an alternate condition. And we will prove
that two links are congruent modulo 2, 1 if and only if the two links are Z2-
link-homologous. (The number of congruence classes of ^.-component links
modulo 2, 1 is just 2W~"1)/2.) As a corollary, we have that all knots are con-
gruent modulo 2, 1.
1. Definitions and Theorems
In this note we only consider a μ-component link L=K1{J ••• \jKμ, that is
an ordered collection of μ disjoint simple closed oriented curves K/s in a three
dimensional oriented sphere S3. Two links are said to be equivalent, if there is
an orientation preserving homeomorphism of S3 onto itself, which maps one
link onto the other preserving the orientation and order of the components.
And such an equivalence class of links is called a link type. A /^-component
link L=K1{J ---{jKμ is called trivial if there exist μ disjoint disks Z^U ••• Uθμ
in S3 with dDi=Ki(i=^l, •••, μ). Especially, we call a 1-component link a knot,
a 1-component link type a knot type, and a 1-component trivial link a trivial
knot.
R.H. Fox introduced the notion of congruence classes of knots in [3], which
can be generalized for links as follows.
DEFINITION. Let n and q be non-negative integers. Two μ-component
link types K and λ are said to be congruent modulo n, q, written κ = X (mod n, q).
Supported in part by Grant-in-Aid for Encouragement of Young Scientists, Ministry of
Education, Science and Culture.
208 Y. NAKANISHI
iff there are /^-component links L
o
, Lly L2y •••, Lh integers cly c2y ••-, ch and trivial
knots mly m2y ••-,#*/ such that
(1) L( _! and m{ are disjoint,
(2) Li is obtained from L ^ by (l/c, ra)-surgery along mi (see [10, 11] for
a/δ-surgery),
(3) the sum of the linking numbers Σ lk(Ki_Xt>n mt ) = 0 (mod g) where
L,-i=-K:«-i.iU - U ^ - , . μ , and
(4) L
o
 represents κy and L7 represents λ.
REMARK. These relations are equivalence relations. Congruence modulo
0, q is just the link equivalence. Any two /^-component link types are con-
gruent modulo 1, q. If the number of components are distinct, then the two
link types are incongruent modulo n> q,
S. Suzuki and the author [9] have studied a necessary condition for con-
gruence modulo n, q in terms of Alexander matrices, Alexander polynomials,
and elementray ideals in the sense of Fox [2]. They gave the condition only
for knot types, but their condition is clearly generalized for link types as in The-
orem 1. From an Alexander matrix A
κ
(tlyt2y •••, tμ) of a /^-component link type
κy we obtain a reduced Alexander matrix Aκ(t) by rewriting t/s ( / = 1 , 2, •••, μ)
in entries of the matrix to the same t. Similarly, we obtain the reduced Ale-
xander polynomial A
κ
(t) and the reduced elementary ideals. In the following,
σ
n
(t) means {\—f)l(l—t) =
Theorem 1. If κ = \ (mod n, q), then, for properly chosen A
κ
(t) and A
λ
(t)y
we have
(l—t) » ^ Λ- 3, —. (1—*)
and hence
., (l-i) f
where il9 •••,/# are all divisors of n and I < ί 1 < </H ί<w. Furthermore, we have
similar statements for the reduced elementary ideals of deficiency greater than 1.
In the above, f(t) = g(t) mod {h^ή, h2(t), •••, hj(t)} means that/( ί ) and g(t)
are in the same class of the quotient ZζtyKh^t), h2(t), •••, hj(t))y where (hλ(t)y
h2(t), •'•> hj(t)) is the ideal generated by hx(t)y h2(t)y •••, hj(t) in Z<f). The proof
of Theorem 1 is parallel to the proof of [9, Theorem 2], so we omit it.
Applying Theorem 1, we can find infinitely many link types that are in-
congruent modulo ny q.
CONGRUENCE CLASSES OF KNOTS 209
Theorem 2. Let n be an integer greater than 1 and q a non-negative integer
such that (n, #)Φ(2, 1), (2, 2). For congruence modulo n, q, there exist infinitely
many distinct classes of μ-component link types for each μ (cf. [9, Theorem 3]).
The proof of Theorem 2 will be given in the next section.
To consider the case (n> ?)=(2, 1), we use the following notion.
DEFINITION. TWO links L=K1\J — \JKμ and L'=Kί\J — \jKl are said
to be Z2-link-homologous if and only if μ=v and lk(Kh Kj) = lk(K'iy Kj) (mod 2)
for every l
Theorem 3. Two given link types are congruent modulo 2, 1 if and only if
two links representing the link types are Z2-link-homologous.
The proof of Theorem 3 will be given in the section 3. Since all knots
are iΓ2-link-homologous, we have the following.
Corollary. All knot types are congruent modulo 2, 1.
This is an answer in the affirmative to one of conjectures in [9]. By his
experiment, the author have not finded the difference between congruence
modulo 2, 1 and modulo 2,2 for knot types. But we can see the difference for
link types as follows.
Proposition 4. The Borromean rings and a 3-component trivial link are con-
gruent modulo 2, 1, but incongruent modulo 2, 2.
Proof. The linking number of each pair of components of the Borromean
rings is 0 and that of a 3-component trivial link is also 0. Since the two links
are ^-link-homologous, the two links are congruent modulo 2, 1 from Theorem
3. On the other hand, a reduced Alexander matrix of the Borromean rings is
o o i
— ί ) 2 θ j '0 (1-
and that of a 3-component trivial link is (0). It can be seen that
Z<f>/(2(l-f), (1-f) (l+<2)) = Z®Z2®Z2,
Therefore, ( l ~ 0 2 * 0 mod {2(1-0, (1—0 (1+t^)}- F r o m Theorem 1, the
Borromean rings and a 3-comρonent link are incongruent modulo 2, 1. We
complete the proof.
Here, we raise the following conjectures which are extensions of the original
conjectures in [9].
CONJECTURE B: If two links are link-homotopic, then the links are con-
210 Y. NAKANISHI
gruent modulo 2,2.
CONJECTURE A: If two links are link-homotopic, then the links are defor-
mable to each other by a finite sequence of operations τ2's, which are shown in
Fig. 1.
t
Fig. 1.
NOTE. Conjecture A is proposed by A. Kawauchi. If Conjecture A is
true, then Conjecture B is true.
For knot types, all Alexander matrices and polynomials are equivalent up
to our condition for congruence modulo 2, 2 as follows.
Theorem 5. For all knot types, their Alexander matrices, which are chosen
properly, are congruent modulo {2(1—t), (1—t) σ2(t2)}. Moreover, their Alexander
polynomials and elementary ideals are also congruent modulo {2(1 — £), (1 — t) σ2(t2)}
(cf. [9, Theorem 4]).
The proof of Theorem 5 will be given in the section 4.
2. Proof of Theorem 2
For a non-negative integer j^N0, let KJ be the connected sum of j copies of
a trefoil knot. We get a ^-component link λ, as the split sum of KJ and a
(μ—l)-component trivial link. Then, the ith reduced elementary ideal 25, (ί) of
Xj is (0) for 1^/^μ—1, ({f-t+lf+i-1) for μ^i^μ+j-l, and (1) for i^μ+j.
As in the proof in [9, §3], their reduced elementary ideals of λ/s are mutually
distinct mod {2(1-0, (1-f) σ
n
(t% (1-f) σ
n
{t*t*% - , (1-ί) σ,(f' x )}. Hence,
by Theorem 1, there exist infinitely many distinct classes for congruence modulo
n, q> completing the proof.
3. Proof of Theorem 3
To prove Theorem 3, we use the notion of Δ-unknotting operation [6] as
follows.
DEFINITION. A Δ-unknotting operation is a local move on a link diagram
as in Fig. 2. If a diagram of a link U is a result of a Δ-unknotting operation
CONGRUENCE CLASSES OF KNOTS 211
on a diagram of L, then we say that U is obtained from L by a Δ-unknotting
operation.
Δ-unknotting
operation
Fig. 2.
DEFINITION. TWO links L=K1\J — UKμ. and L'=K{ U — UK'V are said to
be link-homologous if and only if μ=v and lk{Kh Kj) = lk(K/i, Kj) for every
Proposition 6 ([6]). Two £«;#& Z/wfo are link-homologous if and only if the
two links can be deformed to each other by a finite sequence of A-unknotting opera-
tions.
From the above Proposition 6 and the following Lemma, it can be seen that
if two links are link-homologous then two link types represented by the links
are congruent modulo 2,1.
Lemma. A A-unknotting operation can be realized by (< ^ Λj2)-surgeries
along trivial knots.
xxx>
(a) ( b )
( d )
Fig. 3.
212 Y. NAKANISHI
Proof. Watch Fig. 3. Take three trivial knots ml9 m2, and m3 as in Fig.
3 (a). By —1/2-surgery along ml9 1/2-surgery along m2, and 1/2-surgery along
m3, we obtain (b) from (a). Take a trivial knot mA as in Fig. 3 (c). By 1/2-
surgery along m4y we obtain (d) from (c). Deformation from (a) to (d) is just
a Δ-unknotting operation, completing the proof.
Proof of Theorem 3. Let L=K
γ
 U — \}Kμ and L'=K[ U ••• U-KΓ/L be Z2-
link-homologous, i.e. lk(Kh Kj) = lk(Kiy Kj) (mod 2) for every pair z, j . If
there is a pair z, j such that lk{Kh Kj)^rlk(K\y Kty, then take a trivial knot m
such that a disk bounded by m intersects K\ (and Kj) in a single point respec-
tively. By (l/2^)-surgery along m for some integer c, the result link L*—
Kf\J- UK$ has &(£?, Kf)=lk(Ki, Kj). Therefore, there exists a link L*
such that L* and L are link-homologous, and that two link types represented by
L* and L' are congruent modulo 2, 1. As mentioned in the above, two link
types represented by L* and L are congruent modulo 2,1. Since congruence
is an equivalence relation, two link types represented by L and U are congruent
modulo 2, 1. Conversely, a (l/2c)-surgery along a trivial knot does not change
the linking number of components modulo 2 for every integer c. Therefore, if
two link types represented by two links L and L' are congruent modulo 2, 1,
then L and L' are J^-link-homologous. Hence, we complete the proof.
4. Proof of Theorem 5
In order to show Theorem 5, we use the following.
For a ^-component link L=K
λ
 U ••• UKμ y let E=E(L)=S3—L and take the
universal abelian covering p: E
a
-^>E, associated with the epimorphism 7Γ1(£')->
ζt
u
 •• •,*/*> sending each meridian of K{ to f,.(z=l, •••, μ), where <X, •••, £μ>
is the free abelian group with a basis tlt •••, ^ The first integral homology group
H
λ
{E
a
\Z) is a finitely generated Z<*i, •••, ^ >-module and has a presentation
matrix as a Z " ^ , •••, £μ>-module, written PL(ίi, •••, ίμ).
Concerning presentation matrices, we note the following well-known fact
(cf. [11, pp. 204-205]). For a coefficient ring Λ, the Λ-module presented by
a given matrix P is unchanged, up to Λ-isomorphism, by any one of the fol-
lowing operation on P:
(1) Interchange two rows or two columns.
(2) Add to any row a Λ-linear combination of other rows.
(3) Add to any column a Λ-linear combination of other columns.
(4) Multiply a row or column by a unit of Λ.
/I * ••• *\
0
(5) Replace P with the matrix
CONGRUENCE CLASSES OF KNOTS 213
(6) The reverse of (S).
(7) Adjoin a new row which is a Λ-linear combination of rows of P.
(8) Delete a row which is a Λ-linear combination of other rows.
Proposition 7. Two matrices, with entries in A, present isomorphίc Λ-
module if and only if one can be deformed into the other by a finite sequence of ap-
plications of the above operations (l)-(8).
For a knot type /e, an Alexander matrix A
κ
(t) is very similar to a presen-
tation matrix P
κ
(t) of H
x
{E
a
\ Z) as a Z'<ί>-module. To say more strictly, we
know the following (cf. [2]).
Proposition 8. Two matrices A
κ
(t) and (P
κ
(t) O), where O has only one
column with all entries 0, are equivalent up to fundamental deformations of pre-
sentation matrices. Furthermore, there exists a knot group presentation of K whose
Alexander matrix is (P
κ
(t) O).
D. Rolfsen gave a characterization of P
κ
(t) in [11].
Proposition 9. Up to fundamental deformations of presentation matrices,
P
κ
(t) is equivalent to a matrix of the following form; (tft; (0) where aa{t)=aji(t~ι)
and aij(\) is 0 when iφj, and 1 when i=j.
The entry tftί(0 has the same properties as Alexander polynomials of knots,
so we have aii(t)==±f-l mod {2(1—t), (l—t)σ2(t2)} like as in [9, Theorem 4].
And we can regard that all Λ
ίf (ί) are 1 mod {2(1—t), (1—t) σ2(?)} up to funda-
mental deformations (4). When iφj, the property Λ
ί; ( l ) = 0 implies that aij(t) =
(1—t) h(i) for a certain Laurent polynomial h(i). It can be seen that A (0 = 0,
±f l, or ±f-(l—t) mod {2, 1+*2}. So, we have a
u
(t) = 0, ±f-(l—t), or
±f-(l-t)2 mod {2(l-0> (1-ί) °-2(t2)} for ί φ j .
First, we take the smallest integer i such that α
ί
y(f) =
 =
j
=
£r (l—£) mod
{2(1—t), (1—t) σ2(f)}, and fix i. And we take the smallest integer j such that
aij(t)~±tr'(ί — t) mod {2(1—t), (1—t) σ2{?)}. Adding to the zth row the prod-
uct of the /th row by Tί r (l—i) (which is one of fundamental deformation),
atj(t) becomes 0 mod {2(1—t)y (1—i) σ2{?)}. Then, aH(t) becomes 1, ITf-
(I-*) 2 , or lTf.(l-t)* mod {2(1-/), (1-f) σ2{f)}. Since ( l - * ) 3 = 0 mod
{2(1-0, (l-t)σ2(?)}, lTf.(l-t)3==l mod {2(1 -t)y (1 -t) σ2{?)}. Since
{\-t)2=-t{l-tf = t{\-tf moά {2(l-t),(ί-t)σ2(t2)}f lTts (l-t)2=Ξl-{-
{\—t)2 = t2 mod {2(1—0,(1—t)σ2(#)}. Therefore, new Λf , (ί) can be regarded
to be 1 mod {2(1—1)> (1—t) σ2(?)} up to fundamental deformation (4). We
remark that the other entries aik(t) = 0 or ±£ s (l— tf mod {2(1— t), (ί—t) σ2(t2)}
if k<j. Perform these operations inductively, and we have aij(t) = 0 or i f
( 1 - 0 2 mod {2(1-0, ( ϊ - 0 σ2(f)} for every pair f φ j .
Secondly, for aij(t) = ±f *(\— t)2 mod {2(1—0, (1—0 ^OOh we add to the
214 Y. NAKANISHI
ith row the product of the jth row by ψf (l —t)2, and tf ,,•(£) becomes 0 mod
{2(\-t),{\-t)σ2{?)}. Since (l-ί)4==0 mod {2(l-t),(l-t) cr2(?)}, we can
regard the other entries are unchanged mod {2(1—t), (1—t) <τ2(t2)}.
Hence, we have the fact that (Λ, /(0) *S congruent to the unit matrix mod
{2(1— t), (l—t)σ2(f)} up to fundamental deformations. Furthermore, (# ί ; (0)
is equivalent to (1) by fundamental deformations (6). Since the fundamental
deformations using in the above can be realized by exchanges of knot group
presentations of /c, we have the required Theorem 5.
5. Concluding Remarks
We consider an operation τΛ, whih is a local move cancelling n full-twists
on a link diagram, as shown in Fig. 4. About rn operations, S. Kinoshita
gave results in [4], which is for a special case of congruence modulo 2,2. Here,
we note a μ-variable version as follows. Since the proof is parallel to that in
[4], so we omit it.
n full-twists
-n full-twists
Fig. 4.
T h e o r e m 10. Let two links L and L * be deformable to each other by a finite
sequence of operations τn)s. Then, for properly chosen AL(tιy •••, tμ) and AL*(tly •••,
tμ), we have
, tμ) ΞΞ AL.(tu - , U) mod (\-tt)
(
Furthermore, the Alexander polynomials and elementary ideals of AL and AL* are
congruent modulo {(1-f. ) σ^ί. tj\ (1—f,-) σ
n
{ti tj) (l^ij^
In the previous note [8], the author tried to give the PL-version of the
above, but the proof has gaps. Therefore, we give it here as conjecture. If
this conjecture is valid, then we can show that the Borromean rings and a 3-
component trivial link are never deformed to each other by a finite sequence of
operations τ2>s and link-homotopies.
CONGRUENCE CLASSES OF KNOTS 215
CONJECTURE. Let two links L and L* be deformable to each other by a
finite sequence of operations τ*'s. Then, for properly chosen PL(tlt •--,tμ) and
PAh, —,tμ), we have
( σ
Λ
(t, tj), <r.(ί, tj1),
,
 tμ)
Furthermore, the Alexander polynomials and elementary ideals of PL and PL* are
congruent modulo {σ.(ί, ίy), σ.(ί, ίj1), (1-ί,) ( l - ί y ) σ,_ι(ί, ίy), (1-f,) ( l - ί y ) σn.λ
References
[1] J.W. Alexander: Topological invariants of knots and links, Trans. Amer. Math.
Soc. 30 (1928), 275-306.
[2] R.H. Fox: Free differential calculus, /, Ann. of Math. 57 (1953), 547-560; ibid.,
II, ibid. 59 (1954), 196-210.
[3] R.H. Fox: Congruence classes of knots, Osaka Math. J. 10 (1958), 37-41.
[4] S. Kinoshita: On the distribution of Alexander polynomials of alternating knots and
links, Proc. Amer. Math. Soc. 79 (1980), 644-648.
[5] J. Levine: A characterization of knot polynomials, Topology 4 (1965), 135-141.
[6] H. Murakami and Y. Nakanishi: On a certain move generating link-homology,
Math. Ann. 284 (1989), 75-89.
[7] Y. Nakanishi: Prime links, concordance and Alexander invariants, Math. Sem.
Notes Kobe Univ. 8 (1980), 561-568.
[8] Y. Nakanishi: Fox's congruence classes and Conway's potential functions of knots
and links, Preprint, Kobe Univ., 1986.
[9] Y. Nakanishi and S. Suzuki: On Fox's congruence classes of knots, Osaka J. Math.
24(1987), 217-225.
[10] D. Rolfsen: A surgical view of Alexander's polynomial, in Geometric Topology
(Proc. Park City, 1974), Lecture Notes in Math. 438, 415-423, Springer-Verlag,
Berlin and New York, 1978.
[11] D. Rolfsen: Knots and Links, Math. Lecture Series 7, Publish or Perish Inc.,
Berkeley, 1976.
[12] H. Seifert: Uber das Geschlecht von Knoten, Math. Ann. 110 (1934), 571-592.
Department of Mathematics
Kobe University
Nada-ku, Kobe, 657
Japan



On Fox's congruence classes of knots. II (Dedicated to Professor Akio Hattori on his sixtieth birthday)

http://dlisv03.media.osaka-cu.ac.jp/infolib/user_contents/sugaku/111F0000002-02701-10.pdf

On Fox's congruence classes of knots. II (Dedicated to Professor Akio Hattori on his sixtieth birthday)

Abstract

Similar works

Full text

Available Versions

Osaka City University Repository