We show that if each of $K_1$ and $K_2$ is a trefoil knot or figure eight
knot, the homology 3-sphere defined by the Kirby diagram which is a simple link
of $K_1$ and $K_2$ with framing (0, n) is represented by an n-twisted Whitehead
double of $K_2$ .Comment: 21 pages, 102 figure

Tsuchiya, Masatsuna

English

arXiv

TSUCHIYA, MASATSUNA

Osaka City University Repository

Tsuchiya, M.
Osaka J. Math.
52 (2015), 993–1015
ON HOMOLOGY 3-SPHERES DEFINED BY TWO KNOTS
MASATSUNA TSUCHIYA
(Received February 24, 2014, revised June 30, 2014)
Abstract
We show that if each of K1 and K2 is a trefoil knot or a figure eight knot, the
homology 3-sphere defined by the Kirby diagram which is a simple link of K1 and
K2 with framing (0, n) is represented by an n-twisted Whitehead double of K2.
1. Introduction
We define Wn(K1, K2) to be the 4-dimensional handlebody represented by Fig. 1.1’s
Kirby daigram, the following Kirby diagram, and define Mn(K1, K2) to be (Wn), where
K1 and K2 are knots. Note that Mn(K1, K2) is a homology 3-sphere.
When K1 and K2 are right handed trefoil knots T2.3, Y. Matsumoto asked in [5]
whether M0(T2.3, T2.3) bounds a contractible 4-manifold or not. By Gordon’s result
[3], if n is odd, Mn(T2.3, T2.3) does not bound any contractible 4-manifold. If n is 6,
N. Maruyama [6] proved that M6(T2.3, T2.3) bounds a contractible 4-manifold. If n is 0,
S. Akbulut [1] proved that M0(T2.3, T2.3) does not bound any contractible 4-manifold.
In this note, we show that if each of K1 and K2 is a trefoil knot or a figure
eight knot, the homology 3-sphere defined by Fig. 1.1 is represented by an n-twisted
Whitehead double of K2.
NOTATIONS. (i) Let K be a knot, we define D
C
(K , n) (or D
 
(K , n)) to be the
n-twisted Whitehead double of K with a positive hook (or a negative hook). For ex-
ample, when K is a right handed trefoil knot T2.3, DC(T2.3, n) is the knot represented
by Fig. 1.2, and D
 
(T2.3, n) is the knot represented by Fig. 1.3.
(ii) We define S3
1(K ) to be the 1-surgery along a knot K . For example, when K is
a figure eight knot, S3
C1(DC(K , n)) is represented by Fig. 1.4.
Theorem 1.1. If each of K1 and K2 is a trefoil knot or a figure eight
knot, Mn(K1, K2) is represented by the second column on the following table.
(S3
1(D(K , n))) is the Casson invariant of S31(D(K , n)).
2010 Mathematics Subject Classification. Primary 57R65; Secondary 57M25.
994 M. TSUCHIYA
Fig. 1.1. Wn(K1, K2).
Fig. 1.2. D
C
(T2.3, n). Fig. 1.3. D (T2.3, n). Fig. 1.4. S3
C1(DC(K , n)).
Table 1. Theorem 1.1’s table.
Mn(K1, K2) S3
1(D(K , n)) (S31(D(K , n)))
K1: right handed trefoil, K2: right handed trefoil S3
C1(DC(K2, n))  n
K1: left handed trefoil, K2: right handed trefoil S3
 1(D (K2, n))  n
S3
 1(DC(K2, n))
K1: figure eight knot, K2: right handed trefoil  n
S3
C1(D (K2, n))
K1: right handed trefoil, K2: figure eight knot S3
C1(DC(K2, n))  n
K1: figure eight knot, K2: figure eight knot S3
C1(D (K2, n)) n
We will prove Theorem 1.1 in Section 2.
REMARK. When n is 0, S. Akbulut [1] shows essentially the same result of the
first row on the table by a different method.
Corollary 1.2 (Gordon [3], cf. Y. Matsumoto [7] §3.1). Let Mn(K1, K2) be one
of the manifolds in the above table. If n is odd, Mn(K1, K2) does not bound any con-
tractible 4-manifold.
Proof. A short proof of this result goes as follows:
The Casson invariant, when reduced modulo 2, is the Rohlin invariant:
(Mn(K1, K2))  (Mn(K1, K2)) mod 2.
ON HOMOLOGY 3-SPHERES 995
Fig. 1.5. Mn(T2.3, T2.3).


Fig. 1.6. S3
C1(DC(T2.3, n)).
Fig. 1.7. S3
 1(DC(T2.3, 0)). Fig. 1.8. S3C1(D (T2.3, 0)). Fig. 1.9. S3C1(DC(41, 0)).
By Theorem 1.1, (Mn(K1, K2)) is n or  n. Therefore if n is odd, we have
(Mn(K1, K2))  1 mod 2, and so Mn(K1, K2) does not bound any contractible
4-manifold.
Corollary 1.3 (N. Maruyama [6]). If K1 and K2 are right handed trefoil knots
T2.3, M6(T2.3, T2.3) bounds a contractible 4-manifold.
Proof. By the first row on Theorem 1.1’s table, Mn(T2.3, T2.3) is represented by
S3
C1(DC(T2.3, n)). If n is 6, DC(T2.3, 6) is known to be a slice knot ([8], p. 226). There-
fore by [3], M6(T2.3, T2.3) bounds a contractible 4-manifold.
Corollary 1.4. If n is 0, D
C
(T2.3, 0) is not a slice knot.
Proof. By [1], M0(T2.3, T2.3) does not bound any contractible 4-manifold. There-
fore D
C
(T2.3, 0) is not a slice knot.
REMARK. M. Hedden [4] showed that if n is smaller than 2, D
C
(T2.3, n) is not
a slice knot.
Corollary 1.5. Let T2.3 be a right handed trefoil knot and 41 be a figure eight
knot. The homology 3-spheres S3
 1(DC(T2.3, 0)), S3C1(D (T2.3, 0)) and S3C1(DC(41, 0))
are pairwise diffeomorphic.
Proof. By the third row and the fourth row on Theorem 1.1’s table, if n D 0, the
4-dimensional handlebodies defined by Figs. 1.7, 1.8 and 1.9 have the same boundaries.
996 M. TSUCHIYA
Fig. 1.10. Mn(K1, K2).


Fig. 1.11. S3
 1(DC(K2, n)).


Fig. 1.12. S3
C1(D (K2, n)).
Fig. 1.13. S3
C1(D (T2,3, 6)).
Therefore the homology 3-spheres S3
 1(DC(T2.3,0)), S3C1(D (T2.3,0)) and S3C1(DC(41,0))
are pairwise diffeomorphic.
Corollary 1.6. If K1 is a figure eight knot and K2 is a right handed trefoil knot
(see Fig. 1.10), then M6(K1, K2) bounds a contractible 4-manifold.
Proof. By the third row on Theorem 1.1’s table, Mn(K1, K2) is represented by
S3
 1(DC(K2, n)) and also by S3C1(D (K2, n)). If n is 6, DC(K2, 6) is known to be a
slice knot ([8], p. 226). Therefore by [3], M6(K1, K2) bounds a contractible 4-manifold.
REMARK. By Corollary 1.6, the homology 3-sphere S3
C1(D (T2,3, 6)) bounds a
contractible 4-manifold. The author does not know whether the knot D
 
(T2,3, 6) is a
slice knot or not.
ON HOMOLOGY 3-SPHERES 997
QUESTION. Let V 1n be the 4-dimensional handlebody defined by Fig. 1.12, and V 2n
be the 4-dimensional handlebody defined by Fig. 1.11. Since (V 1n ) is diffeomorphic to
(V 2n ) by Theorem 1.1, we have a closed 4-manifold V 1n [ ( V 2n ). Because DC(T2,3, 6)
is a slice knot, we have a smooth S2 with self intersection  1 in V 26 representing a gen-
erator of H2(V 26 ). Blow down this smooth S2 from the V 16 [ ( V 26 ). Then we are left
with a closed smooth 4-manifold homotopy equivalent to CP2. Is this 4-manifold diffeo-
morphic to CP2?
Proposition 1.7. V 1n [ ( V 2n ) is diffeomorphic to CP2 # CP2.
We show this fact in Section 3.
It seems that Theorem 1.1 is related to [6] Corollary 8 (3), but the author could
not understand the relationship clearly.
The author does not know whether there is an even number n ¤ 0, 6, such that
Mn(T2,3, T2,3) bounds a contractible 4-manifold or not. M. Tange [10] proved that
if n is smaller than 2, Mn(T2,3, T2,3) does not bound any contractible 4-manifold by
computing the Heegaard Floer homology H FC(Mn(T2,3, T2,3)) and the correction term
d(Mn(T2,3, T2,3)).
2. Proof of Theorem 1.1
In this section, first we show that Mn(K1, K2) is represented by S3
1(D(K2, n)).
Next we compute the Casson invariant (S3
1(D(K2, n))).
2.1. Proof of the first row on Theorem 1.1’s table. K1 and K2 are right handed
trefoil knots.
Proof. We show that the 4-manifolds represented by Figs. 2.1 and 2.17 have the
same boundaries by following Kirby calculus:
998 M. TSUCHIYA
Fig. 2.1. Mn(K1, K2).

    !
isotopy
Fig. 2.2.


    !
blow up
Fig. 2.3.


    !
blow up
Fig. 2.4.

    !
isotopy
Fig. 2.5.

    !
isotopy
Fig. 2.6.

      !
handle slide
Fig. 2.7.
ON HOMOLOGY 3-SPHERES 999

      !
handle slide
Fig. 2.8.

     !
surgery
and
cancelling
Fig. 2.9.


      !
blow down
Fig. 2.10.

    !
isotopy
Fig. 2.11.

    !
isotopy
Fig. 2.12.

      !
handle slide
Fig. 2.13.

      !
handle slide
Fig. 2.14.
1000 M. TSUCHIYA


      !
blow down
Fig. 2.15.

    !
isotopy
Fig. 2.16.

    !
isotopy
Fig. 2.17. S3
C1(DC(K2, n)).
2.2. Proof of the second row on Theorem 1.1’s table. K1 is a left handed tre-
foil knot and K2 is a right handed trefoil knot.
Proof. We show that the 4-manifolds represented by Figs. 2.18 and 2.23 have the
same boundaries.
2.3. Proof of the third row on Theorem 1.1’s table. K1 is a figure eight knot
and K2 is a right handed trefoil knot.
Proof. We show that the 4-manifolds represented by Figs. 2.24, 2.29 and 2.35
have the same boundaries.
Fig. 2.30 is the same diagram of Fig. 2.24, but by using the invertibility of the fig-
ure eight knot, we can show that they can be represented by a different doubled knot.
2.4. Proof of the fourth row on Theorem 1.1’s table. K1 is a right handed
trefoil knot and K2 is a figure eight knot.
Proof. We show that the 4-manifolds represented by Figs. 2.36 and 2.47 have the
same boundaries.
ON HOMOLOGY 3-SPHERES 1001
Fig. 2.18. Mn(K1, K2).

    !
isotopy
Fig. 2.19.


    !
blow up
Fig. 2.20.


    !
blow up
Fig. 2.21.

    !
isotopy
Fig. 2.22.


           !
by the same process
except for the sign
of this sign of
the framing
Figs. 2.5–2.17
Fig. 2.23. S3
 1(D (K2, n)).
1002 M. TSUCHIYA
Fig. 2.24. Mn(K1, K2).

    !
isotopy
Fig. 2.25.


    !
blow up
Fig. 2.26.


    !
blow up
Fig. 2.27.

    !
isotopy
Fig. 2.28.


          !
by the same
process except for
the sing of the
framing
Figs. 2.5–2.17
Fig. 2.29. S3
 1(DC(K2, n)).
ON HOMOLOGY 3-SPHERES 1003
Fig. 2.30. Mn(K1, K2).

    !
isotopy
Fig. 2.31.

    !
isotopy
Fig. 2.32.


    !
blow up
Fig. 2.33.


    !
blow up
Fig. 2.34.


          !
by the same
process except for
the sign of the
framing
Figs. 2.4–2.17
Fig. 2.35. S3
C1(D (K2, n)).
1004 M. TSUCHIYA
Fig. 2.36. Mn(K1, K2).

    !
isotopy
Fig. 2.37.


    !
blow up
Fig. 2.38.


    !
blow up
Fig. 2.39.

    !
isotopy
Fig. 2.40.


          !
by the same
process except for
the sign of the
framing
Figs. 2.5–2.11 Fig. 2.41.

    !
isotopy
Fig. 2.42.

       !
handle slide
Fig. 2.43.

       !
handle slide
Fig. 2.44.


      !
blow down
Fig. 2.45.
ON HOMOLOGY 3-SPHERES 1005

    !
isotopy
Fig. 2.46.

    !
isotopy
Fig. 2.47. S3
C1(DC(K2, n)).
2.5. Proof of the fifth row on Theorem 1.1’s table. K1 and K2 are figure
eight knots.
Proof. We show that the 4-manifolds represented by Figs. 2.48 and 2.55 have the
same boundaries.
Since Fig. 2.53 is the same diagram of Fig. 2.51, we can show that they can be
represented by a different double knot.
Next we compute the Casson invariant (Mn(K1, K2)). Now suppose that K , KC
and K0 are links in S3 which have projections which differ at a single crossing of K 
as depicted Fig. 2.56.
REMARK. Our convention in Fig. 2.56 is different from that in [2]. In fact, their
K
C
(resp.K
 
) is our K
 
(resp.K
C
). We adopt our convention as Fig. 2.56 because by
our convention 0(T2,3) is computed to be 1, where T2,3 is a right handed trefoil knot.
While by their convention 0(T2,3) is computed to be  1, contradicting the normaliza-
tion 0(T2,3) D 1 ([2] p148, [9] p. 52).
Lemma 2.1 (see [2], p. 143). Let K
 
be a knot in S3. Let K
C
and K0 be as
above. Then K0 is a two component link and:

0(K
C
)   0(K
 
) D lk(K0)
where 0(K ) is the Casson invariant of a knot K.
Lemma 2.2 (Surgery formula, see [9], p. 52). Let K be a knot in S3. The Casson
invariant (S3
C1(K )) is equal to 0(K ).
By Lemmas 2.1, 2.2 and the second column on Theorem 1.1’s table, we can com-
pute the Casson invariant (Mn(K1, K2)).
1006 M. TSUCHIYA
Fig. 2.48. Mn(K1, K2).


    !
blow up
Fig. 2.49.


    !
blow up
Fig. 2.50.

    !
isotopy
Fig. 2.51.


          !
by the same
process except for
the sign of the
framing
Figs. 2.40–2.47
Fig. 2.52. S3
 1(DC(K2, n)).
ON HOMOLOGY 3-SPHERES 1007
Fig. 2.53.

    !
isotopy
Fig. 2.54.


          !
by the same
process except for
the sign of the
framing
Figs. 2.40–2.47
Fig. 2.55. S3
C1(D (K2, n)).
Fig. 2.56.
1008 M. TSUCHIYA
Fig. 2.57. K
C
. Fig. 2.58. K
 
. Fig. 2.59. K0.
Fig. 2.60. K
C
. Fig. 2.61. K
 
. Fig. 2.62. K0.
2.6. The Casson invariant of the first row on Theorem 1.1’s table.
Proof. K1 and K2 are right handed trefoil knots. By Section 2.1, Mn(K1, K2)
is diffeomorphic to S3
C1(DC(K2, n)). Therefore (Mn(K1, K2)) is equal to
(S3
C1(DC(K2, n))). By Lemmas 2.1 and 2.2, we can compute the Casson invariant
(S3
C1(DC(K2, n))) as Figs. 2.57–2.59.
By Lemma 2.1, 0(K
C
) 0(K
 
) D lk(K0). Since K  is a trivial knot, 0(K ) D 0.
lk(K0) is  n. Therefore, 0(KC) D  n. Then (Mn(K1, K2)) D (S3
C1(DC(K2, n))) D

0(K
C
) D  n.
2.7. The Casson invariant of the second row on Theorem 1.1’s table.
Proof. K1 is a left handed trefoil knot and K2 is a right handed trefoil knot. By
Section 2.2, Mn(K1, K2) is diffeomorphic to S3
 1(D (K2, n)). Therefore (Mn(K1, K2))
is equal to (S3
 1(D (K2, n))). Since (S3 1(D (K2, n))) D  (S3C1(DC(K1,  n))) (see
[9], p. 52, Theorem 3.1.), we compute the Casson invariant (S3
C1(DC(K1,  n))) as
Figs. 2.60–2.62.
By Lemma 2.1, 0(K
C
)   0(K
 
) D lk(K0). Since K  is a trivial knot, 0(K ) D
0. lk(K0) is n. Therefore, 0(KC) D n. Then (Mn(K1, K2)) D (S3
 1(D (K2, n))) D
 (S3
C1(DC(K1,  n))) D  0(KC) D  n.
2.8. The Casson invariant of the third row on Theorem 1.1’s table.
Proof. K1 is a figure eight knot and K2 is a right handed trefoil knot. By Sec-
tion 2.3, Mn(K1, K2) is diffeomorphic to S3
C1(D (K2, n)). Therefore (Mn(K1, K2))
ON HOMOLOGY 3-SPHERES 1009
Fig. 2.63. K
 
. Fig. 2.64. K
C
. Fig. 2.65. K0.
Fig. 2.66. K
C
. Fig. 2.67. K
 
. Fig. 2.68. K0.
is equal to (S3
C1(D (K2, n))). By Lemmas 2.1 and 2.2, we can compute the Casson
invariant (S3
C1(D (K2, n))) as Figs. 2.63–2.65.
By Lemma 2.1, 0(K
C
) 0(K
 
) D lk(K0). Since KC is a trivial knot, 0(KC) D 0.
lk(K0) is  n. Therefore, 0(K ) D n. Then (Mn(K1, K2)) D (S3
C1(D (K2, n))) D

0(K
 
) D n.
2.9. The Casson invariant of the fourth row on Theorem 1.1’s table.
Proof. K1 is a right handed trefoil knot and K2 is a figure eight knot. By 2(iv),
Mn(K1, K2) is diffeomorphic to S3
C1(DC(K2, n)). Therefore (Mn(K1, K2)) is equal to
(S3
C1(DC(K2, n))). By Lemmas 2.1 and 2.2, we can compute the Casson invariant
(S3
C1(DC(K2, n))) as Figs. 2.66–2.68.
By Lemma 2.1, 0(K
C
) 0(K
 
) D lk(K0). Since K  is a trivial knot, 0(K ) D 0.
lk(K0) is  n. Therefore, 0(KC) D  n. Then (Mn(K1, K2)) D (S3
C1(DC(K2, n))) D

0(K
C
) D  n.
2.10. The Casson invariant of the fifth row on Theorem 1.1’s table.
Proof. K1 and K2 are figure eight knots. By Section 2.5, Mn(K1, K2) is diffeo-
morphic to S3
C1(D (K2, n)). Therefore (Mn(K1, K2)) is equal to (S3C1(D (K2, n))).
By Lemmas 2.1 and 2.2, we can compute the Casson invariant (S3
C1(D (K2, n))) as
Figs. 2.69–2.71.
1010 M. TSUCHIYA
Fig. 2.69. K
 
. Fig. 2.70. K
C
. Fig. 2.71. K0.
By Lemma 2.1, 0(K
C
) 0(K
 
) D lk(K0). Since KC is a trivial knot, 0(KC) D 0.
lk(K0) is  n. Therefore, 0(K ) D n. Then (Mn(K1, K2)) D (S3
C1(D (K2, n))) D

0(K
 
) D n.
3. Proof of Proposition 1.7
We show that V 1n [ ( V 2n ) is diffeomorphic to CP2 # CP2.
Proof. By Kirby calculus from Figs. 3.1–3.6, we will show that the Kirby dia-
gram of V 1n [ ( V 2n ) is represented by Fig. 3.6.
4. Appendix
An alternative proof of Corollary 1.2. By [3], if n is odd, Mn(T2,3, T2,3) does not
bound any contractible 4-manifold. In this Section we will give an alternative proof of
this fact. For this purpose, we will prove the following proposition;
Proposition 4.1. The 4-dimensional handlebodies represented by Figs. 4.1, 4.8
and 4.9 have the same boundaries.
Proof. See Figs. 4.1–4.9.
ON HOMOLOGY 3-SPHERES 1011
Fig. 3.1.


          !
by the same
process except for
the sign of the
framing
Fig. 2.5–2.17
Fig. 3.2.

    !
isotopy
Fig. 3.3.


          !
by the same
process except for
the sign of the
framing
Fig. 2.5–2.9 Fig. 3.4.


      !
blow down
Fig. 3.5.


      !
blow down
Fig. 3.6. V 1n [ ( V 2n ).

      !
handle slide
Fig. 3.7. CP2 # CP2.
1012 M. TSUCHIYA
Fig. 4.1. Mn(T2,3, T2,3).

    !
isotopy
Fig. 4.2.


    !
blow up
Fig. 4.3.

    !
isotopy
Fig. 4.4.


    !
blow up
Fig. 4.5.


    !
blow up
Fig. 4.6.

      !
handle slide
Fig. 4.7.


       !
blow up and
down
Fig. 4.8.


       !
blow up and
down
Fig. 4.9.
ON HOMOLOGY 3-SPHERES 1013
Fig. 4.10. Q1.
We give an alternative proof of Corollary 1.2.
Proof of Corollary 1.2. Fig. 4.8 gives a smooth 4-manifold Q1 with intersection
form A.
A D (i j ), i j D ai  a j , 1  i, j  10,
A D
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B

 2 1
2 1
1 2 1
1 1 0 1
1  6 1
1 n   6 1
1 0 1 1
1 2 1
1 2
1  2
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
.
We have Index(A) D 0. Note that A is an even type matrix if n is even.
1014 M. TSUCHIYA
Fig. 4.11. Q2
Fig. 4.9 gives a smooth 4-manifold Q2 with intersection form B.
B D (i j ), i j D bi  b j , 1  i, j  14,
B D
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B

2 1
2 1
1 2 1
1 1 2 1
1 2 1
1 2 1
1 2 1
1 2 1
1 2 1
1 n   5 1
1 0 1 1
1 2 1
1 2
1  2
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
.
We have Index(B) D 8. Note that B is an even type matrix if n is odd.
By Proposition 4.1, we have the Rohlin invariant (Mn(T2,3, T2,3)) as follows:
(Mn(T2,3, T2,3)) 

Index(B)  1 (n is odd)
Index(A)  0 (n is even) mod 2.
Therefore if n is odd, Mn(T2,3, T2,3) does not bound any contractible 4-manifold.
ACKNOWLEDGEMENTS. The author would like to thank Yukio Matsumoto and
Nobuhiro Nakamura for their useful comments and encouragement. The author is very
grateful to the referee for his/her careful reading and useful comments.
ON HOMOLOGY 3-SPHERES 1015
References
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[3] C.McA. Gordon: Knots, homology spheres, and contractible 4-manifolds, Topology 14 (1975),
151–172.
[4] M. Hedden: Knot Floer homology of Whitehead doubles, Geom. Topol. 11 (2007), 2277–2338.
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[7] Y. Matsumoto: On the bounding genus of homology 3-spheres, J. Fac. Sci. Univ. Tokyo Sect.
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[8] D. Rolfsen: Knots and Links, reprinted with corrections by the AMS, Amer. Math. Soc., Provi-
dence, RI, 2003.
[9] N. Saveliev: Invariants for Homology 3-Spheres, Encyclopaedia of Mathematical Sciences 140,
Springer, Berlin, 2002.
[10] M. Tange: Heegaard Floer homology of Matsumoto’s manifolds, http://
www.math.tsukuba.ac.jp/~tange/MatsumotoM.pdf.
Department of mathematics
Gakushuin University
5-1, Mejiro 1-chome, Toshima-ku
Tokyo, 171-8588
Japan
e-mail: tsuchiya@math.gakushuin.ac.jp


On homology 3-spheres defined by two knots

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