GENUS 2 MUTANT KNOTS WITH THE SAME DIMENSION IN KNOT FLOER
AND KHOVANOV HOMOLOGIES
ALLISON MOORE AND LAURA STARKSTON
Abstract. We exhibit an infinite family of knots with isomorphic knot Heegaard Floer homology.
Each knot in this infinite family admits a nontrivial genus 2 mutant which shares the same total
dimension in both knot Floer homology and Khovanov homology. Each knot is distinguished from its
genus 2 mutant by both knot Floer homology and Khovanov homology as bigraded groups, as well
as by the δ-graded version of knot Heegaard Floer homology.
1. Introduction
Genus 2 mutation is an operation on a 3-manifold M in which an embedded, genus 2 surface F is
cut from M and reglued via the hyperelliptic involution τ . The resulting manifold is denoted M τ .
When M is the three-sphere, the genus 2 mutant manifold (S 3 )τ is homeomorphic to S 3 . If K ⊂ S 3
is a knot disjoint from F , then the knot that results from performing a genus 2 mutation of S 3 along
F is denoted K τ and is called a genus 2 mutant of the knot K. The related operation of Conway
mutation in a knot diagram can be realized as a genus 2 mutation or a composition of two genus 2
mutations.
In [21], Ozsváth and Szabó demonstrate that as a bigraded object, knot Heegaard Floer homology
can detect Conway mutation. However, it can be observed that in all known examples [1], the rank of
[
HFK(K)
as an ungraded object remains invariant under Conway mutation. The question of whether
the rank of knot Floer homology is unchanged under Conway mutation, or more generally, genus 2
mutation, remains an interesting open problem. Moreover, while it is known that Khovanov homology
with F2 = Z/2Z coefficients is invariant under Conway mutation [4],[30], the general case is also
unknown. The invariance of the rank of Khovanov homology under genus 2 mutation constitutes a
natural generalization of the question. In this note, we offer an example of an infinite family of knots
with isomorphic knot Floer homology, all of which admit a genus 2 mutant of the same dimension in
[ and Kh, though each pair is distinguished by both HFK
[ and Kh as bigraded vector spaces.
both HFK
1
Theorem 1. There exists an infinite family of genus 2 mutant pairs (Kn , Knτ ), n ∈ Z+ , in which
(1) each infinite family has isomorphic knot Floer homology groups,
∼
[ m (Kn , s) =
HFK
[ m (Knτ , s) ∼
HFK
=
[ m (K0 , s), ∀m, s
HFK
[ m (K0τ , s), ∀m, s,
HFK
[ and Kh,
(2) each genus 2 mutant pair shares the same total dimension in HFK
M
M
[ m (Kn , s) =
[ m (Knτ , s)
dimF2 HFK
dimF2 HFK
m,s
M
m,s
dimQ Khiq (Kn )
=
M
dimQ Khiq (Knτ ),
i,q
i,q
[ and Kh as bigraded groups,
(3) and each genus 2 mutant pair is distinguished by HFK
[ m (Kn , s) ∼
[ m (Knτ , s) for some m, s
HFK
6= HFK
i
Kh (Kn ) ∼
6
Khi (K τ ) for some i, q.
=
q
q
n
1Because we compute HFK
[ and Kh as graded vector spaces over Z/2Z or Q, the theorem has been formulated in
terms of dimension rather than rank.
1
2
MOORE AND STARKSTON
This example suggests that having invariant dimension of knot Floer homology or Khovanov homology
is a property shared not only by Conway mutants, but by genus 2 mutant knots as well, offering
positive evidence towards all the above open questions about total rank.
1.1. Organization. In Section 2 we review genus 2 mutation and describe the infinite family of genus
2 mutant pairs. In Section 3 we show that within each infinite family {Kn } and {Knτ }, the knots
have isomorphic knot Heegaard Floer homology and that these families share the same dimension. In
Section 4 we show that each family also shares the same dimension of Khovanov homology. Section 5
mentions a few observations.
2. Genus 2 Mutation
Figure 1. The genus 2 surface F and hyperelliptic involution τ .
Let F be an embedded, genus 2 surface in a compact, orientable 3-manifold M , equipped with the
hyperelliptic involution τ . A genus 2 mutant of M , denoted M τ , is obtained by cutting M along F
and regluing the two copies of F via τ [8]. The involution τ has the property that an unoriented simple
closed curve γ on F is isotopic to its image τ (γ).
When M = S 3 , any closed surface F ⊂ S 3 is compressible. This implies by the Loop Theorem that
(S 3 )τ is homeomorphic to S 3 [8]. Therefore, if S 3 contains a knot K disjoint from F , mutation along
F is a well-defined homeomorphism of S 3 taking a knot K to a potentially different knot K τ [8]. In
this note, we restrict our attention to surfaces of mutation which bound a handlebody containing K
in its interior. These mutations are called handlebody mutations.
A Conway mutant of a knot K ⊂ S 3 is similarly obtained by an operation under which a Conway
sphere S interests K in four points and bounds a ball containing a tangle. The ball containing the
tangle is replaced by its image under a rotation by π about a coordinate axis. In fact, Conway mutation
of a knot can be realized as a special case of genus 2 mutation. Since S separates K into two tangles,
i.e.
K = T1 ∪S T2
a genus 2 surface F is formed by taking S and tubing along either T1 or T2 . The Conway mutation is
then achieved by performing at most two such genus 2 mutations [8]. Like Conway mutants, genus 2
mutants are difficult to detect and are indistinguishable by many knot invariants [8].
Theorem 2. [5], [16] The Alexander polynomial and colored Jones polynomials for all colors of a
knot in S 3 are invariant under genus 2 mutation. Generalized signature is invariant under genus 2
handlebody mutation.
Theorem 3 (Theorem 1.3 of [26]). Let K τ be a genus 2 mutation of the hyperbolic knot K. Then K τ
is also hyperbolic, and the volumes of their complements are the same.
Theorem 3 is a special case of a more general theorem which shows that the Gromov norm is preserved
under mutation along any of several symmetric surfaces, including the genus 2 surface on which we are
focused here. Ruberman also shows that cyclic branched coverings and Dehn surgeries along a Conway
mutant knot pair yield manifolds of the same Gromov norm. Moreover, it is well-known that Conway
mutation preserves the homeomorphism type of the branched double covering. In light of this, it is
natural to ask whether Σ2 (K) is homeomorphic to Σ2 (K τ ); however, this is not the case. We verify
this by investigating the pair of genus 2 mutant knots in Figure 2, which we call K0 and K0τ and which
are known as 14n22185 and 14n22589 in Knotscape notation.
GENUS 2 MUTANT KNOTS WITH THE SAME DIMENSION IN KNOT FLOER AND KHOVANOV HOMOLOGIES 3
14n22185
14n22589
Figure 2. The genus 2 mutant pair K0 = 14n22185 and K0τ = 14n22589 .
Proposition 4. The branched double covers of K0 and K0τ are not homeomorphic.
Proof. This is a fact which can be checked by computing the geodesic length spectra of Σ2 (K0 ) and
Σ2 (K0τ ) in SnapPy [6] with the following code snippet.
>> M1=M a n i f o l d ( ” 1 4 n22185 . t r i ” ) ; M2=M a n i f o l d ( ” 1 4 n22589 . t r i ” )
>> M1. d e h n f i l l ( ( 2 , 0 ) , 0 ) ; M2 . d e h n f i l l ( ( 2 , 0 ) , 0 )
>> M1. c o v e r s ( 2 , c o v e r t y p e =” c y c l i c ” ) ; M2. c o v e r s ( 2 , c o v e r t y p e =” c y c l i c ” )
>> M1. l e n g t h s p e c t r u m ( c u t o f f = 1 . 5 )
mult l e n g t h
1
(0.618708509882 −0.915396961493 j )
1
(1.02046533287 −2.87373908997 j )
1
(1.19267652219 −1.97573028631 j )
1
(1.2943687184 −0.108601853389 j )
1
(1.4180061001+1.77458043688 j )
topology
mirrored arc
mirrored arc
circle
mirrored arc
circle
parity
o r i e n t a t i o n −p r e s e r v i n g
o r i e n t a t i o n −p r e s e r v i n g
o r i e n t a t i o n −p r e s e r v i n g
o r i e n t a t i o n −p r e s e r v i n g
o r i e n t a t i o n −p r e s e r v i n g
>> M2. l e n g t h s p e c t r u m ( c u t o f f = 1 . 5 )
mult l e n g t h
1
(0.61977975736+1.04574145952 j )
1
(0.946415249278+3.02707626124 j )
1
(1.07345426322+2.11448221051 j )
1
(1.2943687184 −0.108601853389 j )
topology
mirrored arc
mirrored arc
circle
mirrored arc
parity
o r i e n t a t i o n −p r e s e r v i n g
o r i e n t a t i o n −p r e s e r v i n g
o r i e n t a t i o n −p r e s e r v i n g
o r i e n t a t i o n −p r e s e r v i n g
The complex length spectrum of a compact hyperbolic 3-orbifold M is the collection of all complex lengths of closed geodesics in M counted with their multiplicities (Chapter 12 of [15]). SnapPy
demonstrates that the complex length spectra of Σ2 (K) and Σ2 (K τ ) bounded above are different,
therefore these manifolds are not isospectral, and therefore not isometric. Mostow rigidity says that
the geometry of a finite-volume hyperbolic 3-manifold is unique, therefore Σ2 (K) and Σ2 (K τ ) are not
homeomorphic.
Corollary 5. The genus 2 mutant pair K0 and K0τ are not Conway mutants.
Proof. Since Conway mutants have homeomorphic branched double covers, this follows directly from
Proposition 4.
We will continue to explore the pair 14n22185 and 14n22589 . As genus 2 mutants, they share all of the
properties mentioned in Theorems 2 and 3. Moreover, 14n22185 and 14n22589 are also shown in [8] to have
the same HOMFLY-PT and Kauffman polynomials, although in general these polynomials are known
to distinguish larger examples of genus 2 mutant knots [8]. Just as a subtler hyperbolic invariant
was required to distinguish their branched double covers, we require a subtler quantum invariant to
[ and Kh do the trick.
distinguish the knot pair. The categorified invariants HFK
Theorem 6. The genus 2 mutant knots K0 and K0τ are distinguished by their knot Heegaard Floer
homology and Khovanov homology.
See Table 1. Khovanov homology with Z coefficients was computed in [8] using KhoHo [27]. Here,
we include Khovanov homology with rational coefficients computed with the Mathematica program
4
MOORE AND STARKSTON
[
HFK(K
0)
−2 −1 0
3
2
1
0
−1
−2
F2
F2 F3
F F2
F
dim = 17
τ
[
HFK(K
0)
−1 0 1
2
F2
5
1
F F2
0 F2 F4
−1 F2
dim = 17
1 2
F
F2 F
F2
[
δ − graded HFK(K
0)
−2 −1 0 1 2 dim
s − m = −1
F F2 F2 F2 F
8
s−m=0
F F2 F3 F2 F
9
dim = 17
Kh(K0 ; Q) =
Kh(K0τ ; Q)
=
7
6
4
3
3
2
2
1
6
19
5
19
4
19
4
17
4
15
3
17
3
15
τ
[
δ − graded HFK(K
0)
−1 0 1 dim
s − m = −1 F2 F5 F2
9
s − m = 0 F2 F4 F2
8
dim = 17
1
113 19 17 17 13 15 13 13 11 103 201 201 211 113 121 123 125 133 135 137 147 157 1611
dim = 26
13 15 23 13 11 11 201 201 111 113 121 125 135 157 1611
dim = 26
7
113
3
2
2
1
1
1
Table 1. Knot Floer groups are displayed with Maslov grading on the vertical axis
and Alexander grading on the horizontal axis. Computation [7] also confirms that
[ 0) ∼
[ 1 ) and HFK(K
[ 0τ ) ∼
[ 1τ ). For Khovanov homology, Rij
HFK(K
= HFK(K
= HFK(K
denotes Khovanov groups in homological grading i and quantum grading j with dimension R. The underline denotes negative gradings. This notation originated in
[3].
[ is known to detect Conway mutation [21], it is not surprising that knot
JavaKH [9]. Since HFK
[ 0)
Floer homology can distinguish genus 2 mutant pairs. Nonetheless, the knot Floer groups HFK(K
τ
[ 0 ) have been computed using the Python program of Droz [7]. The key observation is
and HFK(K
that although both knot Floer homology and Khovanov homology distinguish the genus 2 mutants as
bigraded vector spaces, in both cases the pairs are indistinguishable as ungraded objects.
Figure 3. The surface of mutation for all Kn . Note the surface bounds a handlebody.
We will derive an infinite family of knots from the pair 14n22185 and 14n22589 . Notice that each of these
can be formed as the band sum of a two-component unlink. Let us call 14n22185 and 14n22589 by K0 and
K0τ , respectively. By adding n right-handed half-twists to the bands of K0 and K0τ , as in Figure 4, we
obtain knots Kn and Knτ . It is visibly clear that that Knτ is the genus 2 mutant of Kn by the same
surface of mutation relating K0 and K0τ , illustrated in Figure 3.
Observe that by resolving a crossing in the twisted band, Kn and Kn−2 fit into an oriented Skein
triple (L+ , L− , L0 ) with L0 equal to the two-component unlink U for all integers n > 1. Moreover,
τ
τ
Kn and Kn−1 fit into an unoriented Skein triple, again with third term the unlink. Knτ , Kn−1
, Kn−2
and U fit into these same oriented and unoriented Skein triples. Similar statements can be made when
left-handed twists are placed in the band.
GENUS 2 MUTANT KNOTS WITH THE SAME DIMENSION IN KNOT FLOER AND KHOVANOV HOMOLOGIES 5
(a) Oriented Skein triple of Kn ,
Kn−2 and U .
(b) Unoriented Skein triple of Kn ,
Kn−1 and U .
Figure 4. Oriented and unoriented Skein triples.
Figure 5. A smooth cobordism illustrating that Kn is slice.
Lemma 7. Let K be any knot formed from the band sum of a two-component unlink. Then K is
smoothly slice.
For example, the knots Kn and Knτ are such knots.
Proof. Recall that a knot K ⊂ S 3 is (smoothly) slice when it bounds a (smoothly) embedded disk in
B 4 . We construct the disk bounding K in S 3 × (0, 1] ⊂ B 4 (where the (0, 1] component represents the
radius from the center of B 4 ). Start with K × [ 34 , 1] and smoothly attach a band near the top crossing
in the column of twists so that the boundary of the result is the 0-resolution as above, lying in S 3 × { 21 }
(thickening the rest of the surface as a product around the band). This resolution is isotopic to the
standard two-component unlink, so perform this isotopy in S 3 × [ 14 , 21 ]. Then smoothly cap of the two
unknotted circles with disks. The resulting surface is a smoothly embedded disk (see Figure 5). To
verify this is a disk, simply compute the Euler characteristic.
3. Knot Floer Homology
Knot Floer homology is a powerful invariant of oriented knots and links in an oriented three manifold
Y , developed originally by Ozsváth and Szabó [18], and independently by Rasmussen [24]. We tersely
paraphrase Ozsváth and Szabó’s construction of the invariant for knots from [18], and refer the reader
to [18] for details of the construction.
3.1. Background from knot Floer homology. To a knot K ⊂ S 3 is associated a doubly pointed
Heegaard diagram (Σ, α, β, w, z). The data of the Heegaard diagram gives rise to a Z2 filtered chain
complex CF∞ (Σ, α, β, w, z) generated by triples [x, i, j], where x ∈ Tα ∩ Tβ is an intersection point
of two Lagrangian submanifolds in Symg (Σ) and i, j ∈ Z. The chain complex is made into an F2 [U ]module by defining an operator U which acts by U [x, i, j] = [x, i − 1, j − 1].
CF∞ (Σ, α, β, w, z) splits into summands CF∞ (Y, K, t) parameterized by t, where t is a Spinc structure
over Y0 (K) which extends s ∈ Spinc (Y ). By fixing a particular Spinc structure t0 , we obtain a filtration
of CF∞ by the j-term. This descends to a filtration of the subcomplex CFK−,∗ (Y, K, t0 ), generated
by triples [x, i, j] with i ≤ 0, and the quotient complex CFK0,∗ (Y, K, t0 ) generated by triples with
6
MOORE AND STARKSTON
i = 0. Since the j-term filtration depended on the choice of t0 , the associated graded complexes
CFK0,∗ (Y, K, t0 ) and CFK−,∗ (Y, K, t0 ) are alternatively graded by elements of Spinc (Y0 (K)). When
Y = S 3 , there is a unique Spinc structure s, and we may identify Spinc (Y0 (K)) with Z. In this case,
the associated graded complexes are graded by integers and identified with
M
M
[ 3 , K, s) and
CFK(S
CFK– (S 3 , K, s).
s∈Z
s∈Z
We describe the indexing by Alexander grading A(x) = s and Maslov grading M (x) = m. Each
summand of the chain complex is generated by intersection points with A(x) = s, and their respective
differentials are given by
X
X
b =
c
#M(φ)
·y
∂x
(1)
y∈Tα ∩Tβ |A(y)=s
µ(φ)
=
1
φ∈π2 (x,y)
nw (φ) = 0, nz (φ) = 0
(2)
∂ − [x, i] =
X
X
µ(φ) = 1 y∈Tα ∩Tβ |A(y)=s
φ∈π2 (x,y) n (φ) = 0
w
c
#M(φ)
· [y, i − nz (φ)],
where φ ∈ π2 (x, y) is a Whitney disk connecting x to y, µ(φ) is the Maslov index of φ, nz (φ) is the
c
algebraic intersection number #(φ ∩ {z} × Symg−1 (Σ)) and #M(φ)
is the number of points in the
moduli space of pseudo-holomorphic representatives of φ modulo an R-action. The associated graded
homology groups
M
M
[ 3 , K) =
[ 3 , K, s) and HFK– (S 3 , K) =
HFK(S
HFK(S
HFK– (S 3 , K, s)
s∈Z
s∈Z
are invariants of K.
We will require the following two theorems of Ozsváth and Szabó, which we state without proof.
Theorem 8 (Theorem 1.1 of [22]). Let L+ , L− and L0 be three oriented links, which differ at a single
crossing as indicated by the notation. Then, if the two strands meeting at the distinguished crossing in
L+ belong to the same component, so that in the oriented resolution the two strands corresponding to
two distinct components a and b of L0 , then there are long exact sequences
fb
g
b
b
h
[ m (L+ , s) −→ HFK
[ m (L− , s) −→ HFK
[ m−1 (L0 , s) −→ HFK
[ m−1 (L+ , s) −→ · · ·
· · · −→ HFK
CFL– (L0 )
f−
g−
h−
· · · −→ HFK– m (L+ , s) −→ HFK– m (L− , s) −→ Hm−1
, s −→ HFK– m−1 (L+ , s) −→ · · ·
U1 − U2
We remark that the Skein exact sequence of Theorem 8 is derived from a mapping cone construction.
Indeed, Ozsváth and Szabó show in Theorem 3.1 of [22] that there is a chain map f : CFK– (L+ ) →
CFK– (L− ) whose mapping cone is quasi-isomorphic to the mapping cone of the chain map U1 − U2 :
CFL– (L0 ) → CFL– (L0 ), which is in turn quasi-isomorphic to the complex CFL– (L0 )/U1 − U2 . Spe[ 0 ).
cializing to U1 = U2 = 0, one obtains the Skein long exact sequence with third term HFK(L
Theorem 9 (Lemma 3.6 of [18]). Let Y be an oriented three-manifold, K ⊂ Y be a knot, and fix a
Spinc structure s ∈ Spinc (Y ). Then, there is a convergent spectral sequence of relatively graded groups
whose E 1 term is
M
[
HFK(Y,
K, t)
{t∈Spinc (Y0 (K))|t extends s}
c
and whose E ∞ term is HF(Y,
s). Moreover, when c1 (s) is torsion, the spectral sequence respects absolute
gradings.
We are concerned with oriented Skein triples (L+ , L− , L0 ) = (Kn , Kn−2 , U), where Kn are knots
and U is the two-component unlink in Y = S 3 . In the case of a knot in S 3 , the spectral sequence
c 3) ∼
induced by the knot terminates at HF(S
= F2 , supported in Maslov grading zero (or at F2 [U ] for
HF– (S 3 )).
[ 0 ) appearing in the Skein exact sequence actually refers to the Floer homology of the
The term HFK(L
f0 in S 2 × S 1 (see Section 2.1 of [18]). When L = U,
knotification of L0 , which is an oriented knot L
GENUS 2 MUTANT KNOTS WITH THE SAME DIMENSION IN KNOT FLOER AND KHOVANOV HOMOLOGIES 7
the associated spectral sequence collapses at the E 1 page, which we make precise with the following
lemma.
Lemma 10. Let U be the two-component unlink in S 3 . U corresponds with the unknot Ue ⊂ S 2 × S 1 ,
whose knot Floer homology is
(3)
(4)
e ∼
[ 3 , U) ∼
[ 2 × S 1 , U)
HFK(S
= HFK(S
= F2 m = 0 ⊕ F2 m = −1
s=0
s=0
–
CFL (U) ∼ [ 2
e ⊗F2 F2 [U ]
H∗
= HFK(S × S 1 , U)
U1 − U2
where in the module F2 [U ], the action of U drops the Maslov grading by two and the Alexander grading
by one.
Proof. Let U ⊂ S 3 . Take a pair of points, with one point lying on each link component. Remove balls
B1 and B2 about each point, and attach an S 2 × I along the two S 2 -boundary components. Since
S 3 − (B1 ∪ B2 ) ∼
= S 2 × I, the result of attaching this S 2 × I is S 2 × S 1 . Form the connect sum of
the two link components via a band embedded in the attached S 2 × I. Since each link component is
isotopic to the unknot, by sliding along the S 2 × I band we see that the connect sum corresponds to
e ⊂ S 2 × S 1 is shown in Figure 6. The diagram
the unknot Ue ⊂ S 2 × S 1 . A Heegaard diagram for U
F
f
G
e
E
D
b
d
c
w
C
a
A
z
B
Figure 6. A genus 1 Heegaard diagram for the unknot in S 2 × S 1 .
contains six intersection points and seven regions. The periodic domain P = A − C + D − F contains
both positive and negative coefficients, therefore the diagram is weakly admissible. Moreover, since
s = 0 corresponds to t, the torsion Spinc structure extending the torsion Spinc structure on the knot
complement, we have that hc1 (t, H(D)i = 0 for any any domain D, where H(D) is the corresponding
element of H2 (Y ; Z). This means the diagram is also strongly t-admissible (Definition 4.10 of [20]).
There are only three orientation-preserving Whitney disks connecting intersection points, and these
are given by the domains A, C, E. The differential of the chain complex is described by
b =a
∂f
b =c
∂d
b =a
∂b
b = 0.
∂e
∼ hf + b, ei. Since the basepoint z is not contained in any of A, C or E,
[ 2 × S 1 , Ue) =
Therefore, HFK(S
–
2
1
∼
e ⊗F2 F2 [U ]. The relative grading difference is evident from
[
then HFK (S × S , U) = HFK(S 2 × S 1 , U)
the diagram and pinned down by the observation that the U ⊂ S 3 fits into a Skein exact sequence with
c 2 × S1) ∼
the unknot. Note that HF(S
= F2 ⊕ F2 can be computed from the same diagram by ignoring
z. Compare with Proposition 3.1 of [19].
Lemma 11. The Ozsváth and Szabó τ invariant and Rasmussen s invariant vanish for all Kn and
Knτ .
Proof. The Ozsváth and Szabó smooth concordance invariant τ (K) is defined as the minimal filtration
level for which the inclusion of chain complexes
CFK0,j≤s (Y, Kn , t0 ) ֒→ CFK0,j (Y, Kn , t0 )
8
MOORE AND STARKSTON
induces a non-trivial map on homology [17]. For knots in S 3 , this corresponds to the filtration level of
the single cycle in Maslov grading zero remaining on the E ∞ page of the spectral sequence mentioned
in Theorem 9. τ (K) provides a lower bound on the four-ball genus (see Corollary 1.3 of [21])
|τ (K)| ≤ g∗ (K).
Similarly, in Khovanov homology Rasmussen’s invariant s(K) ∈ 2Z also gives a lower bound on the
four ball genus (see Theorem 1 of [25])
|s(K)| ≤ 2g∗ (K).
Since all of our knots are slice, we immediately obtain τ = s = 0.
3.2. Knot Floer homology proof. The main objective of this section is to show that each knot in
[ 0 ), and that each knot in the family
the family {Kn } has knot Floer homology isomorphic to HFK(K
τ
τ
[ 0 ). The proof is an application of the Skein exact
{Kn } has knot Floer homology isomorphic to HFK(K
sequence above. The observation that a Skein triple can be used to generate knots with isomorphic
knot homologies is not new, and occurs in the work of the second author [28], Watson [29] and Greene
and Watson [11], to name a few. The generalization in Lemma 12 to include all ribbon knots formed
from the band sum of a two-component unlink (rather than our specific families of interest) is an
observation that is originally due to Matthew Hedden and will soon appear as part of a more general
result in [12].
Lemma 12. Let K be a knot in S 3 formed from the band sum of a two-component unlink, and let
{Kn } denote the family of knots obtained by adding n half-twists to the band. For all m, s ∈ Z and
n ≥ 2 ∈ Z, HFK– m (Kn , s) ∼
= HFK– m (Kn−2 , s).
Proof. The proofs for right and left-handed twists are very similar, so we assume here that the twists
are right-handed. We proceed by induction on n.
Just as with the specific families of knots described above, Kn fits into the Skein triple (Kn , Kn−2 , U).
Theorem 8 applied to the Skein triple gives a long exact sequence
f−
g−
· · · → HFK– m (Kn , s) −→ HFK– m (Kn−2 , s) −→ Hm−1
CFL– (U )
,s
U1 − U2
h−
−→ HFK– m−1 (Kn , s) → · · · .
We will use this sequence in conjunction with information coming from the τ invariant. By Lemma 11,
τ (Kn ) = 0 ∀n. Therefore, we isolate an element in bigrading (0, 0) which ‘survives’ in the spectral
c 3 ). Although the τ invariant is usually defined in terms of HFK,
[ there is
sequence terminating at HF(S
an alternate definition for τ of the mirror of K appearing in [23] which is expressed in terms of HFK– ,
τ (m(K)) = max{s | ∃ x ∈ HFK– (K, s) such that U d x 6= 0 for all integers d ≥ 0}.
Since τ (Kn ) = 0, we have the additional fact that τ (Kn ) = τ (m(Kn )). Using the HFK– formulation
of the definition of τ , we have the existence of a unique element xn ∈ HFK– (Kn , 0) such that U d x 6= 0
∀d and in particular, xn is in bigrading (0, 0).
The third term U of the Skein triple corresponds with Ue ⊂ S 2 × S 1 , which induces a spectral
2
1
sequence terminating
at
the Floer homology of S × S . In this case, there exist two elements
CFL– (L0 )
′
z, z ∈ H∗ U1 −U2 , 0 , ∗ = 0, −1 respectively, such that U d z 6= 0 and U d z ′ 6= 0 ∀d. Each of
xn , xn−2 , z and z ′ generate
anF2 [U ] summand, and in the case of z and z ′ , they generate the en
CFL– (L0 )
tire homology of H∗ U1 −U2 . Since HFK– (U) is supported entirely in the torsion Spinc structure,
∼ HFK– m (Kn−2 , s) in the
the long exact sequence immediately supplies isomorphisms HFK– m (Kn , s) =
c
non-torsion Spin summands and whenever |m| > 1.
0
/ HFK– 1 (Kn )
f−
/ HFK– 1 (Kn−2 )
g−
h−
/ F2 [U ]
HFK– 0 (Kn−2 )
j−
/ F2 [U ]
∈
∈
xn−2 {0,0}
/ z′
{−1,0}
k
−
i−
/
∈
∈
z{0,0}
/ HFK– 0 (Kn )
/ xn {0,0}
/ HFK– −1 (Kn )
ℓ
−
/ HFK– −1 (Kn−2 )
/0
GENUS 2 MUTANT KNOTS WITH THE SAME DIMENSION IN KNOT FLOER AND KHOVANOV HOMOLOGIES 9
In the diagram above, equivariance of the long exact sequence with respect to the action of U implies
that z cannot be in the image of any torsion element. Since HFK– 1 (Kn−2 , 0) is torsion, z is not in the
image of g − , and the map g − = 0. Exactness implies that f − is an isomorphism, and also that h− is
an injection. Since the map h− is degree preserving, z maps isomorphically onto xn . By exactness,
xn ∈ Ker i− . Now xn−2 is neither the image of xn , nor can it be the image of any torsion element (by
equivariance), therefore xn−2 6∈ Im i− . By exactness, xn−2 6∈ Ker j − . Since j − preserves Alexander
grading, xn−2 must map to z ′ . Exactness implies that k − = 0 and ℓ− is an isomorphism. What
remains is an isomorphism of torsion submodules at i− .
Hence, for all (m, s), HFK– m (Kn , s) ∼
= HFK– m (Kn−2 , s).
Corollary 13. Let {Kn } and {Knτ } denote the infinite family of knots derived from 14n22185 and
14n22589 . Then
∼
[ m (Kn , s) =
HFK
τ
[ m (Kn , s) ∼
HFK
=
[ m (K0 , s)
HFK
[ m (K0τ , s).
HFK
Proof. Once a suitable base case has been established, then the result follows directly. There are four
distinct families in our investigation, with base cases K0 , K1 , K0τ and K1τ , for even and odd values
[ of each has been verified computationally with the program of Droz [7].
of n. The hat-version HFK
τ
[
[
[ 0 ) and HFK(K
[ 0τ ), respectively
HFK(K1 ) and HFK(K1 ) have been found to be isomorphic with HFK(K
–
[ by the 5-Lemma.
(see Table 1). The isomorphism of HFK implies isomorphism of HFK
U
/ HFK– m (Kn , s)
/ HFK
/ ···
/ HFK– m (Kn , s)
[ m (Kn , s)
···
∼
∼
=
=
/ HFK
/ HFK– m (Kn−2, s) U / HFK– m (Kn−2, s)
/ ···
[ m (Kn−2, s)
···
This verifies that {Kn }, n ∈ Z+ , is an infinite family of knots admitting a distinct genus 2 mutant of
the same total dimension in knot Floer homology..
4. Khovanov Homology
Khovanov homology is a bigraded homology knot invariant introduced in [13]. The chain complex and
differential of the homology theory are computed combinatorially from a knot diagram using the cube
of smooth resolutions of the crossings. See [3] for an introduction to the theory. Here, we compute the
Khovanov homology of Kn and Knτ over rational coefficients. While our computation of Heegaard Floer
homology was over coefficients in F2 , we need to work over Q to obtain the corresponding results in
Khovanov homology. This is for two reasons. First, Rasmussen’s invariant and Lee’s spectral sequence
are only applicable to Khovanov homology with rational coefficients, and we require these tools for
the computation. Furthermore, Khovanov homology over F2 coefficients is significantly weaker at
distinguishing mutants. Bloom and Wehrli independently proved that Khovanov homology over F2
is invariant under Conway mutation in [4], [30]. While these pairs are not Conway mutants, we can
compute that K0 and K0τ have the same F2 -Khovanov homology (though we have not proven this for
the infinite family). The goal of this section is to provide an infinite family of genus 2 mutants where
the bigraded rational Khovanov homology distinguishes between the knot and its mutant, whereas the
total dimension of the Khovanov homology is invariant under the mutation. Our main result in this
section is the following theorem.
Theorem 14. The Khovanov homology with rational coefficients for Kn respectively Knτ , for n ≥ 8 is
described by the following sequences of the numbers. Here Rij denotes that the Khovanov homology in
homological grading i and quantum grading j has dimension R. This notation originated in [3]
Kh(Kn ) =
n−7 n−6 n−4 n−3 n−3
n−2 n−2
n−1
n−1
10−1 101 11+m
15+m 17+m 17+m 111+m 19+m
111+m 111+m
113+m
1n11+m 1n13+m 1n15+m
n+1
n+2
n+2
n+2
n+3
n+3
n+3
n+4
n+5
n+6
2n+1
15+m 117+m 115+m 117+m 119+m 117+m 119+m 121+m 121+m 121+m 125+m
Kh(Knτ )
n−7 n−6 n−5 n−4 n−4 n−4 n−3 n−3 n−3
n−2 n−2
n−1
= 10−1 101 11+m
15+m 15+m 15+m 17+m 19+m 17+m 19+m 111+m 19+m
211+m 111+m
n−1
n−1
n+1
n+2
n+2
n+3
n+5
n+6
113+m
115+m
1n13+m 1n15+m 1n+1
15+m 117+m 115+m 119+m 119+m 121+m 125+m
10
MOORE AND STARKSTON
where m = 2(n − 7).
The key aspect of this computation to note for the proof is that as n increases by 1, in all but the first
two terms the homological grading increases by 1 and the quantum grading increases by 2. The first
part of the proof will justify the computation for all but the first two terms. The second part of the
proof justifies the computation of the first two terms. Before we give the proof of the computation,
the following corollary highlights the relevant conclusions.
Corollary 15. For all n ≥ 0,
Kh(Kn ) ∼
6 Kh(Knτ )
=
however
dim(Kh(Kn )) = dim(Kh(Knτ )) = 26.
Proof of corollary. For n ≥ 8 it is clear from the theorem that the bigraded Khovanov homology over
Q of Kn and Knτ differ. For example Kn has dimension zero in homological grading n − 5, quantum
grading 5 + m while Knτ has dimension 1 in that grading.
The total dimension of the Khovanov homology in each case is 26, and can be computed by summing
the dimensions over all bidegrees.
For the finitely many cases where 0 ≤ n ≤ 7 this result has been computationally verified using Green’s
program JavaKh [9].
Proof of theorem 14. The method of computing Khovanov homology we use here was previously used
in [28] to find the Khovanov homology of (p, −p, q) pretzel knots. The reader may refer to that paper
or the above cited sources for further background and detail.
There is no difference in the proof for Kn versus Knτ . We will write Kn throughout the proof, but all
statements in the proof hold for Knτ as well.
There is a long exact sequence whose terms are given by the unnormalized Khovanov homology of a
knot diagram and its 0 and 1 resolutions. The unnormalized Khovanov homology is an invariant of a
specific diagram, not of a particular knot. It is given by taking the homology of the appropriate direct
sum in the cube of resolutions before making the overall grading shifts. Let n+ denote the number of
positive crossings in a diagram and n− the number of negative crossings. Let [·] denote a shift in the
homological grading and {·} denote a shift in the quantum grading such that Q(q) {k} = Q(q+k) and
such that Kh(K)[k] has an isomorphic copy of Khi (K) in homological grading i + k for each i. 2 Let
c
Kh(D)
denote the unnormalized Khovanov homology of a knot diagram D. Then
c
Kh(D) = Kh(D)[−n
− ]{n+ − 2n− }.
If D is a diagram of a knot, D0 is the diagram where one crossing is replaced by its 0-resolution and
D1 is the diagram where that crossing is replaced by its 1-resolution. Then, we have the following long
exact sequence (whose maps preserve the q-grading)
(5)
i−1
c
· · · → Kh
i
i
i
c (D) → Kh
c (D0 ) → Kh
c (D1 ){1} → · · · .
(D1 ){1} → Kh
Let D, D0 and D1 be the diagrams for Kn and its resolutions U and Kn−1 as shown in Figure 4b.
Observe that D0 is a diagram for the two component unlink U with 6 + n positive crossings and 7
negative crossings. D1 is a diagram for Kn−1 with 6 + n positive crossings and 7 negative crossings
and D is a diagram for Kn with 7 + n positive crossings and 7 negative crossings. Therefore we have
the following identifications
c 1 )[−7]{n − 8} = Kh(Kn−1 )
Kh(D
c 0 )[−7]{n − 8} = Kh(U)
Kh(D
c
Kh(D)[−7]{n
− 7} = Kh(Kn ).
2There is some discrepancy in the notation for grading shifts. The notation in this paper agrees with that of BarNatan’s introduction [3], though it is the opposite of that used in Khovanov’s original paper [13]. Negating all signs
relating to grading shifts will give Khovanov’s original notation.
GENUS 2 MUTANT KNOTS WITH THE SAME DIMENSION IN KNOT FLOER AND KHOVANOV HOMOLOGIES11
n+6
n+5
n+4
n+3
n+2
n+1
n
n-1
n-2
n-3
n-4
n-5
n-6
n-7
..
.
1
0
1
1
1
1
1
1
1
11+m
13+m
1
1
1
1
2
1
1
1
1
1
1
17+m
19+m
1
1
1
1
1
1
-1
a
1+a
1
b
b
3
···
1+m
3+m
5+m
7+m
9+m
15+m
21+m
23+m
25+m
Table 2. Here m = 2(n−7). When a = b = 0 this table gives the Q-dimensions of the
Khovanov homology of Kn with homological grading on the vertical axis and quantum
grading on the horizontal axis. This is the E1 page of Lee’s spectral sequence.
Note that the Khovanov homology of the two component unlink is Kh0 (U) = Q(−2) ⊕ Q2(0) ⊕ Q(2)
c 0 ). We will inductively
and Khi (U) = 0 for i 6= 0. After applying appropriate shifts we obtain Kh(D
assume the computation in the theorem holds for Kn−1 . The base case is established by computing
Kh(K8 ) using the Green’s JavaKh program [9]. Applying the appropriate shifts from above we thus
c 1 ). Plugging this into the long exact sequence (5) gives the following exact
get the value for Kh(D
sequences
(6)
for i 6= 7, 8, and
0 → Khi−8 (Kn−1 )){8 − n}{1} → Khi−7 (Kn ){7 − n} → 0
0 → Kh−1 (Kn−1 ){9−n} → Kh0 (Kn ){7−n} → Q(6−n) ⊕Q2(8−n) ⊕Q(10−n) → Kh0 (Kn−1 ){9−n} → Kh1 (Kn ){7−n} → 0
which by the inductive hypothesis is the same as
(7)
0 → 0 → Kh0 (Kn ){7 − n} → Q(6−n) ⊕ Q2(8−n) ⊕ Q(10−n) → Q(8−n) ⊕ Q(10−n) → Kh1 (Kn ){7 − n} → 0.
Exactness of line (6) yields isomorphisms
Khj−1 (Kn−1 ){2} ∼
= Khj (Kn )
for all j 6= 0, 1. Inspecting the way the formula for Kh(Kn ) in the theorem depends on n, one can see
that the inductive hypothesis verifies the computation for Khj (Kn ) for j 6= 0, 1.
Exactness of line (7) gives a few possibilities. Analyzing the sequence we must have
Kh0 (Kn ) =
b
Q(−1) ⊕ Q1+a
(1) ⊕ Q(3)
Kh1 (Kn ) =
Qa(1) ⊕ Qb(3)
where a, b ∈ {0, 1}.
Now we use the fact that s(Kn ), vanishes by Lemma 11. Since s(Kn ) = 0, the spectral sequence
given by Lee in [14] converges to two copies of Q, each in homological grading 0, with one in quantum
grading −1 and the other in quantum grading 1, as proven by Rasmussen in [25]. Note that the rth
differential goes up 1 and over r, because of an indexing that differs from the standard indexing for a
spectral sequence induced by a filtration. (See the note in section 3.1 of [28] for further explanation).
th
Let dp,q
page from Erp,q to Erp+1,q+r in Lee’s spectral sequence. Here
r denote the differential on the r
p is the coordinate for the homological grading shown on the vertical axis and q is the coordinate for
the quantum grading shown on the horizontal axis.
See Tables 2 and 3 for the E1 page on which the following analysis is carried out. In order to preserve
one copy of Q(−1) and one copy of Q(1) in the 0th homological grading we must have dr0,−1 = 0 and
d0,1
acting trivially on one copy of Q for every r.
r
12
n+6
n+5
n+4
n+3
n+2
n+1
n
n-1
n-2
n-3
n-4
n-5
n-6
n-7
..
.
1
0
MOORE AND STARKSTON
1
1
1
1
1
1
1
1
1
1
7+m
9+m
1
2
1
1
1
1
1
1
1
13+m
15+m
1
1
1
1
1
-1
a
1+a
1
b
b
3
···
1+m
3+m
5+m
11+m
17+m
19+m
21+m
23+m
25+m
Table 3. Here m = 2(n−7). When a = b = 0 this table gives the Q-dimensions of the
Khovanov homology of Knτ with homological grading on the vertical axis and quantum
grading on the horizontal axis. This is the E1 page of Lee’s spectral sequence.
We may computationally verify another base case where n = 9 and then assume n ≥ 10. By the above
inductive results, we know that Kh2 (Kn ) = 0 when n ≥ 10. Therefore, dr1,1 = 0 for all r ≥ 1. Thus, if
1,1
a 6= 0, an additional copy of Q will survive in E∞
since it cannot be in the image of any dr for r > 0.
This contradicts Lee’s result that there can only be two copies of Q on the E∞ page. Therefore a = 0
and d0,1
r = 0 for all r ≥ 1. Because the row corresponding to the first homological grading has zeros
in quantum gradings greater than 3, d0,3
= 0 for all r ≥ 1. Therefore, if b 6= 0, an additional copy
r
0,3
of Q will survive in E∞
, again contradicting Lee’s result. Therefore a = b = 0, and the Khovanov
homology of Kn and Knτ is as stated in the theorem.
5. Observation and speculation
The families of knots which we have employed in this paper are all non-alternating slice knots, and
in particular, are formed from the band sum of a two-component unlink. Our proof of Theorem 1
works for all such families of knots, and other examples of infinite families of knots abound. For
example, Hedden [12] proves that there are infinitely many knots with isomorphic Floer groups in
a given concordance class, whereas Greene and Watson [11] have worked with the Kanenobu knots.
Certain pretzel knots (see [28]) also share this property. Nor is the non-alternating status of these
knots a coincidence; in fact there can only be finitely many alternating knots of a given knot Heegaard
Floer homology type.
Proposition 16. Let K be an alternating knot. There are only finitely many other alternating knots
[
with knot Floer homology isomorphic to HFK(K)
as bigraded groups.
Proof. Suppose to the contrary that K belongs to an infinite family {Kn }n∈Z of alternating knots
[
[ n) ∼
and knot Floer homology categorizes
sharing the same knot Floer groups. Since HFK(K
= HFK(K)
the Alexander polynomial,
det(Kn ) = |∆Kn (−1)| = |∆K (−1)| = det(K)
for all n. Each knot Kn admits a reduced alternating diagram Dn with crossing number c(Dn ). The
Bankwitz Theorem implies that c(Kn ) ≤ det(Kn ). However, there are only finitely many knots of a
given crossing number, and in particular c(Kn ) is arbitrarily large, which contradicts that c(Kn ) ≤
det(K).
This fact leads to the interesting open question of whether there are infinitely many quasi-alternating
knots of a given knot Floer type. Josh Greene formulates an even stronger conjecture in [10], and
proves the cases where det(L) = 1, 2 or 3.
GENUS 2 MUTANT KNOTS WITH THE SAME DIMENSION IN KNOT FLOER AND KHOVANOV HOMOLOGIES13
Conjecture 17 (Conjecture 3.1 of [10]). There exist only finitely-many quasi-alternating links with a
given determinant.
In Section 4, we mention that {Kn } and {Knτ } have the same Khovanov homology with F2 coefficients.
In fact, (K0 , K0τ ) is one of five pairs of genus 2 mutants appearing in [8], none of which can be
distinguished by Khovanov homology over F2 . Bloom and Wehrli [4],[30] have shown that Khovanov
homology with F2 coefficients is invariant under component-preserving Conway mutation. This leads
to another unanswered question.
Question 18. Is Khovanov homology with F2 coefficients invariant under genus 2 mutation?
A positive answer to question 18 would imply a positive answer to the following:
c 2 (K τ ))?
c 2 (K)) ∼
Question 19. Is HF(Σ
= HF(Σ
This is due to Ozsváth and Szabó’s spectral sequence relating the reduced Khovanov homology of L
over F2 to the Heegaard Floer homology of the branched double cover of −L.
Genus 2 mutation provides a method for producing closely related knots and links, but more generally
it is an operation on three manifolds. This yields yet another unanswered question:
Conjecture 20. Let M be a closed, oriented 3-manifold with an embedded genus 2 surface F . If M τ
is the genus 2 mutant of M , then
τ
c
c
rank HF(M
) = rank HF(M
)
The question of whether total rank is preserved under Conway mutation remains an interesting problem. The evidence that we offer above suggests that total ranks of knot Floer homology and Khovanov
homology are also preserved by genus 2 mutation as well, or at least by genus 2 handlebody mutations. If so, a general proof will not likely result from combinatorial properties of knot diagrams, since
genus 2 mutations are not always obtained diagrammatically. Recently, Baldwin and Levine have
conjectured [2] that the δ-graded knot Floer homology groups
M
[ m (L, s)
[ δ (L) =
HFK
HFK
δ=s−m
are unchanged by Conway mutation, which implies that their total ranks are preserved, amongst
other things. We conclude by remarking that this conjecture cannot be generalized to include genus 2
mutants. The δ-graded Floer groups of the genus 2 mutant pairs Kn and Knτ provide counterexamples.
In particular, the involution τ sends the group in δ-grading ±1 to δ-grading ∓1. The δ-graded groups
are also displayed in Table 1.
6. Acknowledgments
We would like to thank our advisors, Cameron Gordon and Robert Gompf for their guidance and
support. We would also like to thank John Luecke, Matthew Hedden, and Cagri Karakurt for helpful
conversations. The first author was partially supported by the NSF RTG under grant no. DMS0636643. The second author was supported by the NSF Graduate Research Fellowship under grant
no. DGE-1110007.
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Department of Mathematics, The University of Texas, Austin, TX 78712, USA
E-mail address: moorea8@math.utexas.edu
URL: http://www.ma.utexas.edu/∼moorea8
Department of Mathematics, The University of Texas, Austin, TX 78712, USA
E-mail address: lstarkston@math.utexas.edu
URL: http://www.ma.utexas.edu/∼lstarkston