On Conway mutation and link homology

0.45 MB·

by  Peter Lambert-Cole

ON CONWAY MUTATION AND LINK HOMOLOGY
PETERLAMBERT-COLE
7
1
0
2

n Abstract. We give a new, elementary proof that Khovanov homology with Z/2Z–coefficients is
a invariant under Conway mutation. This proof also gives a strategy to prove Baldwin and Levine’s
J conjecture that δ–graded knot Floer homology is mutation–invariant. Using the Clifford module

0 structure on H]FK induced by basepoint maps, we carry out this strategy for mutations on a large
3 classoftangles. LetL′ bealinkobtainedfromLbymutatingthe tangleT. Supposesomerational
closureofT correspondingtothemutationistheunlinkonanynumberofcomponents. ThenLandL′

] haveisomorphicδ–gradedH[FKgroupsoverZ/2ZaswellasisomorphicKhovanovhomologyoverQ.
T
Weapplytheseresultstoestablishmutation–invariancefortheinfinitefamiliesofKinoshita-Terasaka
G and Conway knots. Finally, we give sufficient conditions for a general Khovanov-Floer theory to be
. mutation–invariant.
h
t
a
1. Introduction
m
[ A Conway sphere for a link L ⊂ S3 is a smoothly embedded 2-sphere that intersects the link in

4 points. This separates the link into a pair of tangles (B3,T ) and (B3,T ). The center of the
2 1 2
v mapping class group of S2 with 4 marked points has four elements: the identity and 3 involutions.
0 After identifying S2 with the unit sphere in R3 and the four marked points on the xz–plane, we can
8 identify the 3 involutions with rotations by π around the 3 coordinate axes. A mutation of L is a link
8 obtained by changing the gluing map of (B3,T ) and (B3,T ) by such an involution τ.
0 1 2
0 Let H[FK(L) denote the Z⊕Z–graded knot Floer homology groups associated to the link L with
1. coefficients in F2. The δ–graded knot Floer groups are obtained by collapsing along the diagonals
0 m−a=δ:
7 H[FK (L):= H[FK (L,a)
δ m
1
: mM−a=δ
v
The bigraded knot Floer homology groups can distinguish mutant knots, such as the Kinoshita-
i
X Terasaka and Conway knots [OS04c, BG12]. However, explicit computations showed that for 11-
r and 12-crossing knots, mutation preserves the δ–graded invariant [BG12]. This data, along with a
a
combinatorialmodel for H[FK , led Baldwin and Levine to conjecture that this phenomenon is true in
δ
general.
Conjecture 1.1 (Baldwin-Levine [BL12]). Let L and L′ be a mutant pair of links. Then there is an
isomorphism
H[FKδ(L)∼=H[FKδ(L′)
In this paper, we investigate this conjecture and prove it for mutations on a class of tangles. Let
(B3,T ) denote the tangle consisting of 2 boundary-parallel arcs. A link L is a rational closure of a
0
2010 Mathematics Subject Classification. 57M27;57R58.
Key words and phrases. Mutation, HeegaardFloerhomology,Khovanovhomology.
1
2 P.LAMBERT-COLE
tangle (B3,T) if it can be decomposed in the form
(S3,L)=(B3,T)∪ (B3,T )
1 φ 2 0
for some homeomorphism φ : ∂B3 → ∂B3. Rational closures are not unique: the set of rational
1 2
closures of (B3,T ) itself is the set of 2-bridge links. Let C(T) denote the set of rational closures of T.
0
A mutation of T by the involution τ determines a subset C (T) as follows. A rational closure of T is
τ
in C (T) if the arcs of T connect points of ∂T exchanged by the involution τ. We refer to C (T) as
τ 0 τ
the set of rational closures corresponding to the mutation τ.
In this paper, we show that if the set C (T) contains the unlink on any number of components,
τ
then many link homology theories are preserved by the mutation τ. For example, Kinoshita and
Terasaka defined an infinite family of knots KT for r,n ∈ Z with trivial Alexander polynomial,
r,n
where the Kinoshita-Terasaka knot 11n42 is KT [KT57]. Analogously, there is an infinite family of
2,1
Conway mutants C extending C = 11n34 that are obtained from the Kinoshita-Terasaka family
r,n 2,1
by mutation. We can choosea diagramfor KT so that the numeratorclosure ofthe mutated tangle
r,n
is the unknot (Figure 6). As a second example, if T is the tangle sum of two rational tangles p
q
and r , then the numerator closure of T is the unknot or 2-component unlink if ps+qr ∈{−1,0h,1}i
s
[KL12].
(cid:2) (cid:3)
1.1. Khovanov Homology. The starting point is a new proof of the following theorem.
Theorem1.2(Bloom[Blo10],Wehrli[Weh10]). LetL,L′ bemutantlinks andletKhdenoteKhovanov
homology with Z/2Z-coefficients. There is an isomorphism
Kh(L)∼=Kh(L′)
The tools involved are more elementary than the proofs in [Blo10, Weh10]. In particular, we need
only the following 3 facts1:
(1) the unoriented skein exact triangle
→Kh(L)→Kh(L )→Kh(L )→Kh(L)→
0 1
(2) over Z/2Z and for any two (unoriented) links L ,L , the group Kh(L #L ) is independent,
1 2 1 2
up to isomorphism, of the choice of connected sum of L and L , and
1 2
(3) over Z/2Z and for any two (unoriented) links L ,L , the elementary merge cobordism
1 2
µ:Kh(L ∪L )→Kh(L #L )
1 2 1 2
is surjective for any choice of connected sum.
OurstrategyistofinddiagramsforamutantpairL,L′ instandard form (Figure1). Wethenapply
the unoriented skein exact sequence to relate Kh(L) and Kh(L′).
Over a different coefficient ring or with respect to a different link homology theory, Facts (2) and
(3) may or may not hold. In particular, Facts (2) and (3) are not true in general for Khovanov
homology over Z or Q. Moreover, it is exactly the failure of (2) that leads to Wehrli’s examples of
links distinguishedby Khwith Q coefficients [Weh]. With extraconditions,we canadapt this proofof
mutation invariance to other coefficient rings or link homology theories.
1Facts (2) and (3) themselves are consequences of the fact that over Z/2Z, reduced Khovanov homology does not
dependonthecomponent containingthebasepoint.
ON CONWAY MUTATION AND LINK HOMOLOGY 3
T T T T
1 2 1 2
Figure 1. A mutant pair in standard form
Theorem 1.3. Let L,L′ be links such that L′ is obtained from L by mutating the tangle T by the
involution τ. Let C (T) denote the set of rational closures of T corresponding to the mutation. If
τ
C (T) contains the unlink on any number of components, then for any field F there is a bigraded
τ
isomorphism
KhF(L)∼=KhF(L′)
In particular
KhQ(L)∼=KhQ(L′)
Remark 1.4. Theorem 1.3 applies to all mutations, not simply component–preserving mutations.
However,it only applies to unreduced Khovanovhomology. If L,L′ containa basepointp in T, then a
mutation satisfying the hypotheses can swap the component containing p.
A corollaryto Theorem 1.3 is that all mutant pairs in the Kinoshita-Terasakaand Conway families
have isomorphic Khovanov homology.
Theorem 1.5. For all r,n∈Z and any field F, there is a bigraded isomorphism
KhF(KTr,n)∼=KhF(Cr,n)
1.2. Knot Floer homology. A variant of knot Floer homology, the tilde group H]FK(L,p) of a
pointedlink,satisfiesanunorientedskeinexacttriangle[Man07]. Moreover,theinvariantH[FKsatisfies
a Kunneth formula for any choice of connected sum [OS04a, OS08]. However, the elementary merge
map
µ:H]FK(L ∪L ,p)→H]FK(L #L ,p)
1 2 1 2
is never surjective. In fact, the rank of µ is exactly 1rkH]FK(L ∪ L ,p) = 1rkH]FK(L #L ,p).
2 1 2 2 1 2
Nonetheless, the proofof Theorem1.2 providesa strategyto proveConjecture 1.1. The maintheorem
of this paper is to carry out that strategy for mutations on a large class of tangles. This provides the
first step in the proof of Conjecture 1.1.
Theorem 1.6. Let L,L′ be links such that L′ is obtained from L by mutating the tangle T by the
involution τ. Let C (T) denote the set of rational closures of T corresponding to the mutation. If
τ
C (T) contains the unlink on any number of components, then there is an isomorphism
τ
H[FKδ(L′)∼=H[FKδ(L)
A key technical toolin the proof of Theorem1.6 is the extra algebraicstructure on unreduced knot
Floer homology H]FK(L,p) of a pointed link (L,p). Each basepoint in a multi-pointed Heegaard dia-
gram for (L,p) determines a differential on H]FK(L,p). These basepoint actions have previously been
applied in [BL12, BVVV13, Sar15, BLS, Zem16]. The combined actions, subject to anticommutation
relations described in Subsection 3.3, make H]FK(L,p) a Clifford module over a Clifford algebra Ω .
ν
4 P.LAMBERT-COLE
This Clifford module structure is an algebraic quantization of the exterior algebra structure used in
[BL12, BLS].
Importantly, the maps in the unoriented skein exact triangle are mostly but not fully compatible
with the basepointactions. In particular,the three homologygroups in the skeintriangle are modules
over slightly different Clifford algebras. In Subsection 4.2, we quantify the skein maps’ failure to be
fully Ω -linear. Nonetheless, this weaker equivariance condition is sufficient to prove the theorem.
ν
We remark that the following statement is an easy corollary of Theorem 1.6.
Corollary 1.7. Let L,L′ be links satisfying the hypotheses of Theorem 1.6 and suppose that H[FK(L)
is thin. Then H[FK(L′) is also thin. Moreover, if L,L′ are knots then
(1) g(L′)=g(L),
(2) L′ is fibered if and only if L is fibered, and
(3) τ(L)=τ(L′)
We can apply Theorem 1.6 in a few ways. First, Ozsv´ath and Szabo´ computed the top–degree
H[FK groups for the Kinoshita-Terasaka and Conway families and showed that KT and C are
r,n r,n
distinguished by their bigraded knot Floer groups [OS04c]. This extends an earlier result of Gabai
calculating their genera [Gab86]. However, using Theorem 1.6, we can deduce that KT and C
r,n r,n
have isomorphic δ–graded groups.
Theorem 1.8. For all r,n∈Z, there is a graded isomorphism
H[FKδ(KTr,n)∼=H[FKδ(Cr,n)
Secondly,DeWitandLinksand,independently,Stoimenowgiveenumerationsof11-and12-crossing
mutant cliques (6 alternating and 10 nonalternating 11-crossing pairs; 27 pairs and 2 triples of alter-
nating 12-crossing knots; 43 pairs and 3 triples of nonalternating 12-crossing knots) [DWL07, Sto10].
The δ–gradedknotFloergroupsofanalternatinglinkis determinedbythe determinantandsignature
[OS03]. Since these are mutation–invariant, Conjecture 1.1 holds for alternating links. Moreover, all
nonalternating cliques in the De Wit-Links and Stoimenow enumerations admit a mutation in a mini-
maldiagramonone ofthe tanglesinFigure7. Thenumeratorclosuresofthese 3tanglesareunlinked.
Applying Theorem 1.6, we recover the computational results of [BG12].
Theorem 1.9. Let K,K′ be mutant knots with crossing number at most 12. Then
H[FKδ(K)∼=H[FKδ(K′)
for all δ ∈Z.
1.3. Khovanov-Floer theories. Interestingly, the geometric arguments in the proof of Theorem 1.2
apply more generally to other link homology theories.
Baldwin, Hedden and Lobb introduced the notion of a Khovanov-Floer theory [BHL]. See Section
6 for a definition. This framework encompasses several link homology theories — Heegaard Floer
homology of the double-branched cover Σ(L) [OS05b], singular instanton homology [KM11a], Szabo´’s
cube of resolutions [Sza15], Bar-Natan’s construction of Lee homology over Z/2Z [BN05] — that
admit similar spectral sequences. In particular, the homology groups can be computed from filtered
complexes and the E2–pages of the corresponding spectral sequences are isomorphic to Khovanov
homology. Moreover,many of these theories appear insensitive to Conway mutation.
It is possible, but unknown, that all Khovanov-Floer theories must also satisfy extra elementary
properties of Khovanov homology with Z/2Z–coefficients. In order to give sufficient conditions for a
ON CONWAY MUTATION AND LINK HOMOLOGY 5
Khovanov-Floer theory to be mutation–invariant, we introduce the notion of an extended Khovanov-
Floer theory.
Definition 1.10. An extended Khovanov-Floer theory is a pair A,Ar consisting of an unreduced and
reduced Khovanov-Floertheories, respectively, satisfying the following 3 extra axioms:
(1) A(L)=Ar(L∪U,p) for a basepoint p on the unknot component U,
(2) A and Ar satisfy unoriented skein exact triangles, and
(3) up to isomorphism, Ar(L,p) is independent of the component containing p.
The proof of Theorem 1.2 in Section 2 easily adapts to prove the following theorem. We state
it without reference to grading, although we expect it can be extended to a graded statement by
inspecting the relevant exact triangle.
Theorem1.11. LetA,Ar beanextendedKhovanov-Floer theory. Then AandAr areinvariantunder
Conway mutation.
For example,letΣ(L) denote the double branchedcoverofL⊂S3 andletHF denote the Heegaard
Floer homology with Z/2Z–coefficients. Setting Ar(L,p)= HF(−Σ(L)) for any choice of basepoint p
andA(L)=HF(−Σ(L)#S1×S2)yieldsanextendedKhovanov-Floertheory. Tchemutation–invariance
of Σ(L), and thus HF(−Σ(L)), is a well–known fact [Vir76]. c
c
Another potential extended Khovanov-Floer theory is Szabo´’s geometric spectral sequence [Sza15].
c
For a link L, each link diagram D and decoration t determines a filtered chain complex C(D,t).
The pages of the corresponding spectral sequence are independent of D and t and so are invariants
of L. The complex C(D,t) is constructed via a cube of resolutions, so its homology satibsfies an
unorientedskeintriangle. Moreover,therearenaturallyreducedcomplexesC(D,t),C(D,t)determined
by a basepoint on L.bExtensive computational work by Seed is consistent with the conjecture that
H (C(D,t))=HF(−Σ(L)),thatthe reducedhomologyis independentofthe basepoeint,andthateach
∗
page of the spectral sequence is mutation–invariant [See11]. Thus we conjecture the following:
b c
Conjecture 1.12. Szabo´’s geometric spectral sequence is an extended Khovanov-Floer theory.
Finally, the singular instanton homology groups I#,I♮ defined by Kronheimer and Mrowka are
known to satisfy the Khovanov-Floer axioms [BHL]. In addition, they satisfy an unoriented skein
exact triangle.
Question1.13. Arethesingularinstantonhomology groupsI#(L),I♮(L)anextendedKhovanov-Floer
theory?
A positive answer to Question 1.13 would imply, via Theorem 1.11, that the singular instanton
homology groups I#,I♮ are mutation–invariantover Z/2Z. Regardlessof the answer,however,we can
prove an analogous statement to Theorems 1.3 and 1.5.
Theorem 1.14. Let L,L′ be links such that L′ is obtained from L by mutating the tangle T by the
involution τ. LetC (T)denote theset ofrational closures ofT corresponding tothemutation. IfC (T)
τ τ
contains the unlink on any number of components, then for any field F there is an isomorphism
I#(L)∼=I#(L′)
F F
Consequently, for all r,n∈Z, there is an isomorphism
IF#(KTr,n)∼=IF#(Cr,n)
6 P.LAMBERT-COLE
1.4. Acknowledgements. I would like to thank Matt Hogancamp for many useful discussions on
homological algebra. In addition, several people have helped with apt suggestions, technical details
and their general interest, including John Baldwin, Matt Hedden, Adam Levine, Tye Lidman and
Zoltan Szabo´.
2. Khovanov mutation invariance
In this section, we give a new proof that Khovanov homology with Z/2Z coefficients is invariant
underConwaymutation. Similargeometricargumentswillbeappliedinsuccessivesectionstoestablish
mutation–invariance results for knot Floer homology and other Khovanov-Floer theories.
Bloom provedthat odd Khovanovhomologyis invariantunder mutation, which implies that (even)
Khovanov homology with Z/2Z coefficients is also invariant [Blo10]. In addition, Wehrli showed that
Bar-Natan’sKhovanovbracketoverZ/2Zis invariantunder component–preservingmutation[Weh10].
Conversely,Wehrli has also observedthat Khovanovhomology is not mutation–invariantwith Q coef-
ficients. The links T(2,3)∪T(2,3)andT(2,3)#T(2,3)∪U are mutants but aredistinguished by their
Khovanov groups over Q [Weh].
2.1. Khovanov homology. Khovanov homology is an oriented link invariant obtained by applying
a (1+1)-dimensional TQFT to the cube of resolutions of a link diagram. In this first subsection,
we will sketch a definition of Khovanov homology, state some well–known properties and prove some
elementary properties.
Let D be a planar diagram of an oriented link L with n crossings and fix an enumeration of the
crossings. Each given crossing can be resolved in two ways, the 0–resolution and 1–resolution. The
cube of resolutions of D is the collection of 2n planar diagrams obtained by resolving all the crossings
of D in all possible ways. In particular, for each vertex I ∈ {0,1}n, there is a diagram D of an
I
unlink obtained by resolving the crossings of D according to the vector I. Let l denote the number
I
of link components of D . The edges of the cube are given by ordered pairs I,J of vertices such that
I
I ≤J for k = 1,...,n and |J|−|I| = 1. Geometrically, each edge corresponds to replacing a single
k k
0–resolutionbya1–resolutionandeithermergestwocomponentsintooneorsplitsonecomponentinto
two.
The algebra A:=Z[x]/x2 is a Frobenius algebra with multiplication µ:A⊗A→A defined by
µ(1⊗1):=1 µ(1⊗x):=x µ(x⊗1):=x µ(x ⊗x):=0
1
and comultiplication σ :A→A⊗A defined by
σ(1):=x⊗1+1⊗x σ(x):=x⊗x
To each vertex I of the cube assign the chain group CKh(DI) := A⊗lI, with one copy of A for each
component of D . The chain groupofthe Khovanovcomplex is the direct sum of the chaingroups for
I
each vertex
CKh(D):= CKh(D )
I
I∈M{0,1}n
The differential d is the sum of maps associated to each edge. For each edge I,J of the cube,
Kh
there is a component d of the differential determined up to sign by the Frobenius algebra. If the
I,J
edge is a merge map, then d is defined by applying the multiplication map µ to the A-factors of
I,J
CKh(D )correspondingtothemergedcomponentsandextendingthisbytheidentitytotheremaining
I
components. If the edge is a split map, then d is defined similarly using the comultiplication σ
I,J
instead. Khovanov homology Kh(D) is the homology of the complex (CKh(D),d ).
Kh
ON CONWAY MUTATION AND LINK HOMOLOGY 7
The complex (CKh(D),d ) possess two gradings, the quantum and homological grading, and the
Kh
differentialpreservesthe quantumgradingandincreasesthe homologicalgradingbyone. Thusthe ho-
mologyKh(D)splitsintothedirectsumofbigradedmodulesKhi,j(D)whereidenotesthehomological
grading and j the quantum grading.
Let p be a fixed basepoint in the plane contained in the diagram D. The basepoint p determines a
chain map X :CKh(D)→CKh(D) that squares to 0. The kernel of X is a subcomplex and reduced
p p
Khovanov homology is its homology, with a shift in the quantum grading:
Kh(D,p)=H (Σ0,1kerX )
∗ p
It is an invariant of L up to isotopies supported away from p. While over Z the reduced homology
f
depends on the component containing the basepoint, this is not true over Z/2Z.
Proposition 2.1 ([Shu14, OS05b]). Over Z/2Z, there is a bigraded isomorphism
Kh(L)∼=Kh(L,p)⊗Kh(U)
for every link L and any basepoint p. In particular, Kh(L) is well–defined independent of p.
f
Consequently,whendiscussingreducedKhovanovhfomologyoverZ/2Zwewillsuppressanymention
of the basepoint p.
Khovanov homology satisfies Kunneth–type formulas for disjoint unions and connected sums. The
followingpropertiesarewell–knownandtheproofsareeasydeductionsfromthedefinitionofKhovanov
homology and Proposition 2.1.
Lemma 2.2. Let L ,L be arbitrary links. Over Z/2Z there are isomorphisms
1 2
Kh(L ∪L )∼=Kh(L )⊗Kh(L )
1 2 1 2
Kh(L #L )∼=Kh(L )⊗Kh(L )
1 2 1 2
Kh(L #L )⊗Kh(U)∼=Kh(L )⊗Kh(L )
1 2 1 2
f f f
for any choice of connected sum.
Khovanovhomology satisfies an unoriented skein exact triangle. Fix a crossingof D and let D ,D
0 1
denote the 0– and 1–resolutions of D at this crossing. Since Khovanov homology is computed from a
cube of resolutions, the complex CKh(D) is, up to a grading shift, the mapping cone of a chain map
f :CKh(D )→CKh(D )
0 1
Consequently there is an exact triangle
f∗
Kh(D ) // Kh(D )
0ee❏❏[−❏❏1❏]❏❏❏❏ yyttttttttt 1
Kh(D)
The bigrading shift of f depends on whether the resolved crossing is positive or negative. Let n be
−
the number of negative crossings in D and let n0,n1 be the number of negative crossings in D and
− − 0
D , respectively. Then the Khovanov differential determines chain maps
1
f :CKhi,j(D )→CKhi−c,j−1−3c(D ) f :CKhi−d−1,j−2−3d(D )→CKhi,j(D )
+ 0 1 − 0 1
where c:=n1 −n and d:=n0 −n .
− − − −
8 P.LAMBERT-COLE
2.2. Connected sum and disjoint union. Given any two oriented links L ,L and any choice of
1 2
connected sum, there is an exact triangle
β
(1) Kh(L #L ) //Kh(L #−L )
1 ggP2PPPPµPPPPPPP vv♠♠♠♠♠♠♠σ♠♠♠♠♠ 1 2
Kh(L ∪L )
1 2
where −L denotes reversing the orientation on all components (cf. [Kho00, Section 7.4] and [Ras10,
2
Section 3]). Reduced Khovanov homology satisfies an identical triangle.
Lemma 2.3. Let L ,L be any oriented links. Over Z/2Z, the map β in the skein exact triangle is
1 2
identically 0 and there is a short exact sequence
σ µ
0 //Kh(L #−L ) //Kh(L ∪L ) //Kh(L #L ) // 0
1 2 1 2 1 2
In particular, the merge map
µ:Kh(L ∪L )→Kh(L #L )
1 2 1 2
is surjective and the split map
σ :Kh(L #−L )→Kh(L ∪L )
1 2 1 2
is injective.
Moreover, identical statements hold for reduced Khovanov homology.
Proof. Over Z/2Z, the Kunneth principle implies that
Kh(L ∪L )∼=Kh(L )⊗Kh(L )
1 2 1 2
Furthermore, Lemma 2.2 implies that
Kh(L )⊗Kh(L )∼=Kh(L #L )⊗Kh(U)
1 2 1 2
∼=Kh(L #−L )⊗Kh(U)
1 2
Consequently, since rkKh(U)=2, it follows that
rkKh(L ∪L )=2·rkKh(L #L )=2·rkKh(L #L )
1 2 1 2 1 2
Thus,thisskeintripleisanextremalcaseforthetriangleinequalityandthe triangleunfoldstoashort
f
exact sequence.
Over Z/2Z, we have that rkKh(L)=2·rkKh(L) for all links L by Proposition 2.1. Therefore, the
above argument can be applied mutatis mutandis to reduced Khovanov homology. (cid:3)
f
Lemma 2.3 can be extended with arbitrary field coefficients when L is the unlink on any number
2
of components.
Lemma 2.4. Let L be an arbitrary link and let L = U denote the k-component unlink. Over any
1 2 k
field F, the map β in the skein exact triangle is identically 0 and there is a short exact sequence
σ µ
0 //KhF(L1#−L2) // KhF(L1∪L2) //KhF(L1#L2) //0
In particular, the merge map
µ:Kh (L ∪L )→Kh (L #L )
F 1 2 F 1 2
is surjective and the split map
σ :Kh (L #−L )→Kh (L ∪L )
F 1 2 F 1 2
ON CONWAY MUTATION AND LINK HOMOLOGY 9
is injective.
Proof. IfL =U ,thentheconnectedsumformulasinLemma2.2holdovereveryfieldsinceL #U ∼
2 k 1 k
L ∪U . The arguments from the proof of Lemma 2.3 now carry over mutatis mutandis. (cid:3)
1 k−1
2.3. Mutation. Let (B3,t ) be an abstract 2–tangle, where t is the union of 2 arcs with endpoints
1 1
on ∂B3 and any number of closed components. Throughout this section, we will use T to denote an
abstract tangle and T to denote a given diagram for T in the unit disk with boundary points on the
unit circle.
Suppose that L andL′ aremutants. We saythatdiagramsfor the pair L,L′ are instandard form if
they areas in Figure 1. In particular, L is obtained by connecting diagramsof two tangles T ,T with
1 2
two bands, one untwisted and the other with a single crossing, while L′ is obtained by adding a single
crossing to the first band and leaving the second untwisted. Note that if the mutation is negative,
the strands in the two bands are oriented in opposite directions, while if the mutation is positive they
strands are oriented in the same direction.
Lemma 2.5. If L′ is obtained from L by a single mutation, then L,L′ admit diagrams in standard
form.
Proof. Let L be a link with a Conway sphere bounding the tangle T . Identify the Conway sphere
1
with the unit sphere in R3 so that T is contained in the unit ball. We can assume that L′ is obtained
1
by mutating T by the involution around the y–axis. The projection of T to the xy–plane lies in the
1 1
unit disk. Isotope the exteriortangle T so that its projectionto the xy–plane is outside the unit disk.
2
Now, introduce a pair of canceling crossings in lower half plane, one just inside the unit disk and one
just outside. Let T ,T denote these diagram of T ,T . This gives the link on the left of Figure 1.
1 2 1 2
The corresponding diagram for L′ can be obtained from the diagram for L by the mutation and then
a flype. (cid:3)
Take diagrams for L,L′ in standard form. We can obtain 9 links from the tangle diagrams T ,T
1 2
as follows. Let L be the link obtained by connecting T and T by two bands, each with a single
∞,∞ 1 2
crossing. Each crossing has a 0– and 1–resolution and we obtain 9 links {L } for •,◦∈{∞,0,1} by
•,◦
choosing one of the three forms at each crossing. See Figure 2.
Let N(T ),N(T ) denote the numerator closures of the 2-tangles T ,T . Then we have that
1 2 1 2
(1) L =L and L =L′
∞,1 1,∞
(2) L =N(T )∪N(T )
0,0 1 2
(3) L ,L ,L , and L are connected sums of N(T ) and N(T ).
1,0 0,1 ∞,0 0,∞ 1 2
Lemma 2.6. Suppose that L′ is obtained from L by mutating T around the y–axis. For any rational
1
closure C of T corresponding to this mutation, we can choose diagrams for the mutant pair L,L′ in
1
standard form so that the numerator closure N(T ) is C.
1
Proof. As in the proof of Lemma 2.5, choose a diagramfor L such that the projection of T lies in the
1
unitdiskandtheimageofT liesoutside. LetT ,T denotethediskdiagramsforthetwotangles. IfC
2 1 2
isarationalclosureofT ,thenwecanchooseadiagramforC thatistheunionofT withthediagram
1 1
of some rational tangle p . This rational tangle decomposes into an annular diagram A and a disk
q
diagramN,withthelatthericontainingnocrossings. ThediagramT1′ =T1∪AisalsoaprojectionofT1.
Since C is a rational closure corresponding to the mutation, we can assume that it is the numerator
closure of T′.
1
10 P.LAMBERT-COLE
Figure 2. The 9 links obtained by resolving the two crossings of L (bottom
∞,∞
right). The links L and L are a mutant pair in standard form.
∞,1 1,∞
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
f0
//Kh(L ) //Kh(L ) // Kh(L ) //
0,0 0,1 0,∞
k0 k1
(cid:15)(cid:15) f1 (cid:15)(cid:15) (cid:15)(cid:15)
//Kh(L ) //Kh(L ) // Kh(L ) //
1,0 1,1 1,∞
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
//Kh(L ) // Kh(L ) //Kh(L ) //
∞,0 ∞,1 ∞,∞
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
Figure 3. The commutative diagram of skein maps corresponding to the 9 links in
Figure 2.