Table Of ContentQUANTUM INVARIANTS FOR HANDLEBODY-KNOTS
ATSUHIKO MIZUSAWA AND JUN MURAKAMI
1
1 Abstract. We construct quantum invariants for handlebody-knots in a 3-sphere S3.
0 A handlebody-knot is an embedding of a handlebody in a 3-manifold. These invariants
2 are linear sums of Yokota’s invariants for colored spatial graphs which is defined by
c using Kauffman bracket. We also give a non-trivial example.
e
D
8
1
]
T 1. Introduction
G
Aspatialgraphisaembeddingofafinitegraphintoa3-manifold. Suzuki[5]introduced
.
h the notion of the neighborhood equivalence for spatial graphs. Two spatial graphs are
t
a neighborhood equivalent if they have the same regular neighborhoods up to isotopy of
m
the 3-manifold. Ishii [6] reformulated this notion as a handlebody-knot which is an
[
embedding of a handlebody into a 3-manifold. If the genus of the handlebody is 1, the
2 handlebody-knot is regarded as an ordinary knot.
v
A handlebody-knot is represented by a diagram of a spatial trivalent graph whose
9
1 regular neighborhood is isotopic to the handlebody-knot. Ishii [6] also showed that the
7
two diagrams representing the same handlebody-knot are transformed to each other by
2
. a sequence of six local moves called “Reidemeister moves” for handlebody-knots. Five
2
of them are Reidemeister moves for isotopy of spatial trivalent graphs. Therefor we can
1
1 make an invariant of handlebody-knots from an invariant of trivalent graphs modifying
1
to suffice the new Reidemeister move.
:
v In this paper we construct quantum U (sl ) type invariants for handlebody-knots in a
i q 2
X 3-sphere S3 via Yokota’s invariants [10], which are quantum U (sl ) type invariants for
q 2
r spatial graphs in S3. Yokota’s invariants are generalized to the Relativistic invariants [1]
a
for spaial graphs in any 3-manifolds. The arguments in this paper holds for Relativistic
invariants and our invariants are generalized for any 3-manifolds. In this paper we focus
on Yokota’s invariants since it is essential.
There are various invariants for handlebody-knots. Alexsander ideals are known as
invariants for neighborhood equivalent classes of spatial graphs [3] that are handlebody-
knots. Invariants using Fox coloring [6] and quandle cocycle invariants [6, 7] were also
Date: December 20, 2011.
Key words and phrases. handlebody-knot, quantum invariant, Yokota’s invariant.
1
2 ATSUHIKO MIZUSAWA AND JUN MURAKAMI
defined. However quantum type invariants for handlebody-knots have not been defined
yet1 .
In this paper, we first introduce a handlebody-knot and its Reidemeister moves in
Section 2. We then define quantum invariants for the handlebody-knot via Yokota’s
invariants in Section 3. In Section 4, some examples and properties of the invariants are
shown, which tell us that our invariants are nontrivial.
Throughout this paper we work in the peacewise linear category.
2. Handlebody-knot and Handlebody-link
Ahandlebody-knot isanembeddingofahandlebodyintoa3-manifoldandahandlebody-
link isadisjointunionofembeddingsofhandlebodiesinto3-manifold(Figure1). Through-
out this paper, the notation of a handlebody-link include a handlebody-knot. Two
handlebody-links are equal if there is an isotopy of the 3-manifold that the handlebodies
are embedded such that it transforms one handlebody-link to another.
(cid:44)→
Figure 1. An emmbedding of a genus 2 handlebody into a 3-manifold
We have a map from trivalent graphs to handlebody-links that takes a regular neigh-
borhood of a trivalent graph (Figure 2). Here a trivalent graph may have circle compo-
nents that correspond to genus 1 components of the handlebody-link. These circles don’t
have vertices nor edges. Through this map, we can treat handlebody-links as diagrams
of trivalent spatial graphs. However, there are infinitely many diagrams that represent
the same handlebody-link (Figure 3). In [6], Ishii shows that the diagrams of trivalent
graphs have one to one correspondence to handlebody-links subject to local moves called
Reidemeister moves (for handlebody-links). This means,
{handlebody-links} = {trivalent graphs}/{Reidemeister moves}.
The Reidemeister moves for handlebody-links are as Figure 4. The moves RI-RV are the
Reidemeister moves for spatial trivalent graphs.
3. Quantum invariants for handlebody-links
In this section we define the quantum invariants for handlebody-links in 3-sphere S3.
1A construction of quantum invariants for handlebody-knots via Yetter-Drinfeld module was an-
nounced by [9].
QUANTUM INVARIANTS FOR HANDLEBODY-KNOTS 3
→
Figure 2. A spatial trivalent graph represents a handlebody-knot
→ ≈ ←
Figure 3. The two different trivalent graphs that represent the trivial
genus 2 handlebody-knot
RI RII RIII
RIV RV
RVI
Figure 4. Reidemeister Moves for handlebody-links
3.1. SkeinSpaceandJones-Wenzlidempotent. Todefineinvariantsforhandlebody-
links, we introduce the skein space and its special elements called Jones-Wenzl idempo-
tents.
Definition 3.1. (Skein space) Let F be a connected and oriented 2-manidfold (possibly
has boundaries). A link diagram in F is consisted of finitely many closed curves and
arcs whose end points are at the boundaries of F. These curves may have finitely many
transversecrossingsthathavetheupperorlowercrossinginformation. Twolinkdiagrams
4 ATSUHIKO MIZUSAWA AND JUN MURAKAMI
are regarded as the same if there is an isotopy of F that change one link diagram to
another fixing ∂F.
Let A be a fixed value in C\{0}. Skein space S(F) of F is the vector space of formal
linear sums, over C, of link diagrams in F subject to the next relations,
∪ D = −(A2 +A−2) D,
= A +A−1 ,
here the left-hand side of the first relation is a disjoint union of a trivial circle and a link
diagram D and in the second relation the figures represent the parts of the link diagrams
and the complements of them are the same diagrams.
If F = S2, since ∂F = ∅, all curves in S(F) are closed and become a blank diagram
multiplied by a scalar value in C because of the relations. Thus S(S2) is identified with
C. For each link diagram in S2, we represent the corresponding complex value as a
diagram inside the Kauffman bracket (cid:104) (cid:105). Note that the value of the Kauffman bracket
is an invariant of framed links (it doesn’t change under the RII and RIII moves).
We fix A = e2πi/4r, 3 ≤ r ∈ N for later use. Let D2 be a 2-disc that has 2n points
n
in the boundary. We regard it as a rectangle whose left edge and right edge have just n
points respectively.
Definition 3.2. (Jones-Wenzl idempotent) Following [4], for 0 ≤ n ≤ r −1, we define
special elements JW ∈ S(D2) illustrated as a white box in the diagrams. JW is defined
n n n
recursively as follows,
1 ∆ 1 1
n + 1 n−1
= − ,
n n n
∆
n n (cid:15) 1
where JW is a blank diagram and
0
(cid:42) (cid:43)
(A2(n+1) −A−2(n+1)) sin π(n+1)
∆ = = (−1)n = (−1)n r .
n A2 −A−2 sin π
n r
In this relation the arc attached a number means that number of parallel arcs. Note that
∆ = 0.
r−1
TheelementsJW arecalledJones-Wenzl idempotent becauseofthefollowingrelation,
n
n n
= .
QUANTUM INVARIANTS FOR HANDLEBODY-KNOTS 5
3.2. Definitionoftheinvariants. In[10],Yokotadefinedinvariantsforcolored(weighted)
spatial graphs by using the Kauffman bracket. Let I be a set of colors {0,1,...,r −2}.
A coloring of a spatial graph Γ is a map E(Γ) → I where E(Γ) is a set of edges and
circles of Γ. We represent the coloring by attaching numbers to edges and circles. We
define the invariants for handlebody-links via Yokota’s invariants.
From now on, we regard the i colored edge or circle of trivalent graphs in the diagrams
as i parallel arcs inserted by the idempotent JW and the trivalent vertices adjacent to
i
threecolorededgesasFigure5. Herex = (j+k−i)/2,y = (i+k−j)/2,z = (i+j−k)/2. If
i
i
−→ z y
j k
j x k
Figure 5. A colored trivalent vertex and a corresponding diagram
i,j,k satisfyi ≤ j+k, j ≤ i+k, k ≤ i+j andi+j+k ∈ 2Z, thenwehaveacorresponding
diagram in the right-hand side. Otherwise we couldn’t realize the diagram. Moreover, if
i+j +k ≤ 2(r−2), then the next value is non-zero.
(cid:42) (cid:43) (cid:42) y(cid:43)
i ∆ !∆ !∆ !∆ !
z x+y+z x−1 y−1 z−1
θ(i,j,k) = = = ,
j ∆ !∆ !∆ !
x y+z−1 x+z−1 x+y−1
k
where ∆ ! = ∆ ∆ ···∆ and ∆ ! = 1. The value θ(i,j,k) is independent of the
n n n−1 0 −1
order of triple (i,j,k).
Definition 3.3. (admissible) A triple (i,j,k) ∈ I3 is called admissible if (i,j,k) satisfies
following conditions,
i ≤ j +k j ≤ i+k k ≤ i+j
i+j +k ∈ 2Z i+j +k ≤ 2(r−2).
For an admissible (i,j,k), θ(i,j,k) is non-zero.
Yokota’s invariants for colored spatial graphs in S3 are defined as follows, first they are
defined in the case of trivalent graphs and then they are generalized to arbitrary graphs.
Definition 3.4. (Yokota’s invariant[10]) Let Γ be a spatial trivalent graph embedded
to S3. We fix a coloring to Γ by attaching numbers i ,i ,...,i ∈ I to the edges and the
1 2 n
circlessothatforeachvertexthetripleofcolorsofincidentedges(i ,i ,i )areadmissible.
a b c
This coloring is called a admissible coloring for Γ. Let D(i ,i ,...,i ) be a diagram of
1 2 n
Γ(i ,i ,...,i ) and D(i ,i ,...,i ) be the mirror image of D(i ,i ,...,i ). Then the next
1 2 n 1 2 n 1 2 n
6 ATSUHIKO MIZUSAWA AND JUN MURAKAMI
value is defined,
(cid:10) (cid:11)/ (cid:89)
(cid:104)Γ(i ,i ,...,i )(cid:105) := (cid:104)D(i ,i ,...,i )(cid:105) D(i ,i ,...,i ) θ(i ,i ,i ).
1 2 n Y 1 2 n 1 2 n a b c
tripleofcolors
ofvertices
We call this value Yokota(cid:48)s invariant of the spatial trivalent graph for the admissible
coloring.
Yokota’s invariant are generalized for arbitrary graphs with next relations at vertices,
(cid:42) (cid:43) (cid:42) (cid:43)
(cid:88) i
(1) = ∆
i
Y i Y
for an n-valent vertex (n > 3) where color i moves all admissible colors for the right-hand
side diagram. This relation is independent of the ways extending the edge.
(cid:28) (cid:29) (cid:28) (cid:29)
i j δ i
ij
=
∆
Y i Y
for a 2-valent vertex, and
(cid:42) (cid:43) (cid:42) (cid:43)
i
= δ
i0
Y Y
for a 1-valent vertex.
Theorem 3.5. ([10]) (cid:104)·(cid:105) is invariant under RI - RV moves of diagrams and well-difined
Y
as an invariant of spatial graphs .
Remark 3.6. Originally Yokota’s invariants are defined with a general variable A ∈
C \ {0} and a set of colors N ∪ {0}. In this paper we fix A = e2πi/4r (3 ≤ r ∈ N) and
(cid:10) (cid:11)
restrict the set of colors to I. Since A−1 = A, D = (cid:104)D(cid:105) and the definition of Yokota’s
invariants is rewritten as
(cid:89)
(cid:104)Γ(i ,i ,...,i )(cid:105) = |(cid:104)D(i ,i ,...,i )(cid:105)|2 / θ(i ,i ,i ).
1 2 n Y 1 2 n a b c
tripleofcolors
ofvertices
Now we define the invariants for handlebody-links in S3 by taking a linear sum of
Yokota’s invariants for all possible colorings with some weight.
Definition3.7. (QuantumU (sl )invariantsforhandlebody-links)LetJ beahandlebody-
q 2
link and Γ be a corresponding trivalent spatial graph. We attach colors i to edges and
k
QUANTUM INVARIANTS FOR HANDLEBODY-KNOTS 7
j to circles. Then we define the value (cid:104)·(cid:105) as follows,
l H
(cid:88)
(cid:104)J(cid:105) := ∆ ···∆ (cid:104)Γ(i ,...,i ,j ,...,j )(cid:105)
H i1 in 1 n 1 m Y
i1,...,in
j1,...,jm
= (cid:88) ∆ ···∆ |(cid:104)D(i ,...,i ,j ,...,j )(cid:105)|2/ (cid:89) θ(i ,i ,i ),
i1 in 1 n 1 m a b c
i1,...,in tripleofcolors
j1,...,jm ofvertices
where D(i ,...,i ,j ,...,j ) is a diagram of Γ(i ,...,i ,j ,...,j ) and each i , j moves all
1 n 1 m 1 n 1 m k l
admissible colorings for Γ. Usually we put a diagram of the corresponding graph inside
the bracket instead of the handlebody-link itself as (cid:104)D(cid:105) := (cid:104)J(cid:105) .
H H
We see the well-definedness of (cid:104)·(cid:105) by the next theorem.
H
Theorem3.8. LetD beadiagramofaspatialtrivalentgraphthatrepresentsahandlebody-
link J. Then the value (cid:104)J(cid:105) = (cid:104)D(cid:105) doesn’t change under Reidemeister moves RI - RVI.
H H
Hence (cid:104)·(cid:105) is an invariant for handlebody-links.
H
Proof. The invariance for RI - RV is derived from the invariance of Yokota’s invariants.
We show the invariance for RVI. By relation (1),
(cid:42) a b(cid:43) (cid:42)a b (cid:43) (cid:42) a b (cid:43)
(cid:88) (cid:88) j
∆ i = = ∆ .
i j
d c d c
i Y Y j d c Y
Thus we have,
(cid:42) (cid:43) (cid:42) (cid:43)
a b
(cid:88)
= ∆ ∆ ∆ ∆ ∆ i
a b c d i
d c
H a,b,c,d,i Y
(cid:42) a b (cid:43) (cid:42) (cid:43)
(cid:88) j
= ∆ ∆ ∆ ∆ ∆ = .
a b c d j
a,b,c,d,j d c Y H
(cid:3)
4. Non-triviality of the Invariants
In this section, we show calculations of the quantum invariants in some cases and that
the invariants distinguish a handlebody-knot from the trivial one. We also show some
properties of the invariants.
First we mention that the (cid:104)·(cid:105) don’t distinguish a handlebody-link from its mirror
H
image. This is clear from the definition of (cid:104)·(cid:105) .
H
8 ATSUHIKO MIZUSAWA AND JUN MURAKAMI
4.1. Relations. For convenience, we recall some formula for diagrams including JW .
n
More detail, see [2] and [4].
Tetrahedra’s edge
(cid:42) (cid:43)
(cid:20) i j k (cid:21) F! (cid:88) (−1)z[z +1]!
n i m =: = .
j k l m n E! (cid:81) [z −a ]!(cid:81) [b −z]!
l c≤z≤C s s t t
where
A2n −A−2n
(cid:0) (cid:1)
[n] = = (−1)n−1∆ [n]! = [n][n−1]···[1] [0]! = 1
A2 −A−2 n−1
(cid:89)
F! = [b −a ]! E! = [i]![j]![k]![l]![m]![n]!
t s
s,t
1 1
a = (i+j +k) b = (i+j +l+m)
1 1
2 2
1 1
a = (i+m+n) b = (i+k +l+n)
2 2
2 2
1 1
a = (j +l+n) b = (j +k +m+n)
3 3
2 2
1
a = (k +l+m) c = max{a } C = min{b }.
4 s t
2
Local change relations
(cid:42) (cid:43) (cid:42) (cid:43)
n = (−1)nAn2+2n n .
(cid:42) i (cid:43) (cid:42) (cid:43)
n A2(n+1)(i+1) −A−2(n+1)(i+1) n
= (−1)i .
A2(n+1) −A−2(n+1)
(cid:42) i j (cid:43) (cid:42) (cid:43)
i j
(2) = λij ,
k
k k
where λij = (−1)(i+j−k)/2Ai+j−k+(i2+j2−k2)/2.
k
(cid:42) k (cid:43) (cid:42) (cid:43)
i j θ(i,k,l) i
(3) = δ .
ij
∆
i
l
QUANTUM INVARIANTS FOR HANDLEBODY-KNOTS 9
j k
(cid:42)j k(cid:43) (cid:42) (cid:43)
(cid:26) (cid:27)
(cid:88) i j m
(4) = n ,
m k l n
n
i l
i l
(cid:20) (cid:21)
i j m
∆
(cid:26) i j m (cid:27) k l n n
where = .
k l n θ(i,l,n)θ(j,k,n)
i i i
(cid:42) (cid:43) (cid:42) (cid:43)
(cid:88) ∆
k
(5) = .
θ(i,j,k) k
j k
j j
In the relation (4) and (5), the colors m, k move all admissible colors of the right-hand
side diagrams.
4.2. Some Examples. First, we calculate for genus 2 trivial handlebody-knot 0 by
1
two ways; via a theta curve diagram D and via a hand-cuffs diagram D . Secondly, we
1 2
calculate a genus 2 handlebody-knot called 4 (Figure 6) in the table of [8].
1
Figure 6. 4 handlebody-knot
1
Example 4.1. (theta curve)
(cid:42) (cid:43)
i
(cid:88)
(cid:104)D (cid:105) = ∆ ∆ ∆
1 H i j k j
(i,j,k)
k Y
(cid:42) (cid:43)(cid:42) (cid:43)
(cid:88) ∆ ∆ ∆ i i
i j k
=
θ(i,j,k)2 j j
(i,j,k)
k k
(cid:88) (cid:88)r−2 (cid:88)r−2 (cid:88)a r2
= ∆ ∆ ∆ = ∆ ∆ ∆ = ,
i j k i j k 4sin4 π
(i,j,k) i=0 j=0 k=|i−j| r
i+j+k∈2Z
10 ATSUHIKO MIZUSAWA AND JUN MURAKAMI
where in the summations i, j, k move all colors such that (i,j,k) are admissible, and
a = r−2−|(r−2)−(i+j)|.
Before the calculation for the hand-cuffs diagram, we show the next property of (cid:104)·(cid:105) .
H
Proposition 4.2. (reducible splitting) Let D be a diagram of a spatial trivalent graph.
If D has an edge that connects two disjoint subgraph diagrams that can be separated by a
S curve from each other, then we have a splitting relation
1
(cid:42) (cid:43) (cid:42) (cid:43) (cid:42) (cid:43)
A B A B
= .
H H H
Proof. The relation directly comes from the next relation of Yokota’s invariants (See [10]
Proposition 4.5 and Proposition 4.8)
(cid:42) A j1 i k1 B (cid:43) = δ0i (cid:42) A j1(cid:43) (cid:42)k1 B (cid:43) .
j2 k2 Y ∆j1∆k1 Y Y
(cid:3)
Example 4.3. (hand-cuffs)
(cid:42) (cid:43) (cid:42) (cid:43) (cid:42) (cid:43)
(cid:104)D (cid:105) = =
2 H
H H H
2 2
(cid:42) (cid:43) (cid:42) (cid:43)
(cid:88)r−2 (cid:88)r−2 (cid:88)r−2 (cid:88)r−2 r2
= = ∆2 ∆2 = .
i j i j 4sin4 π
i=0 j=0 i=0 j=0 r
Hence we have (cid:104)D (cid:105) = (cid:104)D (cid:105) .
1 H 2 H
Example 4.4. (4 handlebody-knot)
1
(cid:42) (cid:43) (cid:12)(cid:42) (cid:43)(cid:12)2
(cid:88) (cid:88) ∆ ∆ ∆ (cid:12) (cid:12)
(cid:104)4 (cid:105) = ∆ ∆ ∆ i j = i j k (cid:12) i j (cid:12) .
1 H i j k θ(i,i,k)θ(j,j,k) (cid:12) (cid:12)
(i,i,k) k Y (i,i,k) (cid:12) k (cid:12)
(j,j,k) (j,j,k)
Here,
(cid:42) (cid:43) (cid:42) (cid:43)
i j (=5) (cid:88) ∆l∆m i l k j k
θ(i,k,l)θ(j,k,m) i m
k (i,k,l) k j
(j,k,m)
(=2) (cid:88) ∆l∆m(cid:0)λilk(cid:1)−2(cid:0)λjmk(cid:1)−2 (cid:42)i l k j k(cid:43)
θ(i,k,l)θ(j,k,m) i
(i,k,l) k m j
(j,k,m)
