two generalizations of the multivariable alexander polynomial
TRANSCRIPT
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Two generalizations of the MultivariableAlexander Polynomial
Iva Halacheva(University of Toronto)
Knots in Washington XLIDecember 6, 2015
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Table of Contents
1 The original MVALong virtual knots
2 The Archibald generalizationCircuit algebrasAlexander Half DensitiesThe invariant
3 The Bar-Natan generalizationMetamonoidsThe invariant
4 Some resultsReducing the Archibald invariantMaking the connection
5 Future directions
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Long virtual knots
Setup
We will consider:vMVA=the MVA for regular long virtual knots and links
Ingredients
Given a virtual link L with n components, diagram DL :
Label all the arcs of DL (using a set S). To each componentassociate a variable, {ti}ni=1.
Build the Alexander matrix:
M(DL ) ∈ MS\{a}×S(Z(ti | i = 1, . . . , n)), a = incoming long arc
s3
s1
t2 t1
s2
7→s1 s2 s3
s3 1 − t2 −1 t1
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Long virtual knots
Setup
We will consider:vMVA=the MVA for regular long virtual knots and links
Ingredients
Given a virtual link L with n components, diagram DL :
Label all the arcs of DL (using a set S). To each componentassociate a variable, {ti}ni=1.
Build the Alexander matrix:
M(DL ) ∈ MS\{a}×S(Z(ti | i = 1, . . . , n)), a = incoming long arc
s3
s1
t2 t1
s2
7→s1 s2 s3
s3 1 − t2 −1 t1
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Long virtual knots
Setup
We will consider:vMVA=the MVA for regular long virtual knots and links
Ingredients
Given a virtual link L with n components, diagram DL :
Label all the arcs of DL (using a set S). To each componentassociate a variable, {ti}ni=1.
Build the Alexander matrix:
M(DL ) ∈ MS\{a}×S(Z(ti | i = 1, . . . , n)), a = incoming long arc
s3
s1
t2 t1
s2
7→s1 s2 s3
s3 1 − t2 −1 t1
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Long virtual knots
Setup (cont’d)
s2
t1s3
t2
s1
7→s1 s2 s3
s3 t2 − 1 −t1 1
DefinitionThe Multivariable Alexander Polynomial for a regular long virtuallink L is defined as:
vMVA(L) =∏
i
t−µ(i)/2i
det M′(DL )
tl − 1
where,µ(i) = #{i-th component goes over a crossing}
M′(DL ) = M(DL ) with the column a deleted
tl = variable of the long strand
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Long virtual knots
Setup (cont’d)
s2
t1s3
t2
s1
7→s1 s2 s3
s3 t2 − 1 −t1 1
DefinitionThe Multivariable Alexander Polynomial for a regular long virtuallink L is defined as:
vMVA(L) =∏
i
t−µ(i)/2i
det M′(DL )
tl − 1
where,µ(i) = #{i-th component goes over a crossing}
M′(DL ) = M(DL ) with the column a deleted
tl = variable of the long strand
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Long virtual knots
Additional arcs: We might need to subdivide an arc in laterconstructions.
b−→
b
a7→
a ba −1 1
Lemma
Dividing an arc will not change the value of vMVA(L).
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Circuit algebras
Virtual Tangles
Underlying structure: Circuit Algebras.
DefinitionA (oriented) circuit diagram with k inputs is a disk with k internaldisks removed, and (oriented) pairings among the boundary points.
DefinitionA circuit algebra is a collection of spacesV = {Vn}n∈N andmorphisms F = {Fd}d a circuit diagram.
DefinitionAn oriented circuit algebra is a collection of spacesV = {Vn,m}n,m∈N (n incoming and m outgoing components) andmorphisms F = {Fd}d an oriented circuit diagram.
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Circuit algebras
Virtual Tangles
Underlying structure: Circuit Algebras.
DefinitionA (oriented) circuit diagram with k inputs is a disk with k internaldisks removed, and (oriented) pairings among the boundary points.
DefinitionA circuit algebra is a collection of spacesV = {Vn}n∈N andmorphisms F = {Fd}d a circuit diagram.
DefinitionAn oriented circuit algebra is a collection of spacesV = {Vn,m}n,m∈N (n incoming and m outgoing components) andmorphisms F = {Fd}d an oriented circuit diagram.
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Circuit algebras
Virtual Tangles
Underlying structure: Circuit Algebras.
DefinitionA (oriented) circuit diagram with k inputs is a disk with k internaldisks removed, and (oriented) pairings among the boundary points.
DefinitionA circuit algebra is a collection of spacesV = {Vn}n∈N andmorphisms F = {Fd}d a circuit diagram.
DefinitionAn oriented circuit algebra is a collection of spacesV = {Vn,m}n,m∈N (n incoming and m outgoing components) andmorphisms F = {Fd}d an oriented circuit diagram.
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Circuit algebras
Example: A circuit diagram.
: V1,2 ⊗ V3,2 −→ V1,1
Remarks:
Circuit algebras form a category.
Virtual tangles are a circuit algebra, CA⟨!,"
⟩.
Goal: Invariant of tangles which is a circuit algebra morphism.
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Circuit algebras
Example: A circuit diagram.
: V1,2 ⊗ V3,2 −→ V1,1
Remarks:
Circuit algebras form a category.
Virtual tangles are a circuit algebra, CA⟨!,"
⟩.
Goal: Invariant of tangles which is a circuit algebra morphism.
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Alexander Half Densities
Given S a finite set, R a ring, we’ll denote:
Λk (S) := k -th exterior power of the formal R-module with basis S
Definition
Let X in and Xout denote the labelling sets of the incoming andoutgoing arcs of a tangle (we require n =
∣∣∣X in∣∣∣ =
∣∣∣Xout∣∣∣). The
corresponding Alexander Half Density is:
AHD(X in,Xout) := Λn(Xout) ⊗ Λn(X in ∪ Xout)
Remark: To guarantee that∣∣∣X in
∣∣∣ =∣∣∣Xout
∣∣∣, we might have to breaksome arcs artificially.
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Alexander Half Densities
Given S a finite set, R a ring, we’ll denote:
Λk (S) := k -th exterior power of the formal R-module with basis S
Definition
Let X in and Xout denote the labelling sets of the incoming andoutgoing arcs of a tangle (we require n =
∣∣∣X in∣∣∣ =
∣∣∣Xout∣∣∣). The
corresponding Alexander Half Density is:
AHD(X in,Xout) := Λn(Xout) ⊗ Λn(X in ∪ Xout)
Remark: To guarantee that∣∣∣X in
∣∣∣ =∣∣∣Xout
∣∣∣, we might have to breaksome arcs artificially.
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Alexander Half Densities
The morphisms are defined using interior multiplication:
Definition (The Gluing Map)
Suppose {aj ⊗ bj ∈ AHD(X inj ,X
outj )}j=1,...,m with the elements to be
glued labelled the same. Let G =⋃
j X inj ∩
⋃j Xout
j denote the setof elements to be glued. The gluing map produces:
iG
m∧j=1
aj
⊗ iG
m∧j=1
bj
∈ AHD
⋃j
X inj − G,
⋃j
Xoutj − G
Where iG is interior multiplication.
Proposition
(AHD, gluing) is a circuit algebra.
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Alexander Half Densities
The morphisms are defined using interior multiplication:
Definition (The Gluing Map)
Suppose {aj ⊗ bj ∈ AHD(X inj ,X
outj )}j=1,...,m with the elements to be
glued labelled the same. Let G =⋃
j X inj ∩
⋃j Xout
j denote the setof elements to be glued. The gluing map produces:
iG
m∧j=1
aj
⊗ iG
m∧j=1
bj
∈ AHD
⋃j
X inj − G,
⋃j
Xoutj − G
Where iG is interior multiplication.
Proposition
(AHD, gluing) is a circuit algebra.
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Construction of a tangle invariant:
tMVA : {Regular v-tangles} −→ {Alexander Half Densities}
Given an n-tangle T with a labelled diagram DT :
Construct the Alexander matrix M(DT ) as before:
M(DT ) =
internal Xout X in
internal
Xout
Let w ∈ Λn
(Xout
), a choice of ordering of the labels of Xout.
Let M(DT )i1<...<in = the minor of M(DT ) computed using all“internal columns” and those labelled ci1 , . . . , cin in Xout ∪ X in.
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Construction of a tangle invariant:
tMVA : {Regular v-tangles} −→ {Alexander Half Densities}
Given an n-tangle T with a labelled diagram DT :
Construct the Alexander matrix M(DT ) as before:
M(DT ) =
internal Xout X in
internal
Xout
Let w ∈ Λn(Xout
), a choice of ordering of the labels of Xout.
Let M(DT )i1<...<in = the minor of M(DT ) computed using all“internal columns” and those labelled ci1 , . . . , cin in Xout ∪ X in.
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Construction of a tangle invariant:
tMVA : {Regular v-tangles} −→ {Alexander Half Densities}
Given an n-tangle T with a labelled diagram DT :
Construct the Alexander matrix M(DT ) as before:
M(DT ) =
internal Xout X in
internal
Xout
Let w ∈ Λn
(Xout
), a choice of ordering of the labels of Xout.
Let M(DT )i1<...<in = the minor of M(DT ) computed using all“internal columns” and those labelled ci1 , . . . , cin in Xout ∪ X in.
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Theorem (Archibald)
An invariant of regular, virtual n-tangles can be defined by, for anytangle T:
tMVA(T) =n∏
k=1
t−µ(k)/2k w ⊗
∑i1<...<in
M(DT )i1<...<in ci1 ∧ . . . ∧ cin
In particular:
a2
b1
t2a1
t1
b2
7→
a1
b2
t1a2
t2
b1
7→
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Theorem (Archibald)
An invariant of regular, virtual n-tangles can be defined by, for anytangle T:
tMVA(T) =n∏
k=1
t−µ(k)/2k w ⊗
∑i1<...<in
M(DT )i1<...<in ci1 ∧ . . . ∧ cin
In particular:
a2
b1
t2a1
t1
b2
7→
a1 a2 b1 b2
a1 1 0 −1 0a2 1 − t2 t1 0 −1
a1
b2
t1a2
t2
b1
7→
a1 a2 b1 b2
a1 1 0 −1 0a2 t2 − 1 1 0 −t1
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Theorem (Archibald)
An invariant of regular, virtual n-tangles can be defined by, for anytangle T:
tMVA(T) =n∏
k=1
t−µ(k)/2k w ⊗
∑i1<...<in
M(DT )i1<...<in ci1 ∧ . . . ∧ cin
In particular:
a2
b1
t2a1
t1
b2
7→t−1/21 a1 ∧ a2 ⊗ [b1 ∧ b2 + (t2 − 1)b1 ∧ a1
−(t1)b1 ∧ a2 + b2 ∧ a1 + (t1)a1 ∧ a2]
a1
b2
t1a2
t2
b1
7→t−1/21 a2 ∧ a1 ⊗ [(t1)b2 ∧ b1 − (t1)b2 ∧ a1
+b1 ∧ a2 + (t2 − 1)b1 ∧ a1 + a2 ∧ a1]
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Example: Suppose we are considering the following tangle T .
a1 a2
b1 b2
7→
b1 b2 a1 a2
b1 t2 0 −1 1 − t1b2 0 t1 1 − t2 −1
tMVA(T) = t−1/21 t−1/2
2 b1 ∧ b2 ⊗ [(t1t2)b1 ∧ b2 + (t2(1 − t2))b1 ∧ a1
−(t2)b1 ∧ a2 + (t1)b2 ∧ a1 − (t1(1 − t1))b2 ∧ a2 + (t1 + t2 − t1t2)a1 ∧ a2]
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Example: Suppose we are considering the following tangle T .
a1 a2
b1 b2
7→
b1 b2 a1 a2
b1 t2 0 −1 1 − t1
b2 0 t1 1 − t2 −1
tMVA(T) = t−1/21 t−1/2
2 b1 ∧ b2 ⊗ [(t1t2)b1 ∧ b2 + (t2(1 − t2))b1 ∧ a1
−(t2)b1 ∧ a2 + (t1)b2 ∧ a1 − (t1(1 − t1))b2 ∧ a2 + (t1 + t2 − t1t2)a1 ∧ a2]
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Example: Suppose we are considering the following tangle T .
a1 a2
b1 b2
7→
b1 b2 a1 a2
b1 t2 0 −1 1 − t1b2 0 t1 1 − t2 −1
tMVA(T) = t−1/21 t−1/2
2 b1 ∧ b2 ⊗ [(t1t2)b1 ∧ b2 + (t2(1 − t2))b1 ∧ a1
−(t2)b1 ∧ a2 + (t1)b2 ∧ a1 − (t1(1 − t1))b2 ∧ a2 + (t1 + t2 − t1t2)a1 ∧ a2]
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Example: Suppose we are considering the following tangle T .
a1 a2
b1 b2
7→
b1 b2 a1 a2
b1 t2 0 −1 1 − t1b2 0 t1 1 − t2 −1
tMVA(T) = t−1/21 t−1/2
2 b1 ∧ b2 ⊗ [(t1t2)b1 ∧ b2 + (t2(1 − t2))b1 ∧ a1
−(t2)b1 ∧ a2 + (t1)b2 ∧ a1 − (t1(1 − t1))b2 ∧ a2 + (t1 + t2 − t1t2)a1 ∧ a2]
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Properties of the tMVA:
It is a circuit algebra morphism.
Satisfies the extra “Overcrossings Commute” relation, so is awelded tangle invariant.
Can renormalize for usual tangles to get R1 invariance:
tMVA′ =∏
k
t rot(k)/2k tMVA
Can recover vMVA:
tMVA(
a−→ T
b−→
)ta − 1
= vMVA(
a−→ T
b−→
)(b ⊗ b − a ⊗ a)
Gives easy verification of many local vMVA relations(Conway’s second and third identity, Murakami’s fifth axiom,doubled delta move).
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Metamonoids
DefinitionA meta-monoid is a collection of sets {GX }X=a finite set together withmaps between them:
mxyz : G{x,y}∪X −→ G{z}∪X “multiplication”∗ : GX × GY −→ GX∪Y “union”ηx : G{x}∪X −→ GX “deletion”σx
z : G{x}∪X −→ G{z}∪X “renaming”ex : GX −→ G{x}∪X “identity”
satisfying:1 “Monoid axioms”
mxyz ◦ ex = σ
yz = myx
z ◦ ex
mxyz ◦muv
x = muxz ◦mvy
x
2 A list of “set manipulation” axioms.
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Metamonoids
Examples:
The collection {GX }X=a finite set where G is a monoid andGX = {f : X → G}, with the standard operations, is ameta-monoid.
Pure, regular virtual tangles form a meta-monoid, withGX = {tangles with strand labels in X} and the standardoperations.
Note: A property for monoids that does not, in general, holdfor meta-monoids.
ηxg ∗ ηyg = g, for g ∈ G{x,y}
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Metamonoids
Examples:
The collection {GX }X=a finite set where G is a monoid andGX = {f : X → G}, with the standard operations, is ameta-monoid.
Pure, regular virtual tangles form a meta-monoid, withGX = {tangles with strand labels in X} and the standardoperations.
Note: A property for monoids that does not, in general, holdfor meta-monoids.
ηxg ∗ ηyg = g, for g ∈ G{x,y}
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Metamonoids
The target space:
Definition
For a finite set X , let R = Z(Tx | x ∈ X). Take:
BX = R ×MatX×X (R)
And the operations:λ a b Xa α β θ
b γ δ ε
X φ ψ Ξ
mabc
−−−−−−→µB1−β
ta ,tb→tc
λµ c Xc γ + αδ/µ ε + δθ/µ
X φ + αψ/µ Ξ + ψθ/µ
(λ1 X1
X1 A1,λ2 X2
X2 A2
)∗−−→
λ1λ2 X1 X2
X1 A1 0X2 0 A2
Proposition
BX with the above maps has a meta-monoid structure.
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Metamonoids
The target space:
Definition
For a finite set X , let R = Z(Tx | x ∈ X). Take:
BX = R ×MatX×X (R)
And the operations:λ a b Xa α β θ
b γ δ ε
X φ ψ Ξ
mabc
−−−−−−→µB1−β
ta ,tb→tc
λµ c Xc γ + αδ/µ ε + δθ/µ
X φ + αψ/µ Ξ + ψθ/µ
(λ1 X1
X1 A1,λ2 X2
X2 A2
)∗−−→
λ1λ2 X1 X2
X1 A1 0X2 0 A2
Proposition
BX with the above maps has a meta-monoid structure.
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Metamonoids
The target space:
Definition
For a finite set X , let R = Z(Tx | x ∈ X). Take:
BX = R ×MatX×X (R)
And the operations:λ a b Xa α β θ
b γ δ ε
X φ ψ Ξ
mabc
−−−−−−→µB1−β
ta ,tb→tc
λµ c Xc γ + αδ/µ ε + δθ/µ
X φ + αψ/µ Ξ + ψθ/µ
(λ1 X1
X1 A1,λ2 X2
X2 A2
)∗−−→
λ1λ2 X1 X2
X1 A1 0X2 0 A2
Proposition
BX with the above maps has a meta-monoid structure.
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Theorem (Bar-Natan)There exists a unique meta-monoid morphismZ : {Pure, regular, X-labelled tangles} −→ BX such that:
a b7→
1 a ba 1 1 − tab 0 ta b a
7→
1 a ba 1 1 − t−1
ab 0 t−1
a
a 7→1 aa 1
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Example:
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Example:
cd
ab
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Example:
cd
ab7→
1 a ba 1 1 − t−1
ab 0 t−1
a
∗
1 c dc 1 1 − t−1
cd 0 t−1
c
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Example:
cd
ab
7→
1 a b c da 1 1 − ta 0 0b 0 ta 0 0c 0 0 1 1 − tcd 0 0 0 tc
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Example:
c
ab
7→
1 a b ca tc −(ta − 1)tc 0b 0 ta 0c 1 − tc (ta − 1)(tc − 1) 1
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Example:
c
a
7→
ta + tc − ta tc a c
a −tc−ta−tc+ta tc
(ta−1)tc−ta−tc+ta tc
c ta(tc−1)−ta−tc+ta tc
−ta−ta−tc+ta tc
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The invariant
Properties:
Satisfies “Overcrossings Commute”, so is a welded tangleinvariant.
For a long knot K , produces (λ, 1) where λ is the Alexanderpolynomial of K (modulo units).
For a long link L , the result (λ,A) gives the MVA (modulounits): λ det(A − I)/(1 − tl).
Efficient computationally (polynomial time in the number ofstrands).
Reduction of an invariant:
{ribbon-knotted S2 and S1 in R4} −→ {Free Lie and cyclic words}
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Reducing the Archibald invariant
Goal: Relate the two invariants, “A → B”.Strategy: Reduce the Archibald invariant to matrix form.
The Hodge star operator
Consider again, for∣∣∣Xout
∣∣∣ =∣∣∣X in
∣∣∣ = n:
AHD(X in,Xout) = Λn(Xout
)⊗ Λn
(Xout ∪ X in
)For a fixed w ∈ Λn
(Xout
), apply the Hodge star operator ∗w :
n⊕k=0
Λn−k(Xout
)∧ Λk
(X in
) ∗w−−→
n⊕k=0
Λk(Xout
)∧ Λk
(X in
)defined by: ∗w (α ∧ β) = γ ∧ β⇔ α ∧ γ = w
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Reducing the Archibald invariant
Goal: Relate the two invariants, “A → B”.Strategy: Reduce the Archibald invariant to matrix form.
The Hodge star operator
Consider again, for∣∣∣Xout
∣∣∣ =∣∣∣X in
∣∣∣ = n:
AHD(X in,Xout) = Λn(Xout
)⊗ Λn
(Xout ∪ X in
)For a fixed w ∈ Λn
(Xout
), apply the Hodge star operator ∗w :
n⊕k=0
Λn−k(Xout
)∧ Λk
(X in
) ∗w−−→
n⊕k=0
Λk(Xout
)∧ Λk
(X in
)defined by: ∗w (α ∧ β) = γ ∧ β⇔ α ∧ γ = w
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Reducing the Archibald invariant
TheoremFor an n-tangle T, let:
λ = the zero-graded component of ∗w (tMVA(T))
A = the matrix of one-graded components of ∗w (tMVA(T))
If λ , 0, the pair (λ,A) determines tMVA(T).
This is achieved using the formula:
det M(T){1,...,n}\{i1,...,ik };{j1,...,jk } = (−1)kn+k(k−1)/2−∑
p ipdet A j1,...,jk
i1,...,ik
λk−1
M(T){1,...,n}\{i1,...,ik };{j1,...,jk } = M(T) excluding columns ci1 , . . . , cikfrom Xout and including cj1 , . . . , cjk from X in.A j1,...,jk
i1,...,ik= submatrix of A with columns j1, . . . , jk and rows i1, . . . , ik .
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Reducing the Archibald invariant
TheoremFor an n-tangle T, let:
λ = the zero-graded component of ∗w (tMVA(T))
A = the matrix of one-graded components of ∗w (tMVA(T))
If λ , 0, the pair (λ,A) determines tMVA(T).
This is achieved using the formula:
det M(T){1,...,n}\{i1,...,ik };{j1,...,jk } = (−1)kn+k(k−1)/2−∑
p ipdet A j1,...,jk
i1,...,ik
λk−1
M(T){1,...,n}\{i1,...,ik };{j1,...,jk } = M(T) excluding columns ci1 , . . . , cikfrom Xout and including cj1 , . . . , cjk from X in.A j1,...,jk
i1,...,ik= submatrix of A with columns j1, . . . , jk and rows i1, . . . , ik .
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Reducing the Archibald invariant
Corollary
The pair (λ,A) is an invariant in its own right, with the inducedmaps:
λ a b X in
a α β θ
b γ δ ε
Xout φ ψ Ξ
mabc
−−−−−−→ta ,tb→tc
λ + β c X in
c γ + βγ−αδλ ε + βε−δθ
λ
Xout φ + βφ−αψλ Ξ + βΞ−ψθ
λ(λ1 X in
1Xout
1 A1,
λ2 X in2
Xout2 A2
)∗−−→
λ1λ2 X in1 X in
2Xout
1 λ2A1 0Xout
2 0 λ1A2
Let AX =the space of such pairs for n-tangles labelled by the set X(X in and Xout).
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Reducing the Archibald invariant
Corollary
The pair (λ,A) is an invariant in its own right, with the inducedmaps:
λ a b X in
a α β θ
b γ δ ε
Xout φ ψ Ξ
mabc
−−−−−−→ta ,tb→tc
λ + β c X in
c γ + βγ−αδλ ε + βε−δθ
λ
Xout φ + βφ−αψλ Ξ + βΞ−ψθ
λ(λ1 X in
1Xout
1 A1,
λ2 X in2
Xout2 A2
)∗−−→
λ1λ2 X in1 X in
2Xout
1 λ2A1 0Xout
2 0 λ1A2
Let AX =the space of such pairs for n-tangles labelled by the set X(X in and Xout).
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Making the connection
Theorem
The map BX → AX taking a pair (λ,A) 7→ (λ,−λA) respects theoperations on both sides.
Corollary
The above map induces a partial trace operation on the Bar-Nataninvariant, coming from the Archibald invariant:
λ c Xc α θ
X ψ Ξ
trc−−−−−→
λ(1 − α) XX Ξ + ψθ
1−α
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Making the connection
Theorem
The map BX → AX taking a pair (λ,A) 7→ (λ,−λA) respects theoperations on both sides.
Corollary
The above map induces a partial trace operation on the Bar-Nataninvariant, coming from the Archibald invariant:
λ c Xc α θ
X ψ Ξ
trc−−−−−→
λ(1 − α) XX Ξ + ψθ
1−α
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
Still to investigate:
How are the algebraic structures related (circuit algebra vsmeta-monoid)?
This invariant reduces to the Gassner (resp.) Buraurepresentation on braids.
What is the domain space of the trace map?
Possibly leading to a non-commutative generalization of theMVA.
Possibly categorifiable.
The original MVA The Archibald generalization The Bar-Natan generalization Some results Future directions
The End
Thank you!