qualgebras: a bridge between knotted graphs and ...lebed/lebed_tsukuba.pdfras: qualgeb a ridge b een...

Qualgebras: a bridge between knotted graphs

and axiomatizations of groups

Vi toria LEBED

OCAMI, Osaka

Topology Seminar, University of Tsukuba

June 26, 2014

Based on:

✎ Qualgebras and Knotted 3-Valent Graphs, ArXiv, 2014

✎ (With S. Kamada) Alexander and Markov Theorems for Graph-Braids,

in progress



algebra



algebra

topology

Part 1:

How a Knot Theorist

Would Invent Qualgebras

Quandle olorings

Knot diagrams: from illustration to manipulation

1926 K. Reidemeister:

Knots

∼=Diagrams

/R-moves

Vi toria LEBED (OCAMI, Osaka) Qualgebras 4 / 35

Quandle olorings



Knots

∼=Diagrams

/R-moves

Reidemeister moves:

RII←→

RIII←→

RI←→


Quandle olorings



Knots

∼=Diagrams

/R-moves

Combinatorial knot invariants:

Diagrams

��

� //something

Knots

∼=Diagrams

/R-moves

66mm

mm

mm


Quandle olorings

Quandles as an algebraization of knots

a b

b a�b

olorings

by (S ,�)


Quandle olorings


a b

b a�b

olorings

by (S ,�)

a

b

b�

(a�b)�

a

�

b

RIII←→

a

b

b�

(a� )�(b� )

a

�


Quandle olorings


a b

b a�b

olorings

by (S ,�)

a

b

b�

(a�b)�

a

�

b

RIII←→

a

b

b�

(a� )�(b� )

a

�

RIII ↔ (a�b)� = (a� )� (b� ) (SD)


Quandle olorings


a b

b a�b

olorings

by (S ,�)

a

b

b�

(a�b)�

a

�

b

RIII←→

a

b

b�

(a� )�(b� )

a

�

RIII ↔ (a�b)� = (a� )� (b� ) (SD)

RII ↔ (a�b)�̃b = a= (a�̃b)�b (Inv)

RI ↔ a�a= a (Idem)


Quandle olorings


a b

b a�b

olorings

by (S ,�)

a

b

b�

(a�b)�

a

�

b

RIII←→

a

b

b�

(a� )�(b� )

a

�

RIII ↔ (a�b)� = (a� )� (b� ) (SD)

RII ↔ (a�b)�̃b = a= (a�̃b)�b (Inv)


Quandle

(1982 D. Joy e,

S. Matveev)


Quandle olorings


a b

b a�b

olorings

by (S ,�)

a

b

b�

(a�b)�

a

�

b

RIII←→

a

b

b�

(a� )�(b� )

a

�

RIII ↔ (a�b)� = (a� )� (b� ) (SD)

RII ↔ (a�b)�̃b = a= (a�̃b)�b (Inv)


Quandle

(1982 D. Joy e,

S. Matveev)

knot invariants

olorings

; quandle


Quandle olorings


a b

b a�b

olorings

by (S ,�)

a

b

b�

(a�b)�

a

�

b

RIII←→

a

b

b�

(a� )�(b� )

a

�

RIII ↔ (a�b)� = (a� )� (b� ) (SD)

RII ↔ (a�b)�̃b = a= (a�̃b)�b (Inv)


Quandle

(1982 D. Joy e,

S. Matveev)

knot invariants

olorings

; quandle

Example

Group G ;

Conj(G )= (G , g �h= h

−1gh).


Quandle olorings


a b

b b

−1ab

olorings

by Conj(G )

RIII ↔ (a�b)� = (a� )� (b� ) (SD)

RII ↔ (a�b)�̃b = a= (a�̃b)�b (Inv)


Quandle

(1982 D. Joy e,

S. Matveev)

knot invariants

olorings

; quandle

Example

Group G ;


−1gh).


Quandle olorings


a b

b b

−1ab

olorings

by Conj(G )

p

α

θα

Wirtinger

presentation:

olorings by Conj(G )

l

Rep(π1

(R3\K ),G )

RIII ↔ (a�b)� = (a� )� (b� ) (SD)

RII ↔ (a�b)�̃b = a= (a�̃b)�b (Inv)


Quandle

(1982 D. Joy e,

S. Matveev)

knot invariants

olorings

; quandle

Example

Group G ;


−1gh).


Qualgebra olorings

Knotted 3-valent graphs

Standard and Kinoshita-Terasaka Θ- urves:

Θst

ΘKT

Importan e:

❀ handlebody-knots;

❀ foams ( ategori� ation, 3-manifolds);

❀ form a �nitely presented algebrai system (B knots do not).


Qualgebra olorings

Reidemeister moves for 3-graphs

1989 L.H. Kau�man, S. Yamada, D.N. Yetter:

RIV←→

RVI←→

RV←→


Qualgebra olorings



RIV←→

RVI←→

RV←→

3-Graphs

∼= Diagrams

/RI-RVI


Qualgebra olorings



RIV←→

RVI←→

RV←→

Combinatorial graph invariants:Diagrams

��

� //something

3-Graphs

∼= Diagrams

/RI-RVI

66nn

nn

nn


Qualgebra olorings



RIV←→

RVI←→

RV←→

Combinatorial graph invariants:Diagrams

��

� //something

3-Graphs

∼= Diagrams

/RI-RVI

66nn

nn

nn

Question: How to extend quandle olorings to 3-graphs?


Qualgebra olorings

Quandle olorings for 3-graphs

❀a

b

G is a group;

a

±1b

±1

±1 = 1

generalizations:

G -family of quandles (2012

Ishii-Iwakiri-Jang-Oshiro),

multiple onjugation quandle

(2013 A. Ishii)


Qualgebra olorings


❀a

b

G is a group;

a

±1b

±1

±1 = 1

generalizations:




(2013 A. Ishii)

❀a

±1a

±1

a

±1G is a group;

a

3 = 1

a vertex onstant version

of π1

(R3\Γ)(1995 C. Livingston)


Qualgebra olorings


❀a

b

G is a group;

a

±1b

±1

±1 = 1

generalizations:




(2013 A. Ishii)

❀a

±1a

±1

a

±1G is a group;

a

3 = 1


of π1


❀a

±1a

±1

a

±1S is a symmetri quandle;

∀x , ((x �a)�a)�a= x

(2007

Fleming-Mellor)


Qualgebra olorings


❀a

b

G is a group;

a

±1b

±1

±1 = 1

generalizations:




(2013 A. Ishii)

❀a

±1a

±1

a

±1G is a group;

a

3 = 1


of π1


❀a

±1a

±1

a

±1S is a symmetri quandle;

∀x , ((x �a)�a)�a= x

(2007

Fleming-Mellor)

❀a

b

S is a quandle;

∀x , ((x�±1a)�±1

b)�±1 = x

(2010

M. Niebrzy-

dowski)


Qualgebra olorings

Orientation

Well-oriented 3-graphs: only zip and unzip verti es.


Qualgebra olorings

Orientation


Proposition: Every 3-graph admits a well-orientation.


Qualgebra olorings

Orientation



Unoriented 3-graph 7−→ { its well-orientations }.


Qualgebra olorings

Orientation




Side question: # { well-orientations of Γ } ?


Qualgebra olorings

Orientation




Side question: # { well-orientations of Γ } ?

Theorem (A. Ishii): all well-orientations are onne ted by single-edge

orientation swit hes.


Qualgebra olorings

Qualgebras as an algebraization of 3-graphs

a b

b a�b

olorings

by (S ,�,∗)

a b

a∗b

a∗b

bab b


Qualgebra olorings


a b

b a�b

olorings

by (S ,�,∗)

a b

a∗b

a∗b

bab b

a

b

b∗

b∗

a� (b∗ )

RIVz

←→

a

b

b∗ (a�b)�

a�b


Qualgebra olorings


a b

b a�b

olorings

by (S ,�,∗)

a b

a∗b

a∗b

bab b

a

b

b∗

b∗

a� (b∗ )

RIVz

←→

a

b

b∗ (a�b)�

a�b

RIV ↔ a� (b∗ )= (a�b)�

RVI ↔ (a∗b)� = (a∗ )� (b∗ )RV ↔ a∗b = b∗ (a�b)


Qualgebra olorings


a b

b a�b

olorings

by (S ,�,∗)

a b

a∗b

a∗b

bab b

a

b

b∗

b∗

a� (b∗ )

RIVz

←→

a

b

b∗ (a�b)�

a�b

RIV ↔ a� (b∗ )= (a�b)�

RVI ↔ (a∗b)� = (a∗ )� (b∗ )RV ↔ a∗b = b∗ (a�b)

& (SD),

(Inv),

(Idem)

Qualgebra

Quandle

Algebra


Qualgebra olorings


a b

b a�b

olorings

by (S ,�,∗)

a b

a∗b

a∗b

bab b

RIV ↔ a� (b∗ )= (a�b)�

RVI ↔ (a∗b)� = (a∗ )� (b∗ )RV ↔ a∗b = b∗ (a�b)

& (SD),

(Inv),

(Idem)

Qualgebra

Quandle

Algebra

3-graph invariants

olorings

; qualgebra


Qualgebra olorings


a b

b a�b

olorings

by (S ,�,∗)

a b

a∗b

a∗b

bab b

RIV ↔ a� (b∗ )= (a�b)�

RVI ↔ (a∗b)� = (a∗ )� (b∗ )RV ↔ a∗b = b∗ (a�b)

& (SD),

(Inv),

(Idem)

Qualgebra

Quandle

Algebra

3-graph invariants

olorings

; qualgebra

Example

Group G ; QA(G )= (G , g �h= h

−1gh, g ∗h= gh).


Qualgebra olorings


a b

b a�b

olorings

by (S ,�,∗)

a b

a∗b

a∗b

bab b

3-graph invariants

olorings

; qualgebra

Example

Group G ; QA(G )= (G , g �h= h

−1gh, g ∗h= gh).

p

α

θα

Wirtinger

presentation:

olorings by QA(G )

l

Rep(π1

(R3\Γ),G )


Qualgebra olorings

Computation example

Isos eles olorings:

a a

a∗a a a

a∗a


Qualgebra olorings

Computation example

Isos eles olorings:

a a

a∗a a a

a∗a

x xx ∗x

Θst

x

x

x ∗x

y ∗y y

y

a

b

ΘKT


Qualgebra olorings

Computation example

x x x ∗x

Θst

x

x

x ∗x

y ∗y y

y

a

b

ΘKT

(⋆)

a = x� (y ∗y)= y �x ,

b = x�̃y = y�̃(x ∗x),

= (y ∗y)�x = (x ∗x)�̃y .


Qualgebra olorings

Computation example

x x x ∗x

Θst

x

x

x ∗x

y ∗y y

y

a

b

ΘKT

(⋆)

a = x� (y ∗y)= y �x ,

b = x�̃y = y�̃(x ∗x),

= (y ∗y)�x = (x ∗x)�̃y .

Qualgebra (S4

, g �h= h

−1gh, g ∗h= gh).


Qualgebra olorings

Computation example

x x x ∗x

Θst

x

x

x ∗x

y ∗y y

y

a

b

ΘKT

(⋆)

a = x� (y ∗y)= y �x ,

b = x�̃y = y�̃(x ∗x),

= (y ∗y)�x = (x ∗x)�̃y .

Qualgebra (S4

, g �h= h

−1gh, g ∗h= gh).

Solutions: x = y and x = (123),y = (432) and ...


Qualgebra olorings

Computation example

x x x ∗x

Θst

x

x

x ∗x

y ∗y y

y

a

b

ΘKT

(⋆)

a = x� (y ∗y)= y �x ,

b = x�̃y = y�̃(x ∗x),

= (y ∗y)�x = (x ∗x)�̃y .

Qualgebra (S4

, g �h= h

−1gh, g ∗h= gh).

Solutions: x = y and x = (123),y = (432) and ...

#Col isoS

4

(Θst

)= S

4

#Col isoS

4

(ΘKT

)>#S

4


Part 2:

How an Algebraist

Would Invent Qualgebras

Qualgebras and groups

Group qualgebras

Example 1

Group G ; group qualgebra QA(G )= (G , g �h= h

−1gh, g ∗h = gh).



Group qualgebras

Example 1


−1gh, g ∗h = gh).

abstra t level quandle axioms spe i� qualgebra axioms

topology moves RI-RIII moves RIV-RVI

groups onjugation onjugation-multipli ation intera tion



Group qualgebras

Example 1


−1gh, g ∗h = gh).




B Quandle axioms ⇒ all properties of onjugation.



Group qualgebras

Example 1


−1gh, g ∗h = gh).




B Quandle axioms ⇒ all properties of onjugation.

B Qualgebra axioms ; all properties of onjugation/multipli ation

intera tion.

(b�a)∗ (a�b)= ((a�̃b)�a)∗b

in QA(G ) : a

−1bab

−1ab= a

−1bab

−1ab

in free qualgebras : false



Other qualgebra examples

Example 1


−1gh, g ∗h= gh).

Example 1

′

Group G & X ⊂G ; the sub-qualgebra of QA(G ) generated by X .




Example 1


−1gh, g ∗h= gh).

Example 1

′


E.g., N⊂QA(Z).




Example 1


−1gh, g ∗h= gh).

Example 1

′


E.g., N⊂QA(Z).Remark: For �nite G , sub-qualgebra = sub-group.




Example 1


−1gh, g ∗h= gh).

Example 1

′



Example 0

Trivial qualgebra (S , a�b = a, a∗b), where ∗ is ommutative.




Example 1


−1gh, g ∗h= gh).

Example 1

′



Example 0

Trivial qualgebra (S , a�b = a, a∗b), where ∗ is ommutative.

; Abstra t graph invariants.


More examples

An in�nite family of exoti examples

Example S

n

Set X & ommutative operation ⋆ & zero element 0 (0⋆x = x ⋆0= 0)


More examples


Example S

n

Set X & ommutative operation ⋆ & zero element 0 (0⋆x = x ⋆0= 0) ;

Q

X ,n ={((x

1

, . . . ,xn

),g) ∈X×n×Sn

∣∣x

i

= x

j

= 0 whenever g(i)= j with i 6= j

},

(x ,g)� (y ,h)= (x ·h,h−1gh),

(x ,g)∗ (y ,h)= (x ⋆y ,gh)

is a qualgebra.


More examples


Example S

n


Q

X ,n ={((x

1

, . . . ,xn

),g) ∈X×n×Sn

∣∣x

i

= x

j


},

(x ,g)� (y ,h)= (x ·h,h−1gh),

(x ,g)∗ (y ,h)= (x ⋆y ,gh)

is a qualgebra.

E.g., X

2

= {0,a }, a⋆1

a= 0 or a⋆2

a= a, n= 2, S

2

= { Id,τ },


More examples


Example S

n


Q

X ,n ={((x

1

, . . . ,xn

),g) ∈X×n×Sn

∣∣x

i

= x

j


},

(x ,g)� (y ,h)= (x ·h,h−1gh),

(x ,g)∗ (y ,h)= (x ⋆y ,gh)

is a qualgebra.

E.g., X

2

= {0,a }, a⋆1

a= 0 or a⋆2

a= a, n= 2, S

2

= { Id,τ },

Q

X

2

,2 ={((x

1

,x2

), Id)∣∣x

1

,x2

∈X2

}∐ {

((0,0),τ)}

,

#Q

X

2

,2 = 5


More examples


Example S

n


Q

X ,n ={((x

1

, . . . ,xn

),g) ∈X×n×Sn

∣∣x

i

= x

j


},

(x ,g)� (y ,h)= (x ·h,h−1gh),

(x ,g)∗ (y ,h)= (x ⋆y ,gh)

is a qualgebra.

E.g., X

2

= {0,a }, a⋆1

a= 0 or a⋆2

a= a, n= 2, S

2

= { Id,τ },

Q

X

2

,2 ={((x

1

,x2

), Id)∣∣x

1

,x2

∈X2

}∐ {

((0,0),τ)}

,

#Q

X

2

,2 = 5

; two 5-element qualgebras.


More examples


Example S

n


Q

X ,n ={((x

1

, . . . ,xn

),g) ∈X×n×Sn

∣∣x

i

= x

j


},

(x ,g)� (y ,h)= (x ·h,h−1gh),

(x ,g)∗ (y ,h)= (x ⋆y ,gh)

is a qualgebra.

E.g., X

2

= {0,a }, a⋆1

a= 0 or a⋆2

a= a, n= 2, S

2

= { Id,τ },

Q

X

2

,2 ={((x

1

,x2

), Id)∣∣x

1

,x2

∈X2

}∐ {

((0,0),τ)}

,

#Q

X

2

,2 = 5

; two 5-element qualgebras.

❀ Commutative. ❀ Asso iative.

❀ Not an ellative ⇒ do not ome from groups:

((0,0),τ)∗i

((x1

,x2

), Id)= ((0,0),τ), i = 1,2.


More examples

Towards a lassi� ation

Example 3

Size É 3: only trivial qualgebras.


More examples


Example 3


Example 4

Non-trivial qualgebra stru tures on Q = {p,q,r ,s}:

put p = q,q = p,r = r ,s = s;

x� r = x , x�y = x for other y ;

∗ is ommutative, x ∗y = x ∗y ,

r ∗x = r for x 6= r ,

r ∗ r = s ∗ s = p∗q = s , p∗p,p∗ s ∈ {p,q,s}


More examples


Example 3


Example 4


put p = q,q = p,r = r ,s = s;



r ∗x = r for x 6= r ,

r ∗ r = s ∗ s = p∗q = s , p∗p,p∗ s ∈ {p,q,s}

; 3∗3= 9 stru tures.


More examples


Example 3


Example 4


put p = q,q = p,r = r ,s = s;



r ∗x = r for x 6= r ,

r ∗ r = s ∗ s = p∗q = s , p∗p,p∗ s ∈ {p,q,s}


❀ Not an ellative ⇒ do not ome from groups.


More examples


Example 3


Example 4


put p = q,q = p,r = r ,s = s;



r ∗x = r for x 6= r ,

r ∗ r = s ∗ s = p∗q = s , p∗p,p∗ s ∈ {p,q,s}



❀ Two are asso iative (sub-qualgebras of Q

X

2

,2: omit ((a,a), Id)).


More examples


Example 3


Example 4


put p = q,q = p,r = r ,s = s;



r ∗x = r for x 6= r ,

r ∗ r = s ∗ s = p∗q = s , p∗p,p∗ s ∈ {p,q,s}




X

2

,2: omit ((a,a), Id)).❀ Three have neutral elements, none are unital asso iative.


More examples


Example 4


put p = q,q = p,r = r ,s = s;



r ∗x = r for x 6= r ,

r ∗ r = s ∗ s = p∗q = s , p∗p,p∗ s ∈ {p,q,s}




X

2

,2: omit ((a,a), Id)).❀ Three have neutral elements, none are unital asso iative.

Question: Continue the lassi� ation.


More examples

Computation example

Standard and Hopf u� graphs:

a

b

C

st

a

b

a

′

′

a

′ = ((a�̃ )�a

′ = �a

C

H

#ColQ

(Cst

)=#{(a,b, ) ∈Q |b∗a= a, b∗ = } = 18,

#ColQ

(CH

)=#{(a,b, ) ∈Q |b∗a= a� , b∗ = �a} = 14.


Symmetri qualgebras

Qualgebras with inversion

Question: How far qualgebras are from groups?


Symmetri qualgebras



Step 1: inversion.


Symmetri qualgebras



Step 1: inversion.

Good involution: ρ : S → S s.t.

ρ(ρ(a))= a

ρ(a)�b = ρ(a�b)a�ρ(b)= a�̃b

(a∗b)∗ρ(b)=ρ(b)∗ (b∗a)= a

Symmetri

quandle

(1996

S. Kamada)

Symmetri

qualgebra


Symmetri qualgebras



Step 1: inversion.


ρ(ρ(a))= a


(a∗b)∗ρ(b)=ρ(b)∗ (b∗a)= a

Symmetri

quandle

(1996

S. Kamada)

Symmetri

qualgebra

Example

Group G ; QA

∗(G )= (G , g �h= h

−1gh, g ∗h= gh, ρ(h)= h

−1).


Symmetri qualgebras



Step 1: inversion.


ρ(ρ(a))= a


(a∗b)∗ρ(b)=ρ(b)∗ (b∗a)= a

Symmetri

quandle

(1996

S. Kamada)

Symmetri

qualgebra

Example

Group G ; QA

∗(G )= (G , g �h= h

−1gh, g ∗h= gh, ρ(h)= h

−1).

Properties:

❀ Maps a 7→ a∗b and a 7→ b∗a are bije tions

; ∗ is a Latin square (= pseudo-sudoku).


Symmetri qualgebras



Step 1: inversion.


ρ(ρ(a))= a


(a∗b)∗ρ(b)=ρ(b)∗ (b∗a)= a

Symmetri

quandle

(1996

S. Kamada)

Symmetri

qualgebra

Example

Group G ; QA

∗(G )= (G , g �h= h

−1gh, g ∗h= gh, ρ(h)= h

−1).

Properties:

❀ Maps a 7→ a∗b and a 7→ b∗a are bije tions

; ∗ is a Latin square (= pseudo-sudoku).

❀ ρ is de�ned uniquely.


Symmetri qualgebras

Topologi al motivation

Question: When are qualgebra invariants independent of orientations?


Symmetri qualgebras



a ρ(a)←→


Symmetri qualgebras



a ρ(a)←→

ρ(ρ(a))= a


(a∗b)∗ρ(b)=ρ(b)∗ (b∗a)= a

Symmetri

quandle

(1996

S. Kamada)

Symmetri

qualgebra


Symmetri qualgebras



a ρ(a)←→

ρ(ρ(a))= a


(a∗b)∗ρ(b)=ρ(b)∗ (b∗a)= a

Symmetri

quandle

(1996

S. Kamada)

Symmetri

qualgebra

abstra t level good involution axioms

topology unoriented 3-graphs

groups onjugation- and multipli ation-inversion intera tions


Symmetri qualgebras

Symmetri qualgebras: examples

Example 0

Symmetri trivial qualgebras ←→ Latin squares whi h

❀ are symmetri w.r.t. the main diagonal, and

❀ ontain a row σ ∈Bij(S) i� ontain a row σ−1.


Symmetri qualgebras


Example 0

Symmetri trivial qualgebras ←→ Latin squares whi h

❀ are symmetri w.r.t. the main diagonal, and

❀ ontain a row σ ∈Bij(S) i� ontain a row σ−1.

Example 3

S = {x ,y ,z }, a�b = a, ∗ is ommutative.

∗ x y z

x x y z

y y z x

z z x y

ρ x z y

Z/3Z

∗ x y z

x x z y

y z y x

z y x z

ρ x y z

∗ x y z

x y x z

y x z y

z z y x

ρ x y z

not groups


Symmetri qualgebras


Example 4

❀ Non-trivial qualgebras: not symmetri (⇐= not an ellative).


Symmetri qualgebras


Example 4

❀ Non-trivial qualgebras: not symmetri (⇐= not an ellative).

❀ Trivial qualgebras:

∗ x y z w

x x y z w

y y z w x

z z w x y

w w x y z

ρ x w z y

←QA(Z/4Z)

QA(Z/2Z×Z/2Z)→

∗ x y z w

x x y z w

y y x w z

z z w x y

w w z y x

ρ x y z w

∗ x y z w

x z y x w

y y z w x

z x w z y

w w x y z

ρ x w z y

not

groups

∗ x y z w

x x y w z

y y x z w

z w z y x

w z w x y

ρ x y z w


Symmetri qualgebras

Asso iative qualgebras



Symmetri qualgebras



Step 2: asso iativity (for ∗).


Symmetri qualgebras



Step 2: asso iativity (for ∗).

symmetri qualgebras

groups

asso iative qualgebras

qualgebras


Qualgebraizability

From quandles to qualgebras: theory

Question: Can a quandle (S ,�) be qualgebraized into (S ,�,∗)?


Qualgebraizability



Right translations T

b

: a 7→ a�b.


Qualgebraizability




b

: a 7→ a�b.

Properties (quandle ase):


Qualgebraizability




b

: a 7→ a�b.


1

T

b

is an automorphism of the quandle (S ,�).


Qualgebraizability




b

: a 7→ a�b.


1

T

b


2

T

b

∣∣S

b

= Id, where S

b

is the sub-quandle of S generated by b.


Qualgebraizability




b

: a 7→ a�b.


1

T

b


2

T

b

∣∣S

b

= Id, where S

b


3

One has a quandle morphism

T : (S ,�)−→Conj(Aut(S)),

b 7−→T

b

.

That is, T

b�

=T

−1

T

b

T

.


Qualgebraizability




b

: a 7→ a�b.


1

T

b


2

T

b

∣∣S

b

= Id, where S

b


3

One has a quandle morphism

T : (S ,�)−→Conj(Aut(S)),

b 7−→T

b

.

That is, T

b�

=T

−1

T

b

T

.

4

T (S) is a sub-quandle of Conj(Aut(S))


Qualgebraizability




b

: a 7→ a�b.

Properties (qualgebra ase):

1

T

b

is an automorphism of the qualgebra (S ,�,∗).

2

T

b

∣∣S

b

= Id, where S

b

is the sub-qualgebra of S generated by b.

3

One has a qualgebra morphism

T : (S ,�,∗)−→QA(Aut(S)),

b 7−→T

b

.

That is, T

b�

=T

−1

T

b

T

and T

b∗ =T

b

T

.

4

T (S) is a sub-qualgebra of QA(Aut(S))


Qualgebraizability




b

: a 7→ a�b.


1

T

b


2

T

b

∣∣S

b

= Id, where S

b


3


T : (S ,�,∗)−→QA(Aut(S)),

b 7−→T

b

.

That is, T

b�

=T

−1

T

b

T

and T

b∗ =T

b

T

.

4

T (S) is a sub-qualgebra of QA(Aut(S)); ne essary qualgebraizability ondition.


Qualgebraizability




b

: a 7→ a�b.


1

T

b


2

T

b

∣∣S

b

= Id, where S

b


3


T : (S ,�,∗)−→QA(Aut(S)),

b 7−→T

b

.

That is, T

b�

=T

−1

T

b

T

and T

b∗ =T

b

T

.

4

T (S) is a sub-qualgebra of QA(Aut(S)); ne essary qualgebraizability ondition.

Question: A good qualgebraizability riterion?


Qualgebraizability

From quandles to qualgebras: examples

Non-uniqueness

Trivial quandle (S , a�b = a) & any ommutative ∗ ; qualgebra.


Qualgebraizability


Non-uniqueness


#S = n ; n

n(n+1)2

qualgebraizations:


Qualgebraizability


Non-uniqueness


#S = n ; n

n(n+1)2

qualgebraizations:

n= 2: 8 qualgebraizations (4 equivalen e lasses);

n= 3: 729 qualgebraizations (129 equivalen e lasses).


Qualgebraizability


Non-uniqueness


#S = n ; n

n(n+1)2

qualgebraizations:



Question: Count qualgebraizations up to qualgebra isomorphism?


Qualgebraizability


Non-uniqueness


#S = n ; n

n(n+1)2

qualgebraizations:




Non-existen e

Alexander quandle (M , a�b =αa+ (1−α)b), M ∈R

Mod, α ∈R∗.


Qualgebraizability


Non-uniqueness


#S = n ; n

n(n+1)2

qualgebraizations:




Non-existen e


Mod, α ∈R∗.

Qualgebraizable ⇐⇒ trivial (α= 1) (⇐= T

a

◦Tb

∉T (M)).


Qualgebraizability


Non-uniqueness


#S = n ; n

n(n+1)2

qualgebraizations:




Non-existen e


Mod, α ∈R∗.

Qualgebraizable ⇐⇒ trivial (α= 1) (⇐= T

a

◦Tb

∉T (M)).

Uniqueness

QA(Sn

) is the unique qualgebraization of Conj(Sn

) (⇐= T is inje tive).


Qualgebraizability


Size 3

1

Trivial quandle: 129 qualgebraizations.


Qualgebraizability


Size 3

1


2

Alexander quandle (Z/3Z,a�b = 2b−a): no qualgebraizations.

; Fox olorings are not extensible to 3-graphs.


Qualgebraizability


Size 3

1


2



3

Q

′ = {p,q,r }, p� r = q, q� r = p, x�y = x otherwise.


Qualgebraizability


Size 3

1


2



3

Q


T (Q ′)= { Id,(pq) } ⊂Aut(Q ′)∼= S

3

is a sub-qualgebra,


Qualgebraizability


Size 3

1


2



3

Q


T (Q ′)= { Id,(pq) } ⊂Aut(Q ′)∼= S

3

is a sub-qualgebra,

B but Q

′is not qualgebraizable:


Qualgebraizability


Size 3

1


2



3

Q


T (Q ′)= { Id,(pq) } ⊂Aut(Q ′)∼= S

3

is a sub-qualgebra,

B but Q


r ∗ r �xed by T

r

=⇒ r ∗ r = r =⇒ T

r

=T

r∗r =T

r

T

r

= Id, ontradi tion.


Qualgebraizability


Size 3

1


2



3

Q


T (Q ′)= { Id,(pq) } ⊂Aut(Q ′)∼= S

3

is a sub-qualgebra,

B but Q


r ∗ r �xed by T

r

=⇒ r ∗ r = r =⇒ T

r

=T

r∗r =T

r

T

r

= Id, ontradi tion.

Size 4

Quandle Q: 9 qualgebraizations.


Qualgebraizability


Size 3

1


2



3

Q


T (Q ′)= { Id,(pq) } ⊂Aut(Q ′)∼= S

3

is a sub-qualgebra,

B but Q


r ∗ r �xed by T

r

=⇒ r ∗ r = r =⇒ T

r

=T

r∗r =T

r

T

r

= Id, ontradi tion.

Size 4

Quandle Q: 9 qualgebraizations.

B Its sub-quandle Q

′is not qualgebraizable.


Part 3:

Variations of

Qualgebra Ideas

Bran hed braids

Alexander-Markov theorem

losure

Theorem (1923 Alexander; 1935 Markov)

❀ Surje tivity.

❀ Kernel: moves M1 and M2.

B

B

′

M1

←→B

′

B

B

M2

←→B

M2

←→B


Bran hed braids

Braid and knot invariants

Knot invariants out of braid invariants:

Braids

��

� //something

Knots

∼=Braids

/M1,M2

77oo

oo

oo


Bran hed braids



Braids

��

� //something

Knots

∼=Braids

/M1,M2

77oo

oo

oo

1925 E. Artin:

Braids

∼= Diagrams

/RII-RIII


Bran hed braids



Braids

��

� //something

Knots

∼=Braids

/M1,M2

77oo

oo

oo

Combinatorial braid invariants:

Braid diagrams

��

� //something

Braids

∼=Diagrams

/RII-RIII

66mm

mm

mm


Bran hed braids



Braid diagrams

��

� //something

Braids

∼=Diagrams

/RII-RIII

66mm

mm

mm

Many knot invariants adapt to the braid ase, with possible enhan ements.

Example: quandle invariants.


Bran hed braids



Braid diagrams

��

� //something

Braids

∼=Diagrams

/RII-RIII

66mm

mm

mm



❀ Weaker stru ture: ra k (= quandle without a�a= a) ⇐= no RI.


Bran hed braids



Braid diagrams

��

� //something

Braids

∼=Diagrams

/RII-RIII

66mm

mm

mm



❀ Weaker stru ture: ra k (= quandle without a�a= a) ⇐= no RI.

❀ Operator invariants instead of ounting invariants:

braids with n strands −→Aut(S×n),

β 7−→ ( olors of upper ar s 7→ olors of lower ar s ).


Bran hed braids

Bran hed Alexander-Markov theorem

Question: : A losure pro edure for 3-graphs?


Bran hed braids



Bran hed

braids

losure


Bran hed braids



Bran hed

braids

losure

Theorem (2010 K. Kanno - K. Taniyama; 2014 S. Kamada - L.)

❀ Surje tivity.



Bran hed braids


bran hed braids

losure

−→ 3-graphs


❀ Surje tivity.


B

B

′

M1

←→B

′

B

B

M2

←→B

M2

←→B


Bran hed braids


bran hed braids

losure

−→ 3-graphs


❀ Surje tivity.


B

B

′

M1

←→B

′

B

B

M2

←→B

M2

←→B

Generalizations

❀ Graph-braids (verti es of arbitrary valen e).

❀ Virtual and welded versions.


Bran hed braids

Bran hed braid and 3-graph invariants

3-Graph invariants out of B-braid invariants:

B-braids

��

� //something

3-Graphs

∼= B-braids

/M1,M2

66nn

nn

nn


Bran hed braids



B-braids

��

� //something

3-Graphs

∼= B-braids

/M1,M2

66nn

nn

nn

Combinatorial B-braid invariants:

B-braid diagrams

��

� //something

B-braids

∼=Diagrams

/RII-RVI

66mm

mm

mm


Bran hed braids



B-braids

��

� //something

3-Graphs

∼= B-braids

/M1,M2

66nn

nn

nn

Combinatorial B-braid invariants:

B-braid diagrams

��

� //something

B-braids

∼=Diagrams

/RII-RVI

66mm

mm

mm

B-braid invariants

olorings

; weak qualgebra(omit a�a= a)


Co y le invariants

Getting more out of qualgebra olorings

diagrams: D

R-move

D

′,

olorings: C C′,

oloring sets: ColS

(D)bij

←→ ColS

(D ′).


Co y le invariants


diagrams: D

R-move

D

′,

olorings: C C′,

oloring sets: ColS

(D)bij

←→ ColS

(D ′).

Counting invariants:

#ColS

(D)=#ColS

(D ′).


Co y le invariants


diagrams: D

R-move

D

′,

olorings: C C′,

oloring sets: ColS

(D)bij

←→ ColS

(D ′).


#ColS

(D)=#ColS

(D ′).

Question: Extra t more information?

ω(C )=ω(C ′),{ω(C )

∣∣C ∈ ColS

(D)}=

{ω(C ′)

∣∣C ′ ∈ ColS

(D ′)}

.


Co y le invariants


diagrams: D

R-move

D

′,

olorings: C C′,

oloring sets: ColS

(D)bij

←→ ColS

(D ′).


#ColS

(D)=#ColS

(D ′).

Question: Extra t more information?

ω(C )=ω(C ′),{ω(C )

∣∣C ∈ ColS

(D)}=

{ω(C ′)

∣∣C ′ ∈ ColS

(D ′)}

.

Inspiration: Quandle o y le invariants of knots (1999 Carter-

Jelsovsky-Kamada-Langford-Saito).


Co y le invariants

Qualgebra o y le invariants for 3-graphs

a b

b a�b

7→χ(a,b)

b

a�b

bab b

7→ −χ(a,b)

a∗b

bab b

7→λ(a,b)

a b

a∗b

7→ −λ(a,b)


Co y le invariants


a b

b a�b

7→χ(a,b)

b

a�b

bab b

7→ −χ(a,b)

a∗b

bab b

7→λ(a,b)

a b

a∗b

7→ −λ(a,b)

a

b

a∗b

(a∗b)�

−λ(a,b)

χ(a∗b, )

RVIz

←→

a

b

(a� )∗ (b� )

b�

a

� −λ(a� ,b� )

χ(b, )

χ(a, )


Co y le invariants


a b

b a�b

7→χ(a,b)

b

a�b

bab b

7→ −χ(a,b)

a∗b

bab b

7→λ(a,b)

a b

a∗b

7→ −λ(a,b)

a

b

a∗b

(a∗b)�

−λ(a,b)

χ(a∗b, )

RVIz

←→

a

b

(a� )∗ (b� )

b�

a

� −λ(a� ,b� )

χ(b, )

χ(a, )

RIV ↔ χ(a,b∗ )=χ(a,b)+χ(a�b, )RVI ↔ χ(a∗b, )+λ(a� ,b� ) =χ(a, )+χ(b, )+λ(a,b)RV ↔ χ(a,b)+λ(a,b) =λ(b,a�b)


Co y le invariants


a b

b a�b

7→χ(a,b)

b

a�b

bab b

7→ −χ(a,b)

a∗b

bab b

7→λ(a,b)

a b

a∗b

7→ −λ(a,b)

a

b

a∗b

(a∗b)�

−λ(a,b)

χ(a∗b, )

RVIz

←→

a

b

(a� )∗ (b� )

b�

a

� −λ(a� ,b� )

χ(b, )

χ(a, )


Qualgebra

2- o y le

RI-RIII are automati .


Co y le invariants


a b

b a�b

7→χ(a,b)

b

a�b

bab b

7→ −χ(a,b)

a∗b

bab b

7→λ(a,b)

a b

a∗b

7→ −λ(a,b)


Qualgebra

2- o y le


3-graph invariants

olorings

;weights

qualgebra & 2- or 3- o y le


Co y le invariants


a b

b a�b

7→χ(a,b)

b

a�b

bab b

7→ −χ(a,b)

a∗b

bab b

7→λ(a,b)

a b

a∗b

7→ −λ(a,b)


Qualgebra

2- o y le


3-graph invariants

olorings

;weights

qualgebra & 2- or 3- o y le

Qualgebra o y le invariants ⊇ qualgebra ounting invariants.


Co y le invariants

Towards qualgebra ohomology

Qualgebra 2- oboundaries:

φ : S →Z ; χ(a,b)=φ(a�b)−φ(a), ; trivial

λ(a,b)=φ(a)+φ(b)−φ(a∗b) graph invariants


Co y le invariants





Question: : De�ne and study qualgebra ohomology in higher degrees?


Co y le invariants






Solution: for rigid qualgebras (←→ rigid 3-graphs, i.e., omit RV).


Co y le invariants







Cf. the ohomology of multiple onjugation quandles (Carter-Ishii-

Saito-Yang).


Co y le invariants








Saito-Yang).

Enhan ements

❀ Region oloring and shadow o y le invariants ;qualgebra 3- o y les.


Co y le invariants








Saito-Yang).

Enhan ements

❀ Region oloring and shadow o y le invariants ;qualgebra 3- o y les.

❀ Distinguish zip- and unzip-verti es:

a∗b

a

b

7→λ(a,b)

a b

a∗b

7→λ(a,b)


Co y le invariants

Qualgebra o y les: example

Example 4

Q = {p,q,r ,s} p = q,q = p,r = r ,s = s;



r ∗x = r for x 6= r ,

r ∗ r = s ∗ s = p∗q = s , p∗p,p∗ s ∈ {p,q,s}

❀ Z

2(Q)∼=Z8

❀ B

2(Q)∼=Z4

❀ H

2(Q)∼=Z/2Z⊕Z4


qualgebras: a bridge between knotted graphs and ...lebed/lebed_tsukuba.pdfras: qualgeb a ridge b een...

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