Knotted 3-Valent Graphs, Branched Braids, and Multiplication-Conjugation Relations in a Group (Intelligence of Low-dimensional Topology)

Knotted 3-Valent

Graphs,

Branched

Braids,

and Multiplication-Conjugation Relations in

a

Group

Victoria

Lebed

Osaka City University Advanced Mathematical Institute

This

survey

is devotedto

a

new

algebraicstructurecalled qualgebra. Our topological

mo-tivation isthestudyofknotted 3-valentgraphsandcloselyrelated branched braidsvia

com-binatorially defined coloringinvariants. From

an

algebraicviewpoint,

our

structure

a

part of

an

alternativeaxiomatization ofgroups, describingthe properties ofconjugationoperation

anditsinteractionswiththe

group

multiplication. Qualgebras

can

thusbe metaphorically

seen as a

widening ofthe bridge between algebra andtopologyformed by the quandle

struc-ture,popular

among

knottheorists;

_see

Table 1 tobetter understand how this bridge works. Only

a

brief and rather informal exposition ofdifferent facetsofqualgebrasis givenhere.

For

more

details,comments,and proofs,

see

[20, 15]. However, Sections2.1,2.3, 2.4, and3. 1 contain

some

recentunpublishedresults,which will be thoroughly treated elsewhere.

1

How

aknot theorist would invent qualgebras

1.1 Quandles

as

an

algebraizationofknots

Diagram coloring techniquescount amongthemostpowerful combinatorial tools in Knot Theory. Afamous exampleis given byFoxcolorings,which

are

a

particular

case

of quandle colorings. Inthissection

we

brieflyrecall the latter.

Take

a

set$S$ and

a

binaryoperation $\triangleleft on$it. An$(S, \triangleleft)$-coloringof

a

knot diagram$D$is

an

assignment ofan element of$S$toeach

arc

_$ofD$in such

away

that the condition

on

Figure$1\circ A$

(motivatedbelow) issatisfied around each crossing. Unoriented

arcs

in

our

diagrams

mean

that the diagrams should be considered for all coherent orientations of such

arcs.

$\swarrow_{\grave{a\triangleleft}b}^{\backslash }OAab$

${}_{\theta_{a}1^{>^{p}oB}}C^{1}\alpha$ $a_{C}bY_{c^{\pm 1}b^{\pm 1}a^{\pm 1}=1}^{\copyright}$ $a_{Y_{b}^{b},a*}$

Figure1: Coloringrules and their topologicalmotivations

Now, wewantcoloringstosaysomething about theknot$K_{D}$representedby$D$, indepen-dently of the diagram chosen. Therefore,

we

wantReidemeister

moves

(Figure 2) toinduce only local coloring changes, keeping fixed all the colors outside the smallball where the

move

is realized. This happens if and only if operation$\triangleleft$satisfies the following properties:

RIII $arrow$一 self-distributivity $(a\triangleleft b)\triangleleft c=(a\triangleleft c)\triangleleft(b\triangleleft c)$, $(Q_{SD})$

RII – invertibility $\forall b,$ $a\mapsto a\triangleleft b$isinvertible, _{$(Q_{In\nu})$}

$p^{1}$ $\underline{RI}$ $)_{\backslash }^{/}($ _{$\underline{RII})’($}

Figure2: Reidemeistermovesfor knot diagrams

Data$(S, \triangleleft)$ satisfying

$(Q_{SD})-(Q_{Idem})$is called

a

quandle. This structure has_beenactively

studiedsincethe pioneer 1982

papers

ofD. JoyceandS.V Matveev[14, 23]. _Theargument

above_{implies that the number of colorings of}

_a

_{knot diagram by}

_a

_{quandleis stable by}

Rei-demeistermoves,and thus defines

an

invariantof the underlying knot:

Suchquandleinvariantsturn out tobe extremely efficient in practice.

The central example ofquandleisgiven by

a

group

$G$and operation$g\triangleleft h=h^{-1}gh$

on

it; it is the conjugationquandleof$G$, denoted by Conj_(G)

.

Nowfix

a

diagram$D$ of

a

knot$K_{D}.$

Recall Wirtingerpresentation of the knot

group

$\pi_{1}(\mathbb{R}^{3}\backslash K_{D})$, with

one

generator$\theta_{\alpha}$ for each

arc

$\alpha$of$D$,

as

shown

on

Figure

1OB

(point$p$is chosen in front ofthediagram).Around each

crossing,

compare

the relationsimposed

on

the $\theta_{\alpha}$with thecoloringrule from Figure

1OA.

One readily identifies Conj(G)-coloringsof$D$with representations of the knot

group

_in$G$

:

Quandle_{invariantsthusgeneralizethe classical study of knot}

_groups.

1.2 Extendingquandlecoloringsto3-graphs

Knotted3-valentgraphs(simplycalled 3-graphs in whatfollows;_{cf. Figures 5 and 6 for typical} examples) _{have recently attracted}

_a

_{lot of}attention,

among

others due to applicationsto

handle-body classification and to foams (a _{particular type of surfaces appearing in}

_some

categorificationconstructionsandin3-manifoldstudies)._Accordingto [19,26,27],the study of such graphs

up

toisotopyis equivalenttothe study of theirdiagramsuptoReidemeister

moves

I-VI (Figures2 and3), _{opening the}

_way

_{to combinatorialinvariants.}

$\underline{RV})_{\backslash }\lambda$ $\underline{RVI}$

Figure3;_{Additional Reidemeistermovesfor knotted 3-valent graph diagrams}

Ageneralization of the (very powerful)_{quandle colorings}tographsis

a

possible

source

of

suchcombinatorial invariants. Themain challenge istocomplete the coloring rule around

crossings (Figure $1\circ A$) with

a

rule around trivalentvertices. Wirtinger presentation of the

graph

group

suggests

a

solution when colors

come

from

a

conjugation quandle; itis given

to

more

general quandles in [22, 11, 24, 12, 13]. In [20]

we

proposed

an

alternativesolution,

whichconsistsin enriching thenotionof quandle in

a

particularway.

Ourmethod works for well-oriented 3-graphs–thatis,having only zip and unzip vertices

(Figure 4). Since every3-graphiswell-orientable,

our

method also allows to

compare

two

unoriented3-graphs by considering all their well-oriented versions.

$Y$

zip

A

unzip

Figure 4: Zipandunzipvertices for3-graphs

Now,

suppose

our

quandle $(S, \triangleleft)$ tobe endowed with

a

secondbinaryoperation$*$, and

use

itto define

a

coloringrulearoundtrivalent vertices

as

shown

on

Figure 1$OD$

.

_Asusual,

one

checks if this rule forces Reidemeister

moves

toinduceonly local changes in diagrams’

colorings. Ithappens ifandonly if operations$\triangleleft and*are$compatiblein the following

sense:

RIV – translationcomposability $a\triangleleft(b*c)=(a\triangleleft b)\triangleleft c,$ _{$(QA_{Comp})$}

RVI –

distributivitr

$(a*b)\triangleleft c=(a\triangleleft c)*(b\triangleleft c)$, $(QA_{D})$

RV – semi-commutativity. $a*b=b*(a\triangleleft b)$

.

$(QA_{Comm})$

Data $(S, \triangleleft, *)$ satisfying $(Q_{SD})-(Q_{Idem})$ and $(QA_{Comp})-(QA_{Comm})$ is called

a

qualgebra.

This termconsistsof words“quandle” and“algebra” zipped together, which underlines the

presence

and the importance oftwodifferent operations in the story Notethataxiom$(Q_{SD})$

can

be omitted since it follows from $(QA_{Comp})$ and $(QA_{Comm})$

.

Ourchoice ofaxioms

guar-anteesthat the number of colorings of

a

graph diagram$D$by

a

qualgebrais stable by

Reide-meistermoves, and thus defines

an

invariantof the underlying well-oriented 3-graph$\Gamma_{D}$

:

The central example of qualgebra is given,

once

again, by

a group

$G$,withconjugation and

multiplicationoperations: $g\triangleleft h=h^{-1}gh,$_$g*h=gh$

.

It is the

group

qualgebraof$G$,denoted

by$QA(G)$

.

Thecoloring rule from Figure

1

recovers

in this

case

the

one

from Figure

1\copyright ,

prescribed by Wirtingerpresentation ofthe graph

group.

Onethat gets

We finish this sectionwith

a

computation example. Here instead of counting all colo-rings of

a

diagram by

a

qualgebra$S$,

we

restrict ourselvesto isosceles colorings. This

means

that both incoming (oroutcoming) edges ofanyzip (respectively,unzip) vertexarecolored

bythe

same

element of$S$; inotherwords,

one

imposes$a=b$inFigure $1OD$

.

Reidemeister

moves

do not change the property of beingisosceles, hence the number of isosceles colo-rings $\# Cot_{S}^{iso}(D)$ is

a

graph invariant. The 3-graphs

we are

interested in

are

standard and

Kinoshita-Terasaka$\Theta$-curves,with diagrams given

on

Figure 5. An isosceles coloring of$\Theta_{st}$

is entirelydeterminedby the choice of$x\in S$_,

so

$\# Cot_{S}^{iso}(\Theta_{st})=\# S$

.

Any other well-orientation of$\Theta_{st}$ leadstothe

same

result. For$\Theta_{KT}$, the choice of$x,y\in S$ determines everything,but

$(\star)\{\begin{array}{l}a = x\triangleleft(y*y)=y\triangleleft x,b = X\triangleleft\triangleleft\sim_{y=y^{\sim}(x*x)},c = ()\triangleleft X=(X*X)^{\sim}.\end{array}$

Here

we

use

notion$x\triangleleft y\sim$,classicalinQuandleTheory: it standsfor

theunique$z\in S$satisfying

$z\triangleleft y.=x$ (cf. axiom $(Q_{Inv}.)$). Now, for

any

$x$, the choice _$y=x$provides

a

solutionto ($\star$),

so

$\# Cot_{s}^{\iota so}(\Theta_{KT})\geq\# S=\# Cot_{s}^{lSO}(\Theta_{st})$

.

To separatethese quantities _(andthus_to distinguish_the

two $\Theta$

-curves), take

as

$S$the

group

qualgebra of the symmetric

group

$S_{4}$

.

One checks that

$x=(123)$ and$y=(432)\neq x$satisfy($\star$),giving

$\# Cot_{QA(S_{4})}^{iso}(\Theta_{KT})>\# QA(S_{4})=\# Cot_{QA(S_{4})}^{iso}(\Theta_{st})$

.

$\Theta_{KT}$

Figure5: Isosceles colorings for diagrams of standard and Kinoshita-Terasaka$\Theta$-curves

2

How

an

algebraist would invent qualgebras

2.1 An_{abstraction ofthe conjugation-multiplcation interaction in}

_agroup

Let

us

return to quandles

once

again. Besides Knot Theory, they

appear

in another setting, completely algebraic thistime. We

saw

abovethatconjugation operation defines

a

quandle structure

on a

group,

and thus satisfiesaxioms $(Q_{SD})-(Q_{Idem})$

.

Infact,

one can

say

more:

if

a

propertyinvolvingonly conjugation holdstruein

every

group, thenitis

a

consequence

of thesethreeaxioms. The

reason

lies in thestructureofthe freequandle

on

a

set$X$,which

can

be

seen

inside the freegroup

on

$X$

.

Quandlethusprovide

an

axiomatizationofconjugation.

In

a

similarway, conjugationandmultiplication operations define

a

qualgebrastructure

on a

group, andthussatisfy all qualgebraaxioms. Moreover, _axioms $(QA_{Comp})-(QA_{Comm})$

captureall essential relations between conjugation and multiplication (cf. Table 1).

How-ever,formalizingthisidea isnot

so easy

Forinstance,relation

$(b\triangleleft a)*(a\triangleleft b)=((a\triangleleft b)\sim\triangleleft a)*b$

holds in

any group

qualgebra(both_{sides equal}$a^{-1}bab^{-1}$_{ab),butfails in the free qualgebra}

on

twoelements–andthus doesnotfollow from qualgebraaxioms.

The remainder of this section is devoted to various examples of qualgebras. Algebraic

properties of

some

of them

are

very

different from those of

groups.

This confirms that the

interest ofqualgebras

goes

beyond the realm of

groups.

One

more

conclusion is that

con-jugation and multiplication operations do not suffice for

an

alternativeaxiomatization

_of

groups;

themissingingredients will bedetermined inSection2.3.

$\bullet$ The

first exampleisstill closeto

groups.

Considersub-qualgebrasof

a

group

qualge-bra $QA(G)$ – thatis, subsets stable by conjugationandmultiplication. If$G$is finite,

one

getsonly subgroups of$G$

.

For infinite $G$

new

examples

appear:

forinstance, the

sub-qualgebra$\mathbb{N}$of_{$QA(\mathbb{Z}, +)$} contains

no

inverses,and thusisnot

a

group

qualgebra.

$\bullet$

Now,considerqualgebras$(S, \triangleleft, *)$with$a\triangleleft b=a$,called trivialqualgebras.Inthis case,

the onlycondition imposed$on*by$axioms$(QA_{Comp})-(QA_{Comm})$is the commutativity.

Colorings by trivial qualgebras donotdistinguish over-crossings from under-crossings, and thus donot capturetheknottedness of 3-graphs. Such qualgebras thus yield only abstractgraphinvariants.

$\bullet$

A

more

sophisticated example

can

be constructed

as

follows. Take

a

set $X$ equipped

with

a

commutative operation $\star$ and

a

distinguished

zero

element$0$ (this

means

that

$0\star x=x\star 0=0$for all$x$). Fix

an

$n\in \mathbb{N}$

.

Extend operation$\star$to$X^{xn}$coordinate-wise, and

denote by$\cdot$the

usual right action of thesymmetric

group

$S_{n}$

on

$X^{xn}$

.

Now,theset

$Q_{X,n}=${$((x_{1},\ldots,x_{n}),g)\in X^{xn}xS_{n}|x_{i}=x_{j}=0$whenever_$g(i)=j$with$i\neq j$}

can

be endowed with the following qualgebra structure:

$\ulcorner x,g)\triangleleft(\overline{y}, h)=(\overline{x}\cdot h, h^{-1}gh)$, $(\overline{x},g)*(\overline{y}, h)=(\overline{x}\star\overline{y},gh)$

.

Consider the simplest example$X_{2}=\{0, a\}$

.

Then

$Q_{X_{2},2}=\{ ((x_{1\prime}x_{2}), Id) |x_{1},x_{2}\in X_{2}\}\coprod\{((0,0),\tau)\}$

(where$\tau$isthenon-trivial element of$S_{2}$) consistsof five elements. $7Wo$operations $\star_{1}$

and$\star_{2}$

can

be fedinto

our

machine:$0$is

a zero

element forboth,and

we

have$a\star_{1}a=0$

and$a\star_{2}a=a$

.

Thetworesultingqualgebras

are

non-isomorphic. Their operations$*i$

are

commutative, associative,butnon-cancellative:

$((0,0),\tau)*\iota((0, a),Id)=((0,0),\tau)*i((a,0),Id)=((0,0),\tau) , i=1,2.$

2.2 Towards

a

classification of qualgebras: the 4-element

case

Up tosize3, all qualgebras

are

trivial. Thingschangein size 4. In [20]

we

_{classified all}

non-trivia14-e1ementqualgebrasup to qualgebra isomorphism. Here

we

describe all the 9

iso-morphismclasses. Ontheset$Q=\{p, q, r,s\}$,consider the involutionexchanging$p$and$q$

:

$\overline{p}=q,\overline{q}=p,\overline{r}=r,\overline{s}=s.$

Put$x\triangleleft r=X$, and_{$x\triangleleft y=x$}for other$y$

.

Asfor the second operation,take the commutative

$\bullet$

$\overline{x}*\overline{y}=\overline{x*y}$for_{$aJ1x,y\in Q$};

$\bullet$

$r$enjoys theabsorptionproperty:

$r*x=r$for all$x\neq r$;

$\bullet$

one

has

_{$r*r=s*s=p*q=s$}

;

$\bullet$

$q*q$and$q*s$

are

any

elements chosen in$\{p,$ $q,$$sI.$

Thealternatives inthelast point leadto$3\cross 3=9$pairwise_{non-isomorphicstructures.}

The absorptionpropertyfor$r$preventsthese qualgebras from being cancellative with

re-spect to$*$ and,

a

fortiori,fromembeddinginto

a group.

Further,

outofthese ninestructures,

preciselytwo

are

associative. They

are

in fact the sub-qualgebras of$Q_{X_{2},2}$ from Section2. 1

obtained byomittingthe element$((a, a)$_,Id$)$;thetwopossible operations

$\star_{1}$ and$\star_{2}$give

non-isomorphic_{structures. Lastly, three qualgebras out of thenine haveneutralelements, and}

none

are

unitalassociative. Thus

even

in this smallsizequalgebras

can

exhibit

a

wide

range

of algebraicbehavior,confirmingtheinterestofthis structure.

To illustrate topological applications_{of 4-element qualgebras, consider the diagrams of}

standardand Hopf cuff graphs depicted

on

Figure 6, _{Analyzing the colors around trivalent}

vertices,

one

getsfor

any

qualgebra$S$thefollowingbijections:

$Co(_{S}(C_{st})\{(a,b,c)\in S^{x3}|b*a=a, b*c=c\}\underline{bij},$

$C\circ t_{S}(C_{H})\{(a, b, c)\in S^{x3}|b*a=(a^{\sim}\triangleleft c)\triangleleft a\underline{bij}, b*c=c\triangleleft a\}.$

For

a

trivial qualgebra$S$, these sets coincide. _However,

for the

non-trivia14-e1ement

qual-gebra $Q$ above with _$q*q=s$ and

$q*s=q$,

one

gets$\# Cot_{Q}(C_{st})=18$ (and the

same

value

for anywell-orientation of$C_{st}$, due to the commutativity_$of*$), and

$\# Cot_{Q}(C_{H})=14$

.

This

distinguishesthe two cuffgraphs.

$a’=((a\triangleleft c)\triangleleft a\sim$

$c’=c\triangleleft a$

$C_{H}$

Figure6:Qualgebracoloringsfor diagrams of standard and Hopf cuff graphs 2.3 Getting closerto

groups:

symmetric qualgebras

We

now

turn to distinctionsbetween the notions of

group

and qualgebra. Above

were

given examples of qualgebras which

are

not associative and/or_{not cancellative. Here thesetwo}

properties willbe showntobe essentially theonly

ones

needed for

a

qualgebratobe

a group.

The notionof symmetric quandle,introduced in 1996by S. Kamada([17]), _{shouldfirstbe}

recalled. Itis

a

quandle $(S, \triangleleft)$ endowed with

an

involution

$\rho:Sarrow S$ (called

a

good

involu-tion), _{compatible with operation}$\triangleleft in$thefollowing

sense:

$\rho(a)\triangleleft b=\rho(a\triangleleft b)$, (1)

The

topological

role

of

a

good

involution

is

to render quandle

invariants independent

of

orientations. Concretely, if $(S, \triangleleft,\rho)$ is

a

symmetric quandle, then

a

bijection $Cot_{S}(D)rightarrow$

$Cot_{S}(-D)$, where diagrams $D$ and $-D$ _{differ by the orientation only,}

can

_{be given by the}

rule $a\downarrow-\uparrow\rho(a)$ Now,

we

want the

same

kind of ruleto induce

a

bijection between the

$(S, \triangleleft, *)$-coloringsetsof3-graph diagrams which differ by the orientation of

some

edges only

($aJ1$thegraphs involved

are

supposedwell-oriented). Forthistohold,

$\rho$should be

a

good

in-volution for the quandle$(S, \triangleleft)$, compatiblewith $*$ in thefollowing

sense:

$(a*b)*\rho(b)=\rho(b)*(b*a)=a$

.

(3)

The resulting structure$(S, \triangleleft, *,\rho)$ iscalled

a

symmetric qualgebra.

As

one

would expect, the central example is given by

group

qualgebras, for which the

inversion$\rho(g)=g^{-1}$ defines

a

goodinvolution.Table 1

can now

be continuedwithTable2.

Table 2: Different viewpointsonsymmetric qualgebras

In

a

symmetricqualgebra,

maps

$a\mapsto a*b$and$a\mapsto b*a$

are

_bijectionsfor_all$b$,according

toaxiom(3). _{Consequently,}

$\bullet$

forany$b$,property(3) defines$\rho(b)$uniquely; good involutions

can

thus be safely

omit-ted from thedescription of

a

symmetric qualgebra;

$\bullet$ the multiplication table for $*$ is

a

Latinsquare(i.e.,

every

element

occurs

exactly

once

in each column and in eachrow).

Usingtheseobservations, _{symmetric trivial qualgebras}

_are

particularly

easy

todescribe.

They correspondtoLatinsquareswhich

1.

are

symmetric withrespect tothe maindiagonal, and

2. together with

a row

correspondingto

a

permutation$\sigma$necessarily contain

a row

corre-spondingto$\sigma^{-1}$

(thetwo

rows

can

coincide).

Let

us now

turn toexamples.

$\bullet$ Among3-e1ement

qualgebras (which

_are

_necessarilytrivial), there

are

precisely3

sym-metricones,

as

usualuptosymmetricqualgebra isomorphism:

$QA(\mathbb{Z}/3\mathbb{Z})$

–

notgroups

$\bullet$ Among

$QA( \mathbb{Z}\int 4\mathbb{Z})$ $QA( \mathbb{Z}\int 2\mathbb{Z}\cross \mathbb{Z}/2\mathbb{Z})$

not

groups

$\bullet$

Non-trivia14-e1ementqualgebras

are

notcancellative and thusnotsymmetric.

Even though good involutions bring the structure of qualgebra closer to that ofgroup,

the examples above show that symmetric qualgebra stay

more

general than

groups.

The

missingproperty turns out to betheassociativity:

group

qualgebras

are

preciselysymmetric

qualgebraswhichareassociative(i.e., theiroperation $*$ isassociative);seeFigure 7.

Figure 7: Qualgebrasversus groups

In particular, this allows to deduce the non-associativity of two 3-element and two

4-element symmetric qualgebras above from the absence of neutral 4-elements for their

opera-tions $*$ (andthus theirfailuretobe

group

qualgebras),which is mucheasiertocheck.

2.4 Fromquandlestoqualgebras

Above

we

analyzed how far the notion of qualgebraisfromthat of

group.

Acomparison of thenotions of qualgebra and quandle will be given here.

Let

us

first discuss when

a

quandle $(S, \triangleleft)$ isqualgebraizable -that is, admits

a

second

operation $*$ turning itinto

a

qualgebra. For this, considerright translations $T_{b}:a\mapsto a\triangleleft b,$

writtenhere

on

the right of their arguments. Axioms $(Q_{SD})-(Q_{Idem})$ imply that

2’. the

map

$T:b\mapsto T_{b}$ is

a

quandle morphism from $(S, \triangleleft)$ to Conj$(Aut(S))$ –thatis,

one

has $T_{b\triangleleft c}=T_{c}^{-1}T_{b}T_{c}$;

3’. theimage of$T$is

a

sub-quandleof Conj_$(Aut(S))$

.

Now, if$(S, \triangleleft, *)$is

a

qualgebra, thenin addition

1.

maps

$T_{b}$

are

automorphisms of the qualgebra$(S, \triangleleft, *)$;

2. the

map

$T:b\mapsto T_{b}$is

a

qualgebra morphism from$(S, \triangleleft, *)$ to_$QA(Aut(S))$–thatis,

one

has $T_{b*c}=T_{b}T_{\mathcal{C}/}.$

3. $T(S)$ is

a

sub-qualgebra of$QA(Aut(S))$,andisin particular stable under composition;

4. therestrictionof$T_{b}$tothe sub-qualgebra of$S$generatedby$b$istheidentity

map.

Property3 is

an

important

necessary

qualgebraizability condition, which is unfortunately

not sufficient (a counter-exampleis given below). _{Neither does it give estimations for the}

number of qualgebraizations of

a

given quandle: the relatedproperty 2 determines $b*c$

only modulo $Ker(T)$, which

can

beverylarge. Asfor now,

no

satisfying qualgebraizability

criterionis knowntothe author.

We

now

give

some

exampleswhere qualgebraizations

are

unique,

are

numerous,

_or

do

notexistatall.

$\bullet$

As shown above, the qualgebraizationsof

a

trivial quandle

are

given by commutative $n(n+1)$

operations $*$

.

Forthe trivial quandle with _$n$ elements, thisgives $n\overline{2}$

qualgebraiza-tions.However,countingthese qualgebraizations

up

toqualgebraisomorphismismuch

more

difficult. Forinstance, for$n=2$ _{these8 structuresfallinto4 equivalence}classes,

and for$n=3$the729structuresform 129 classes.

$\bullet$

Consider

an

Alexanderquandle$(M, a\triangleleft b=\alpha a+(1-\alpha)b)$,where$M$is

a

module

over a

ring$R$, and$\alpha$is

a

fixed invertible element from$R$

.

Onecalculates

($a$)$T_{b}T_{c}=(a\triangleleft b)\triangleleft c=a^{2}a+\alpha(1-\alpha)b+(1-\alpha)c.$

Our quandle is qualgebraizable only if $T_{b}T_{c}$ equals $T_{d}$

:

$a\mapsto\alpha a+(1-a)d$ for

some

$d\in M$

.

_{Butthis would imply that the value of}

a

$a-\alpha a$doesnotdepend

on

$a$

.

Since$a$is

invertible, the value of$\alpha a-a$isalso

a

constant, and thus$a\triangleleft b=\alpha a+b-ab=a$

.

_One

concludes that

among

Alexander quandles, only the trivial

ones are

qualgebraizable.

$\bullet$

There

are

3 quandlesofsize 3:

-Thetrivial

one was

showntoadmit 4 qualgebraizations.

-The Alexander quandle $(\mathbb{Z}/3\mathbb{Z}, a\triangleleft b=2b-a)$(thecolorings bywhich

are

precisely

thefamousFoxcolorings)

was

provednot tobequalgebraizable.

-Thesub-quandle$Q’=\{p, q, r\}$ofthequandle$Q$fromSection2.2 satisfies

necessary

algebraizability condition 3above,since $T(Q’)$_is

a

_{2-element subgroup}(hence

sub-qualgebra) of$Aut(Q’)$

.

However, $Q’$ isnot qualgebraizable. Indeed, according to property4 above, element $r*r$ should befixed by$T_{\Gamma/}$ implying

$r*r=r$ ; butthis contradictsproperty2,whichgives $T_{r*r}=T_{r}T_{r}=Id\neq T_{r}.$

$\bullet$

Thequandle$Q$from Section2.2admits9 qualgebraizations

(upto isomorphism).Note

that above

we

showed itssub-quandle$tp,$$q,$$r$} not tobe qualgebraizable.

$\bullet$

The

group

qualgebra$QA(S_{n})$is

a

qualgebraization ofthe_conjugation

quandleConj$(S_{n})$

.

This qualgebraization isuniquefor$n\geq 3$, since_{in this}

case

_the

map

$T$is injective.

3

Variations

ofqualgebra

ideas

3.1 Towards qualgebra

_cohomology

Fix

a

qualgebra $(S,\triangleleft, *)$

.

InSection 1.2,

we

$sa\mathcal{W}$ that

any

Reidemeister

move

induces

a

bi-jection betweenthe sets$Cot_{S}(D)$ and$Cot_{S}(D’)$ of$(S, \triangleleft, *)$-coloringsof the two

well-oriented

3-graphdiagrams involved.Theconclusion

was

that thecardinality$\# Cot_{S}(D)$of such

a

_setis

a

3-graphinvariant. However,

a

lot ofinformation_{is lost when passing from the coloringset}

toits cardinality.Here

we

show how toretrieve

some

ofit,imitatingwhat

was

done for

quan-dle coloringsof knots by

_{Carter-Jelsovsky-Kamada-Langford-Saito}

_{in 1999 (cf. the original}

papers

[1, 2],

a

_{verypedagogic}

_survey

_{[16], and}

_numerous

_{relatedpublications).}

Thebasicidea istoassociate to

every

$S$-coloring$\mathscr{C}$of

a

diagram $D$

a

quantity invariant

underReidemeister

moves.

Developingthe approach of[1],

we

_{look for}

_suantities

_of

_a

par-ticular form. Taie two

maps

$\chi,\lambda:SxSarrow \mathbb{Z}$, evaluate them

on

all the crossings and trivalent

verticesof$D$colored according_to$\mathscr{C}$,

as

shown

on

Figure8,_and

_sum

_up

_{the valuesobtained.}

The result is called the$(\chi,\lambda)$-weight of$\mathscr{C}$,

denotedby$\omega_{\chi,\lambda}(\mathscr{C})$

.

a

$b$ $b$ $a\triangleleft b$

$a*b$

$\swarrow_{a\triangleleft}^{\backslash _{\searrow}}\mapsto\chi(a, b)b a\searrow^{\mapsto-\chi(a,b)} aA\mapsto\lambda(a, b) a_{Y_{b}^{b}\mapsto-\lambda(a,b) ,a*}$

Figure 8: Qualgebra 2-cocycle $\sim$ weight

The

invariance

oftheweight$\omega_{\chi,\lambda}(\mathscr{C})$ underReidemeister

moves

is equivalent

tothe

fol-lowing relationsfor$\chi$and$\lambda$

:

RIV – _{$\chi(a, b*c)=\chi(a, b)+\chi(a\triangleleft b,c)$,}

(4)

RVI

_$-$

$\chi(a*b,c)+\lambda(a\triangleleft c, b\triangleleft c)=\chi(a,c)+\chi(b, c)+\lambda(a, b)$_{, (5)}

RV

_$-$

$\chi(a, b)+\lambda(a, b)=\lambda(b, a\triangleleft b)$

.

₍₆₎

Therelationsfor theremaining

moves

follow fromthepresented

ones

and

are

omitted.

Apair of

maps

$\chi,\lambda:S\cross Sarrow \mathbb{Z}satisfi^{r}ing(4)-(6)$ is_called

a

qualgebra 2-cocycle for S. As

shown above,_{for such}

_a

_{pair themulti-set ofweights} $\{\omega_{\chi,\lambda}(^{\zeta}\mathscr{E})|\mathscr{C}\in Cot_{S}(D)\}$ defines

an

in-variant of the_{underlying well-oriented3-graph}$\Gamma_{D}$:

The

same

qualgebrathusgivesriseto

a

wholefamily ofso-called cocycle invariants.In

partic-ular,

_{one recovers}

_{the qualgebrainvariantsfrom Section 1.2 when taking}

_zero

_maps

$\chi$and $\lambda.$

Theterm qualgebra 2-cocycle”

was

chosento stresstheanalogy with quandle 2-cocycles

from [1], which

are

indeed 2-cocycles for the celebratedquandle cohomologytheory. Asfor

now,

_no

_{qualgebra cohomology}theoryis known. Topological arguments suggest what it

should look like in small degrees, but its continuation tohigher degrees remains

mysteri-ous.

Thegeneral braidedcohomologytheory from[21] _yields

a

_{cohomology theory for} rigid

qualgebras (withaxiom $(QA_{Comm})$ omitted from thedefinition); topologically, these

corre-spondtorigid 3-graphs(for_{which graph vertices}

are

viewed

as

disks,not

as

points, excluding Reidemeister

move

V). However, this approach doesnotwork for general qualgebras.

Let

us

describe

some

properties of 2-cocycles for

our

qualgebra$S$

.

Theyform

an

Abelian

group

$Z^{2}(S)$ under point-wise coordinate-wise addition. A subgroup $B^{2}(S)$ is formed by

qualgebra2 $coboundaries-$thatis,2-cocycles builtoutof

maps

$\psi:Sarrow \mathbb{Z}$

as

follows:

$\chi(a, b)=\phi(a\triangleleft b)-\phi(a)$,

$\lambda(a, b)=\phi(a)+\phi(bI-\phi(a*b)$

.

Such 2-cocycles

are

useless for distinguishinggraphs,giving

zero

weights only The quotient

$H^{2}(S)=Z^{2}(S)lB^{2}(S)$is

a

natural candidate for the title degree 2 cohomology of$S.$

Inordertoshow that the definitions from thissection

are

not empty,

we

present

compu-tations for the 4-elementqualgebrasfrom Section2.2. All the9 qualgebras describedthere

exhibit the

same

homological behavior. Namely, they satisfy

$Z^{2}(Q)\cong \mathbb{Z}^{8}, B^{2}(Q)\cong \mathbb{Z}^{4}, H^{2}(Q)\cong \mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}^{4}.$ The torsionappearing in the quotient is particularly interesting.

We finish with two directionscontinuingthe color-and-weight”ideas.

1. Together with diagramarcs,

one

can

color diagram regions with elements of

our

qualge-bra(orof

a

more

generalqualgebramodule). The philosophy of weights then naturally leadsto

a

notion of qualgebra3-cocycles,andto

a

generalizations of shadowcocycle

in-$\iota/$ariants,constructed in the

case

of quandles in[17, 3].

2. The evaluation rules for trivalentverticesfrom Figure 8

are

the simplest

ones

making things work. One

can

add

a

third

map

V: $SxSarrow \mathbb{Z}$tothe initialdata,

use

it for

evalu-ations

on

zipvertices, andwrite down the compatibility conditions forV, $\lambda$ and

$\chi$

im-posed by Reidemeister

moves.

This could leadto

a

richer family of3-graph invariants.

3.2 Weak qualgebras and branched braids

Manycombinatorial knot invariants directly generalizetolinks, braids,tangles and other

1-dimensional topological objects. In the

case

ofbraids

one can

often obtain

even

stronger

results,since

some

flexibility is gained byexcludingReidemeister

move

Ifrom the story. For

example, when extending quandleinvariantstobraids,

one

gets twoenhancements for free:

1.

a

weakerstructurecalled rack$(=$data($S, \triangleleft)$satisfying$(Q_{SD})-(Q_{Inv})$ only)

can serve as a

coloringset;

2. the $S$-colors of the $n$

upper

arcs

of

a

braid $\beta$with $n$ strands uniquely determine the

colors of all remainingarcs, _{in particular of the} $n$ lower arcs; this defines

a

map $B_{\beta}$

:

In the oppositedirection,Alexanderand Markovtheoremspresentknots

as

certain equiva-lenceclassesofbraids,viatheclosureoperation. Hencebraid invariantsprovide

a

potential

source

of knotinvariants. Inthis section

we

introduce

a

topological notionwhichplays for 3-graphs the

same

role

as

braidsplay forknots, and present

a

weak version of qualgebras

sufficientfor_{producing invariantsof these}

_new

_objects.

The closure

map

for braids is recalled

on

Figure 9. Alexander theoremassertsits

surjecti-vityby presenting

every

link

as

the closure of

some

braid. Markov theorem describes its

kernel byshowing that

any

twobraids with isotopicclosures

are

connected by

a

finite

se-quence of Reidemeister

moves

II-III and Markov

moves

1-2 (see Figure 10; thick lines here

replace

an

_{arbitrary number of}strands).

Figure9:Braid$\sim$ link

Figure 10:Markovmoves

When studying 3-graphs, braids should be replaced with branched braids. These

are

knotted graphs in$\mathbb{R}^{2}\cross[0$

,1$]$with$n$univalentvertices

on

thetop, _$m$univalent

vertices

on

the

bottom,

some

trivalentverticesin between, and

no

cups or caps

(with _{respect to the third}

coordinate projection$\mathbb{R}^{2}\cross[0, 1]arrow[0$,1 The closure operation is still

definedforbranched

braids with$n=m$,

as

shown

on

Figure11.

Figure11: Branched braid$\sim 3$-graph

K. Kanno and K. Taniyama ([18]) _{proved that a113-graphs}

_are

_{obtained this way, giving}

an

_{Alexander-type theoremforbranched}braids;

see

also [25] _for

_a

_{related result for}

theta-curves.

AMarkov-type theoremforbranchedbraids

was

established byS.Kamada and the author([15]):

_we

_showed

_any

_{twobranchedbraids withisotopicclosurestobe connected by}

a

finite

sequence

ofReidemeister

moves

II-VI and Markov

moves

1-2. This resultgeneralizes

Table3: Alexander-and Markov-typetheorems in differentsettings

Onthe level ofinvariants, the twotheoremsimplythat

a

branched braid invariantstable under Markov

moves

automatically gives riseto

a

3-graphinvariant.

In the opposite direction, qualgebra colorings work well for branched braids. Among

thetwoenhancements mentioned above for quandle colorings ofbraids, only the first

one

adapts to this setting. Indeed,

a

weak qualgebra $(=$ data ($S, \triangleleft, *)$ satisfying $(Q_{SD})-(Q_{Inv})$

and$(QA_{Comp})-(QA_{Comm})$ only)

can

serve as

a

coloringsetfor branchedbraid diagrams:

However, contrarytothe

case

of usualbraids,here

upper

colors donotdetermine lower colors$*$

because of unzipvertices: the knowledge of$a*b$_{does not give}

you

$a$and$b$

.

Hence

one

hasto contentoneselfwith counting(weak) _{qualgebra colorings, possibly with weights.}

3.3 Qualgebras in SetTheory

Besidesthetopological and algebraic settingsdescribedabove,_axioms$(QA_{Comp})-(QA_{Comm})$

also

emerge

in

a

completelydifferent set-theoretical context. Namely, together with the

as-sociativity of $*$ and the existence of

a

neutral element 1 $for*satisfi^{r}ing$

moreover

$1\triangleleft a=1$

and$a\triangleleft 1=a$forall $a$, they define$a(right-)$distributivemonoid(or,in othersources, RD

al-gebra). Examplesincludeelementaryembeddings,Lavertables,and extended braids. Allof

them admitrich distributivemonoidstructures, motivating

an

extensive study of the

con-cept (see _forinstance [4, 9, 10, 5],

or

_{Chapter XI of}[6] _for

a

comprehensive exposition). $A$

weaker augmented (right-)distributive system structure ofP Dehornoy obeys only axioms $(Q_{SD})$, _{$(QA_{Comp})$},and$(QA_{D})$

.

Themajor example here is that of parenthesized braids([7,8

Ourqualgebras

are

particular

cases

of augmented distributive systems.

Acknowledgements

Theauthor is gratefultoSeiichiKamada,Patrick Dehornoy and Atsushi Ishii for stimulating

discussions,andtothe researchers andsecretariesof OCAMI andRIMSfortheir hospitality The author

was

supported by

a

ISPS

Postdoctral FellowshipForForeign Researchers and by

JSPS

KAKENHIGrant$25\cdot 03315.$

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