An algorithm for

comparing Legendarium knotsI. Dynnilwv

Jie-

K.M

.Prasolov and V

. Shastri - n

Legendrian = tangent to the

standard contact structure

{E Ker (xdytdz)

classification of classification ofhlgendrian links

I non . simjlifi ablerectangular ( aka grid)diagrams of links

Front projections

1) no § 2) overrun der acc .

to slope :

I

2-

ran

.

Reidemister moves

I ⇒ -←I

←I ¥

⇒ ←I

← IF

Sta bi l i za t i ons

- →L

-→#

equivalently .

i

> → →

< →⇐

classical invariants

Thurston - Bennetfuin number tb

¥ > s+L -I -11 - L - Yz -Yz

Rotation ( Maslow ) number r

t s t 7+ k + K -Yz - Yar

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i an atlas

I. D.,M.P

.

2017 :

x

I D.,M.P

.2016,2017 ; I. D. , V. Sh .

2018

F an annulus A c 5,3 si.

A- is tangent to Bsf along OA ,but

OA = Lsu La, Lax Lz as Lege adrian boots

disproving a claim in arXiv : 0909.9326

Rectangular diagram of a link

•

>• •-•

• • ⑧-•

• • •- - -•

. .

.

.

..

. ,¥tHyl• ⑥ •- a

• • •-I

• • / •- -•

• • •-•

>

R I

orientation

Oooo mo

t t0-700

BAD O

Moves

ex:*:p : .IE. T.IE."÷ Is;

stabilizationatios :

Type I TypeI- -Bo O

• • ! •• • •

• •O ⑧, • O

O O O ••O o

• O

⑧ O O Oo •

O O

←→

→ ⇐subtypes : I I II I

For links,

' subtype of a stabilization'

also includes a choice of a connected

component

stabilization of the same subtypegive the same result up to

exchange moves

Rectangular diagram → two Legendarium links

- LI CR )'

÷..

a. mirror

Exchange moves preserve LI and LI .

Type I stabilization preserve LI andstabilize LII⇐I

Moves → morphisms

X ,↳ Xz as a tomeo ( 83 Xp ) → k$3

,X. )

preserving'

orientations,

viewed up to isotopy .

Symmetry group Sym CE) = {morphisms Kirk}

Main results

①"commutation of type I and type II moves :

R.EEK

,⇒ think , EEE Ra

,

same morphismonly

② ↳ CRY =L# ( Rz ) , R¥E¢ Rz =3

only Iany self . Re Rz -712

,induces

a nontrivial elem - of Sym ( RT )

③ F an algorithm for constructinga generating set of Sym (K )

The algorithmQ : LIIR ,

) -- LI ( Rz ) ?-

① check whether R .and Ra rep r .

the same Topol . type .

② find R±EEI, r.fr#iIp ,③ compute a gen -

set for Sym ( RT )91 , . . .

- sgm

① for each stab. subtype t of type I , let

left)=h¥q?,m # of f - stab .

in a realization efgi)and apply k Cf ) stabilization,

of subtype t to R,

'and ke

ki -112 ,

"

,Ra -712.

'

only

⑤ Check whether R,

"m5R,

'

-

Tools i can rectangular calculus forsurfaces in $3

ie) Giroux convex surfaces