Legendrian = tangent to the
standard contact structure
{E Ker (xdytdz)
classification of classification ofhlgendrian links
I non . simjlifi ablerectangular ( aka grid)diagrams of links
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
Yu.
Cheka nor 199 7- i
Myx
Ya.
Eliashbery ,
D. Fuchs,
L. Ng ,
P.Pushkar '
,
P.Ozsvaith
,
Z.
Szabo'
,D. Thurston
,
14. Fraser
,
Y. Etnyre ,
K.
Honda,D. Lafountain
,
B. To sun,
V. Vertesi
D.Benne quin ,
E.
Giroux
W. Chong chit male ,
L. Ng .
i an atlas
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
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
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,