i A is a finite quiver Q that embeds in dimemodelw.boundanI a - - PDF document

121812020

Dimemodelswithboundarydrassmanniandustercategoriesffriet

i

Einasiumaniaigmes

A dimemodelw.boundanI

is

a finite quiver Q that embeds

in

a

surface S

s t

each connected

component of S Q

is simply connectedand

bounded by

an oriented

cycle

These cycles are

the unicycles

Arrows of

Q

are internal

if they are

contained

in

2 faces bounday

if

they

belong to oneface only

• r

The vertices

incident

w boundary

µ

f

f

• arrows

are

boundaryvertices

If

e

Example

Quivers arising from triangulations of disks ofsurfaces

Verdes fr diag's boundary

segments

arrows for rotations

filings of surfaces

insideD's tiles

alternating strand diagrams Postnikovl

• n disk
• rtpYghna.cn

y

vertices.gg

q

ex

k.nl diagrams

• n disk

Scott

degeneration ofthem arrowsfrom

a

Fent ofstrands

j

• r
• triangulations of disks ofsurfaces

n

hextijaaimakadninius

It

filings of surfaces

I is

ScottBranin subsetof

Correspondences

• n Pn

trianguisltigings

I

divterannadiMaiagrams

ftp.lapadmeut

maybe

graphs

zagpaths

toffee

road

P

eta

plabicgraphs

dEoiwqqk4 reduced

nous

amies

EA

I

akin

tiling

7

Images

under

and

are

connected

fully

reduced

them The plabic graphs arising from flings of Pat have

A black node

n white nodes on

bdy other

nodes

are blade

deg a 23

t

every closed face is

a quadrilateral

adf.edpfffateaawwqteuamddfatoao.ie

Alton

Strand

diagram

r

E

Sh

permutation

at

asana grown

permutations

is 5 1918

dn2

appendixby M Glick in BMartin 18

arXiv 1601.05080

Algebasfrandim

emodels

Q dime model

w boundary

Wa

E

i S

u

the

unit

cycles of Q

rest t

5,8

WECzt

Cut

Cg

iii

r

E

t

For QQ

basis given by all paths in Q

Trivialpaths b

re Q vertex

multiplication by concatenation of paths

idempotent

from

trivial

A a

clime algebra

• f Q

f eat ten

paths atbdy

vertices

Ba

i

e Aae

the

boundary

algebra

• f Q

i

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