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The sound of Knots and 3-manifoldsThe sound of Knots and 3The sound of Knots and 3--manifoldsmanifolds
Sergei GukovSergei Gukov
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George Salmon(1819-1904)
Arthur Cayley (1821-1895)
2,875 lines (degree-1) on a quintic 3-fold
27 lines on a general cubic surface
Sheldon Katz, 1986
609,250 conics (degree-2) on a quintic 3-fold
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317,206,375 degree-3 curves on a quintic 3-fold
“Suddenly, in late 1990, Candelas and his collaborators Xenia de la Ossa, Paul Green, and Linda Parkes (COGP) saw a way to use mirror symmetry to barge into the geometers’ garden....
Geir Ellingssrud and Stein Arild Stromme, May 1991
2,682,549,425 degree-3 curves on a quintic 3-fold
Candelas, who liked calculating things, thought maybe it was in fact tractable using some algebra and a home computer.”
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New Opportunities
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Knots
=?
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FIGURE EIGHT
4_1
HOMFLY-PT COLORED
ALEXANDER
JONES AMPHICHEIRAL
SL(3) GENUS
1
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TQFT ~ 30 yrs old
1
Why integer coefficients?
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TQFT in the 21st centuryTQFT in the 21st century
Darling,
I want a new TQFT
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Numbers
Vector spaces
Surfaces
4-manifolds
3-manifolds
Knots
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Numbers
Vector spaces
Surfaces
4-manifolds
3-manifolds
Knots
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Knot homology circa 2003
• Only a few homological knot invariants:– Khovanov homology (N=2, combinatorial)
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Knot homology circa 2003
• Only a few homological knot invariants:– Khovanov homology (N=2, combinatorial)
– Knot Floer homology (N=0, symplectic)
• Categorification of HOMFLY not expected
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Knot homology circa 2003
• Only a few homological knot invariants:– Khovanov homology (N=2, combinatorial)
– Knot Floer homology (N=0, symplectic)
• Categorification of HOMFLY not expected
• Higher rank not computable
• No SO/Sp groups
• No colors
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Knot homology circa 2003
• Only a few homological knot invariants:– Khovanov homology (N=2, combinatorial)
– Knot Floer homology (N=0, symplectic)
• Categorification of HOMFLY not expected
• Higher rank not computable
• No SO/Sp groups
• No colors
• Many unexplained patterns
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“Connecting the dots”
=BPS spectrum(Q-cohomology)
Knot Homology(Khovanov,…)
BPS
[S.G., A.Schwarz, C.Vafa, 2004]
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Fivebrane Setup
BPS BPSdoubly graded triply graded
fixed N packages all ranks
duality
[H.Ooguri, C.Vafa, 1999]
(+ flux)
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3d-3d interpretation
BPS
3d N = 2 theory
(time) x x
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duality
New Bridges
Enumerative invariants of Calabi-Yau 3-folds and combinatoricsof skew3-dimensionalpartitions
Refined Chern-Simons,DAHA, …
[M.Aganagic, S.Shakirov, 2011][I.Cherednik, 2011]
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LG interfaces and KLR algebras
“brane intersection” perspective on (time) x (knot):2d Landau-Ginzburg models & interfaces
D.Roggenkamp J.Walcher
(time) x x
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Reduction to gauge theory
xx
-1
[S.G., 2007]
[P.Kronheimer, T.Mrowka, 2008]
Singly-graded, familiar PDE’s, unknot-detector
Singly-graded, familiar PDE’s,new wall crossing phenomena
[E.Witten, 2011]
Doubly-graded,Haydys-Witten PDE’s in five dimensions
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• HOMFLY homology & new bridges:
What physics gives us?
knot contacthomology
HKRN
KhA-polynomial
HFK
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• HOMFLY homology & new bridges:
• New structural properties (differentials, recursion relations, …)
What physics gives us?
knot contacthomology
HKRN
KhA-polynomial
HFK
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• HOMFLY homology & new bridges:
• New structural properties (differentials, recursion relations, …)
• New computational techniques
What physics gives us?
knot contacthomology
HKRN
KhA-polynomial
HFK
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All knots up to 10 crossings
in one day!
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P = q2+ q5 - q7+ q 8 - q9- q10 + q 11
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q4N+3-q3N+4-q3N+3-q3N+1-q3N+q2N+4+q2N+2+q2N+1+q2N-2P =
sl(N)sl(2)
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-1-2-3
4
0
2
3
321 4
q
a
q4N+3-q3N+4-q3N+3-q3N+1-q3N+q2N+4+q2N+2+q2N+1+q2N-2P =
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Differentials
-1-2-3
4
0
2
3
321 4
q
a
deg(a,q,t) = (-1,-2,-3)
deg(a,q,t) = (-1,1,-1)
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-1-2-3
4
0
2
3
321 4
q
a
q4N+3-q3N+4-q3N+3-q3N+1-q3N+q2N+4+q2N+2+q2N+1+q2N-2P =
deg(a,q,t) = (-1,1,-1)
Differentials
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-1-2-3
4
0
2
3
321 4
q
a
q4N+3-q3N+4-q3N+3-q3N+1-q3N+q2N+4+q2N+2+q2N+1+q2N-2P =
deg(a,q,t) = (-1,-2,-3)
Differentials
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-1-2-3
4
0
2
3
321 4
q
a
0
3
q4N+3-q3N+4-q3N+3-q3N+1-q3N+q2N+4+q2N+2+q2N+1+q2N-2P =
deg(a,q,t) = (-1,-2,-3)
Differentials
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-1-2-3
4
0
2
3
321 4
q
a
0
3
2
q4N+3-q3N+4-q3N+3-q3N+1-q3N+q2N+4+q2N+2+q2N+1+q2N-2P =
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-1-2-3
4
0
2
3
321 4
q
a
0
3
2
5
q4N+3-q3N+4-q3N+3-q3N+1-q3N+q2N+4+q2N+2+q2N+1+q2N-2P =
![Page 37: The sound of Knots and 3The sound of Knots and 3 ...lh33/workshop/Gukov.pdf · The sound of Knots and 3The sound of Knots and 3---manifoldsmanifolds Sergei Gukov](https://reader031.vdocuments.us/reader031/viewer/2022020216/5c60cbd709d3f20a6c8c1c13/html5/thumbnails/37.jpg)
-1-2-3
4
0
2
3
321 4
q
a
0
3
2
5
4
q4N+3-q3N+4-q3N+3-q3N+1-q3N+q2N+4+q2N+2+q2N+1+q2N-2P =
![Page 38: The sound of Knots and 3The sound of Knots and 3 ...lh33/workshop/Gukov.pdf · The sound of Knots and 3The sound of Knots and 3---manifoldsmanifolds Sergei Gukov](https://reader031.vdocuments.us/reader031/viewer/2022020216/5c60cbd709d3f20a6c8c1c13/html5/thumbnails/38.jpg)
-1-2-3
4
0
2
3
321 4
q
a
0
3 3
2 2
6
5 5
4
q4N+3-q3N+4-q3N+3-q3N+1-q3N+q2N+4+q2N+2+q2N+1+q2N-2P =
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Numbers
Vector spaces
Surfaces
4-manifolds
3-manifolds
Knots
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3-manifold homology in 2015
“monopole Floer homology”
based on Seiberg-Witten equations
“embedded contact homology”
homology version of SW=Gr
“Heegaard Floer homology”
(“Atiyah-Floer conjecture”)
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N10-1 2 3-2-3 4
? ?
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WRT( )
Knot Homology
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3-manifold q-series
Homology
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3-manifold
labeled byAbelian flatconnection on
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DW = SW(a)Sa
basic classes
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reducibleirreducible
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Emile Borel(1871-1956)
Resurgence
cut
pole pole
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Emile Borel(1871-1956)
Borel plane
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Emile Borel(1871-1956)
Borel plane
1) position of singularities 2) “residues”
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position of singularities:
Theorem [G-Marino-Putrov]:
Corollary: WRT( ) =
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From knots to 3-manifolds
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“Laplace transform”cf.A.Beliakova,C.Blanchet,T.T.Q.Le
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A (x,y)super ^ ^^
F (unknot;x) = 0
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Numbers
Vector spaces
Surfaces
4-manifolds
3-manifolds
Knots
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A-model��(1)
LG model W = x + x3
LG model W = x3
LG modelx
W = e + e-x
dim ( H ) = 2
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A-model��(1)
LG model W = x + x3
LG model W = x3
LG modelx
W = e + e-x
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Numbers
Vector spaces
Surfaces
4-manifolds
3-manifolds
Knots
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Unification of different theories
N10-1 2 3-2-3 4
also
Khovanov-Rozansky
Khovanov
trivial
trivial
HFKKhovanov-Rozansky
In categorification of quantum group invariants of knots,
HFK is an oddball … Will play an important role in
categorification of 3-manifold invariants.