2012/11/26 by Sergei Gukov, Ingmar Saberi · 17 citations
Mathematics · Medicine · Physics and Astronomy · #Biology #Botulinum Toxin and Related Neurological Disorders #Conjecture #Floer homology #Gene #Genetics #Geometric and Algebraic Topology #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Jones polynomial #Khovanov homology #Knot (papermaking) #Knot invariant #Knot polynomial #Knot theory #Mathematics #Pure mathematics #Symplectic geometry #hep-th #math.AG #math.GT #math.QA
paper · pdf · doi:10.1090/conm/613/12235
published in Contemporary mathematics - American Mathematical Society, 41-78 (American Mathematical Society) · 48 pages, 9 figures. Comments are welcome
arxiv created 2012/11/26 · openalex publication_date 2014/01/01 · arxiv updated 2016/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, it allows one to answer questions like <italic> Is there a direct relation between Khovanov homology and the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -polynomial of a knot? </italic> which would not have been asked otherwise. We will explain that the answer to this question is “yes” and introduce a certain deformation of the planar algebraic curve defined by the zero locus of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -polynomial. This novel deformation leads to a categorified version of the Generalized Volume Conjecture that completely describes the “color behavior” of the colored <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German s German l left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak sl(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> knot homology and, eventually, to a similar conjecture for the colored HOMFLY homology. Furthermore, this deformation is strong enough to distinguish mutants, and its most interesting properties include relations to knot contact homology and knot Floer homology.