Japanese version
Recently, Professor Lieven le Bryun is describing about Arithmetic Toplogy on his BLOG, progressing now. The articles I'm interested in are listed as follows. I would like to translate them into Japanese with other topics.
I add the preprint "Remarks on the Alexander Polynomials" by B. Mazur. I translated into Japanese only the introduction of it.
An episode of arithmetic topology My original
Mumford's treasure map
Japanese version
Manin's geometric axis
Japanese version
Mazur's knotty dictionary
Japanese version
Gukov on Arithmetic Topology and Gauge Theory (extra)
Japanese version
Grothendieck's functor of points
Japanese version
What is the knot associated to a prime?
Japanese version
Remarks on the Alexander Polynomials by Barry Mazur
Who dreamed up the primes=knots analogy?
Japanese version
the birthday of the primes=knots analogy
Japanese version
Manin’s three-space-2000
Japanese version
Artin-Verdier duality came from Tate duality and generalizes it.
ReplyDeleteon en.wiki
In mathematics, Artin–Verdier duality is a duality theorem for constructible abelian sheaves over the spectrum of a ring of algebraic numbers, introduced by Artin and Verdier (1964), that generalizes Tate duality.
In mathematics, Tate duality or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field, introduced by Tate (1962) and Poitou (1967).
Local Tate duality says there is a perfect pairing of finite groups
H^r(k,M)\times H^{2-r}(k,M')\rightarrow H^2(k,G_m)=Q/Z
where M is a finite group scheme and M′ its dual Hom(M,G_m).
"Remarks on the Alexander Polynmials" by Barry Mazur in 1995 is very interesting preprint, which propose Arithmetic Topology and connect Alexander Polynomials with Iwasawa Theory. Firstly, I knew it by Lieven's BLOG.
ReplyDeleteI wonder why Alexander polynomials are quantum invariants. There is the same question on MathOverFlow.
ReplyDeleteWhy is the Alexander polynomial a quantum invariant?
When we think of quantum invariants, we usually think of the Jones polynomial or of the coloured HOMFLYPT. But (arguably) the simplest example of a quantum invariant of a knot or link is its Alexander polynomial. From the beginning, the central problem in the study of quantum invariants has been what do they mean topologically? The Alexander polynomial has clear algebraic topological meaning as the order of the Alexander module (first homology of the infinite cyclic cover as a module over the group of deck transformations). Can people conceptually explain (in terms of both physics and mathematics) why the representation theory of certain small quantum groups naturally gives rise to this quantity? Computationally I can understand it, but not conceptually.
A somewhat related question was already asked here.
as a related problem:
HOMFLY and homology; also superalgebras
My understanding is that an analogy along the following lines is (roughly) true:
"The Alexander polynomial is to knot Floer homology is to gl(1|1)
as the Jones polynomial is to Khovanov homology is to sl(2)
as a-lot-of-other-specializations-of-HOMFLY are to Khovanov-Rozansky homology are to sl(n)."
1) To what extent is it possible to add another line that starts with the (unspecialized) HOMFLY polynomial? I think there is a triply-graded complex that I can put here (and that maybe this is what I should be calling Khovanov-Rozansky homology? or at least is also due to them?), but is there an analogous object to put in place of the Lie (super-)algebras appearing above?
2) Why is gl(1|1) here? That seems weird.
Update: I posted on this question here and here. See also this question.
By the way, Khovanov Homology would unify the Alexander Polynomial and other quantum knot invariants.