82 research outputs found
On an alternative stratification of knots
We introduce an alternative stratification of knots: by the size of lattice
on which a knot can be first met. Using this classification, we find ratio of
unknots and knots with more than 10 minimal crossings inside different lattices
and answer the question which knots can be realized inside and
lattices. In accordance with previous research, the ratio of
unknots decreases exponentially with the growth of the lattice size. Our
computational results are approved with theoretical estimates for amounts of
knots with fixed crossing number lying inside lattices of given size.Comment: 12 page
Commutative subalgebras from Serre relations
We demonstrate that commutativity of numerous one-dimensional subalgebras in
algebra, i.e. the existence of many non-trivial integrable
systems described in recent arXiv:2303.05273 follows from the subset of
relations in algebra known as Serre relations. No other relations are needed
for commutativity. The Serre relations survive the deformation to the affine
Yangian , hence the commutative subalgebras do as
well. A special case of the Yangian parameters corresponds to the
-deformation. The preservation of Serre relations can be thought of a
selection rule for proper systems of commuting -deformed Hamiltonians.
On the contrary, commutativity in the extended family associated with
``rational (non-integer) rays" is {\it not} reduced to the Serre relations, and
uses also other relations in the algebra. Thus their
-deformation is less straightforward.Comment: 13 page
Topological recursion for monotone orbifold Hurwitz numbers: a proof of the Do-Karev conjecture
We prove the conjecture of Do and Karev that the monotone orbifold Hurwitz numbers satisfy the Chekhov-Eynard-Orantin topological recursion
Commutative families in , integrable many-body systems and hypergeometric -functions
We explain that the set of new integrable systems generalizing the Calogero
family and implied by the study of WLZZ models, which was described in
arXiv:2303.05273, is only the tip of the iceberg. We provide its wide
generalization and explain that it is related to commutative subalgebras
(Hamiltonians) of the algebra. We construct many such
subalgebras and explain how they look in various representations. We start from
the even simpler contraction, then proceed to the one-body
representation in terms of differential operators on a circle, further
generalizing to matrices and in their eigenvalues, in finally to the bosonic
representation in terms of time-variables. Moreover, we explain that some of
the subalgebras survive the -deformation, an intermediate step from
to the affine Yangian. The very explicit formulas for the
corresponding Hamiltonians in these cases are provided. Integrable many-body
systems generalizing the rational Calogero model arise in the representation in
terms of eigenvalues. Each element of algebra gives rise to
KP/Toda -functions. The hidden symmetry given by the families of
commuting Hamiltonians is in charge of the special, (skew) hypergeometric
-functions among these.Comment: 43 page
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