55 research outputs found
On stable sl3-homology of torus knots
The stable Khovanov-Rozansky homology of torus knots has been conjecturally
described as the Koszul homology of an explicit non-regular sequence of
polynomials. We verify this conjecture against newly available computational
data for sl(3)-homology. Special attention is paid to torsion. In addition,
explicit conjectural formulae are given for the sl(N)-homology of (3,m)-torus
knots for all N and m, which are motivated by higher categorified Jones-Wenzl
projectors. Structurally similar formulae are proven for Heegard-Floer
homology.Comment: 16 pages, 3 table
Cable links and L-space surgeries
An L-space link is a link in on which all sufficiently large integral
surgeries are L-spaces. We prove that for m, n relatively prime, the
r-component cable link is an L-space link if and only if K is an
L-space knot and . We also compute HFL-minus and HFL-hat of
an L-space cable link in terms of its Alexander polynomial. As an application,
we confirm a conjecture of Licata regarding the structure of HFL-hat for (n,n)
torus links.Comment: 27 pages, 6 figures, 4 tables; v2: Resolved m=1 case in Theorem 1;
minor revisions throughout. This is the version to appear in Quantum Topolog
Evaluations of annular Khovanov--Rozansky homology
We describe the universal target of annular Khovanov-Rozansky link homology
functors as the homotopy category of a free symmetric monoidal category
generated by one object and one endomorphism. This categorifies the ring of
symmetric functions and admits categorical analogues of plethystic
transformations, which we use to characterize the annular invariants of Coxeter
braids. Further, we prove the existence of symmetric group actions on the
Khovanov-Rozansky invariants of cabled tangles and we introduce spectral
sequences that aid in computing the homologies of generalized Hopf links.
Finally, we conjecture a characterization of the horizontal traces of Rouquier
complexes of Coxeter braids in other types.Comment: 41 page
Rational Parking Functions and LLT Polynomials
We prove that the combinatorial side of the "Rational Shuffle Conjecture"
provides a Schur-positive symmetric polynomial. Furthermore, we prove that the
contribution of a given rational Dyck path can be computed as a certain skew
LLT polynomial, thus generalizing the result of Haglund, Haiman, Loehr, Remmel
and Ulyanov. The corresponding skew diagram is described explicitly in terms of
a certain (m,n)-core.Comment: 14 pages, 8 figure
On the set of L-space surgeries for links
It it known that the set of L-space surgeries on a nontrivial L-space knot is
always bounded from below. However, already for two-component torus links the
set of L-space surgeries might be unbounded from below. For algebraic
two-component links we provide three complete characterizations for the
boundedness from below: one in terms of the -function, one in terms of the
Alexander polynomial, and one in terms of the embedded resolution graph. They
show that the set of L-space surgeries is bounded from below for most algebraic
links. In fact, the used property of the -function is a sufficient condition
for non-algebraic L-space links as well.Comment: 28 pages, 13 figures; v2: Major revision, we prove a complete
characterization of algebraic links with bounded below sets of L-space
surgerie
Hilbert schemes and -ification of Khovanov-Rozansky homology
We define a deformation of the triply graded Khovanov-Rozansky homology of a
link depending on a choice of parameters for each component of ,
which satisfies link-splitting properties similar to the Batson-Seed invariant.
Keeping the as formal variables yields a link homology valued in triply
graded modules over . We conjecture that
this invariant restores the missing symmetry of the
triply graded Khovanov-Rozansky homology, and in addition satisfies a number of
predictions coming from a conjectural connection with Hilbert schemes of points
in the plane. We compute this invariant for all positive powers of the full
twist and match it to the family of ideals appearing in Haiman's description of
the isospectral Hilbert scheme
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