475 research outputs found
HOMFLY-PT polynomial and normal rulings of Legendrian solid torus links
We show that for any Legendrian link in the -jet space of the
-graded ruling polynomial, , is determined by the
Thurston-Bennequin number and the HOMFLY-PT polynomial. Specifically, we
recover as a coefficient of a particular specialization of the
HOMFLY-PT polynomial. Furthermore, we show that this specialization may be
interpreted as the standard inner product on the algebra of symmetric functions
that is often identified with a certain subalgebra of the HOMFLY-PT skein
module of the solid torus.
In contrast to the -graded case, we are able to use -graded ruling
polynomials to distinguish many homotopically non-trivial Legendrian links with
identical classical invariants.Comment: 30 pages, 9 figure
The Bennequin number, Kauffman polynomial, and ruling invariants of a Legendrian link: the Fuchs conjecture and beyond
We show that the ungraded ruling invariants of a Legendrian link can be
realized as certain coefficients of the Kauffman polynomial which are
non-vanishing if and only if the upper bound for the Bennequin number given by
the Kauffman polynomial is sharp. This resolves positively a conjecture of
Fuchs. Using similar methods a result involving the upper bound given by the
HOMFLY polynomial and 2-graded rulings is proved.Comment: 17 pages, 9 figure
Equivalence classes of augmentations and Morse complex sequences of Legendrian knots
Let L be a Legendrian knot in R^3 with the standard contact structure. In
[10], a map was constructed from equivalence classes of Morse complex sequences
for L, which are combinatorial objects motivated by generating families, to
homotopy classes of augmentations of the Legendrian contact homology algebra of
L. Moreover, this map was shown to be a surjection. We show that this
correspondence is, in fact, a bijection. As a corollary, homotopic
augmentations determine the same graded normal ruling of L and have isomorphic
linearized contact homology groups. A second corollary states that the count of
equivalence classes of Morse complex sequences of a Legendrian knot is a
Legendrian isotopy invariant.Comment: 28 pages, 17 figure
2007 Founder\u27s Day Correspondence
Letters of correspondence from Dan Rutherford to Barbara Todd about the 2007 Founder\u27s Day.https://ir.library.illinoisstate.edu/founding/1203/thumbnail.jp
The cardinality of the augmentation category of a Legendrian link
We introduce a notion of cardinality for the augmentation category associated
to a Legendrian knot or link in standard contact R^3. This `homotopy
cardinality' is an invariant of the category and allows for a weighted count of
augmentations, which we prove to be determined by the ruling polynomial of the
link. We present an application to the augmentation category of doubly
Lagrangian slice knots.Comment: 15 page
- …