On the Lagrangian fillability of almost positive links

In this paper, we prove that a link which has an almost positive diagram with a certain condition is Lagrangian fillable.Comment: 14 pages, 10 figures, 1 tabl

The maximal degree of the Khovanov homology of a cable link

In this paper, we study the Khovanov homology of cable links. We first estimate the maximal homological degree term of the Khovanov homology of the ($2k+1$, $(2k+1)n$)-torus link and give a lower bound of its homological thickness. Specifically, we show that the homological thickness of the ($2k+1$, $(2k+1)n$)-torus link is greater than or equal to $k^{2}n+2$. Next, we study the maximal homological degree of the Khovanov homology of the ($p$, $pn$)-cabling of any knot with sufficiently large $n$. Furthermore, we compute the maximal homological degree term of the Khovanov homology of such a link with even $p$. As an application we compute the Khovanov homology and the Rasmussen invariant of a twisted Whitehead double of any knot with sufficiently many twists.Comment: 40 pages, 26 figures, I shorten some proof

On Euclidean designs and the potential energy

We study Euclidean designs from the viewpoint of the potential energy. For a finite set in Euclidean space, We formulate a linear programming bound for the potential energy by applying harmonic analysis on a sphere. We also introduce the concept of strong Euclidean designs from the viewpoint of the linear programming bound, and we give a Fisher type inequality for strong Euclidean designs. A finite set on Euclidean space is called a Euclidean a-code if any distinct two points in the set are separated at least by a. As a corollary of the linear programming bound, we give a method to determine an upper bound on the cardinalities of Euclidean a-codes on concentric spheres of given radii. Similarly we also give a method to determine a lower bound on the cardinalities of Euclidean t-designs as an analogue of the linear programming bound.Comment: 14 page