Related papers: Slopes of links and signature formulas
We prove a "splicing formula" for the LMO invariant, which is the universal finite-type invariant of rational homology $3$-spheres. Specifically, if a rational homology $3$-sphere $M$ is obtained by gluing the exteriors of two framed knots…
We construct analogs of Khovanov-Jacobsson classes and the Rasmussen invariant for links in the boundary of any smooth oriented 4-manifold. The main tools are skein lasagna modules based on equivariant and deformed versions of…
For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse…
The purpose of this paper is to present a certain combinatorial method of constructing invariants of isotopy classes of oriented tame links. This arises as a generalization of the known polynomial invariants of Conway and Jones. These…
We study the $S_n$-equivariant log-concavity of the cohomology of flag varieties, also known as the coinvariant ring of $S_n$. Using the theory of representation stability, we give computer-assisted proofs of the equivariant log-concavity…
To a link L in the 3-sphere, we associate a spectral sequence whose E^2 page is the reduced Khovanov homology of L and which converges to a version of the monopole Floer homology of the branched double cover. The pages E^k for k > 1 depend…
We define a new invariant of links in the $3$-sphere and call it the parabolic Dijkgraaf-Witten (DW) invariant. This invariant is a generalization of the reduced DW invariant derived by Karuo. In this paper, we compute the invariant of…
We study Heegaard Floer homology and various related invariants (such as the $h$-function) for two-component L-space links with linking number zero. For such links, we explicitly describe the relationship between the $h$-function, the…
We introduce a homology surgery problem in dimension 3 which has the property that the vanishing of its algebraic obstruction leads to a canonical class of \pi-algebraically-split links in 3-manifolds with fundamental group \pi . Using this…
A classical result of variational analysis, known as Attouch theorem, establishes the equivalence between epigraphical convergence of a sequence of proper convex lower semicontinuous functions and graphical convergence of the corresponding…
We provide several results on splice-quotient singularities: a combinatorial expression of the dimension of the first cohomology of all `natural' line bundles, an equivariant Campillo-Delgado-Gusein-Zade type formula about the dimension of…
In this paper, we discuss the (co)homology theory of biquandles, derived biquandle cocycle invariants for oriented surface-links using broken surface diagrams and how to compute the biquandle cocycle invariants from marked graph diagrams.…
We construct efficient topological cobordisms between torus links and large connected sums of trefoil knots. As an application, we show that the signature invariant $\sigma_\omega$ at $\omega=\zeta_6$ takes essentially minimal values on…
By using the cohomology theory of quandles, quandle cocycle invariants and shadow quandle cocycle invariants are defined for oriented links and surface-links via broken surface diagrams. By using symmetric quandles, symmetric quandle…
A signed graph $(G, \sigma)$ is a graph $G$ along with a function $\sigma: E(G) \to \{+,-\}$. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A…
The slope variety of a graph is an algebraic set whose points correspond to drawings of a graph. A complement-reducible graph (or cograph) is a graph without an induced four-vertex path. We construct a bijection between the zeroes of the…
We use Lee's work on the Khovanov homology to define a knot invariant s. We show that s(K) is a concordance invariant and that it provides a lower bound for the slice genus of K. As a corollary, we give a purely combinatorial proof of the…
We study the space of slice-torus invariants. In particular we characterize the set of values that slice-torus invariants may take on a given knot in terms of the stable smooth slice genus. Our study reveals that the resolution of the local…
The current work will appear in a Celebratio Mathematica volume in honor of Walter Neumann. We summarize results and methods from our long-time collaboration with Neumann, especially the motivation for the introduction of splice diagrams to…
We prove a basic inequality for the d-invariants of a splice of knots in homology spheres. As a result, we are able to prove a new relation on the rank of reduced Floer homology under maps between Seifert fibered homology spheres, improving…