相关论文: Simplified Khovanov-Rozansky Chain Complexes of Op…

We introduce the notion of a Khovanov-Floer theory. Roughly, such a theory assigns a filtered chain complex over Z/2 to a link diagram such that (1) the E_2 page of the resulting spectral sequence is naturally isomorphic to the Khovanov…

We study the twisted Alexander polynomial from the viewpoint of the SL(2,C)-character variety of nonabelian representations of a knot group. It is known that if a knot is fibered, then the twisted Alexander polynomials associated with…

We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov homology from links to arbitrary tangles, not necessarily even. For every plane diagram of…

We study a certain type of braid closure which resembles the plat closure but has certain advantages; for example, it maps pure braids to knots. The main results of this note are a Markov-type theorem and a description of how Vassiliev…

We revisit Rozansky's construction of Khovanov homology for links in $S^2\times S^1$, extending it to define Khovanov homology $Kh(L)$ for links $L$ in $M^r=#^r(S^2\times S^1)$ for any $r$. The graded Euler characteristic of $Kh(L)$ can be…

We put a new spin on Khovanov--Rozansky homology. That is, we equip $\Lambda^n$-colored $\mathfrak{sl}_{2n}$ Khovanov--Rozansky homology with an involution whose $\pm 1$-eigenspaces are link invariants. When $n=1,2,3$ (and assuming…

We give a simple recursion which computes the triply graded Khovanov-Rozansky homology of several infinite families of knots and links, including the $(n,nm\pm 1)$ and $(n,nm)$ torus links for $n,m\geq 1$. We interpret our results in terms…

The 2-bridge knots are a family of knots with bridge number 2. In this paper, we compute the Kauffman polynomials of 2-bridge knots using the Kauffman skein theory and linear algebra techniques. Our calculation can be easily carried out…

We prove the conjecture of Przytycki and Sazdanovic that the Khovanov homology of the closure of a 3-stranded braid only contains torsion of order 2. This conjecture has been known for six out of seven classes in the Murasugi-classification…

In this paper we present two different ways for computing the Kauffman bracket skein module of $S^1\times S^2$, ${\rm KBSM}\left(S^1\times S^2\right)$, via braids. We first extend the universal Kauffman bracket type invariant $V$ for knots…

In the present work, we realize the space of 2-string links $\mathcal{L}$ as a free algebra over a colored operad denoted $\mathcal{SCL}$ (for "Swiss-Cheese for links"). This result extends works of Burke and Koytcheff about the quotient of…

We construct a braid conjugacy class invariant $\kappa$ by refining Plamenevskaya's transverse element $\psi$ in Khovanov homology via the annular grading. While $\kappa$ is not an invariant of transverse links, it distinguishes some braids…

We construct an algebra of non-trivial homological operations on Khovanov homology with coefficients in $\mathbb Z_2$ generated by two Bockstein operations. We use the unified Khovanov homology theory developed by the first author to lift…

We prove that the $(2n,1)$-cable of the figure-eight knot is not smoothly slice when $n$ is odd, by using the real Seiberg-Witten Fr{\o}yshov invariant of Konno-Miyazawa-Taniguchi. For the computation, we develop an $O(2)$-equivariant…

In the Khovanov homology of links, presence of $\mathbb{Z}_2$-torsion is a very common phenomenon. Finite number of examples of knots with $\mathbb{Z}_n$-torsion for $n>2$ were also known, none for $n>8$. In this paper, we prove that there…

In this paper, we study a special family of $(1,1)$ knots called constrained knots, which includes 2-bridge knots in the 3-sphere $S^3$ and simple knots in lens spaces. Constrained knots are parameterized by five integers and characterized…

For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doubly-graded cohomology theory whose Euler characteristic is the sl(n) link polynomial. We use Lagrangian Floer cohomology on some…

Ribbon concordance gives a partial order on knot types, and applying a knot homology functor to a ribbon concordance gives an inclusion of the homologies. The question of the existence of global ribbon minima in each concordance class is a…

In the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links, that is links whose Khovanov homology is supported on two adjacent diagonals, are known to only contain $\mathbb{Z}_2$ torsion. In…

Kanenobu has given infinite families of knots with the same HOMFLY polynomials. We show that these knots also have the same sl(n) and HOMFLY homologies, thus giving the first example of an infinite family of knots undistinguishable by these…