Related papers: Tangle solutions for composite knots: applications…
We develop techniques for describing the derived moduli spaces of solutions to the equations of motion in twists of supersymmetric gauge theories as derived algebraic stacks. We introduce a holomorphic twist of N=4 supersymmetric gauge…
We prove that any finite collection of polygons of equal area has a common hinged dissection. That is, for any such collection of polygons there exists a chain of polygons hinged at vertices that can be folded in the plane continuously…
We solve a strong version of Problem 3.6 (D) in Kirby's list, that is, we show that for any integer $n$, there exist infinitely many mutually distinct knots such that $2$-handle additions along them with framing $n$ yield the same…
We introduce a 4-dimensional analogue of the rational Seifert genus of a knot $K\subset Y$, which we call the rational slice genus, that measures the complexity of a homology class in $H_2(Y\times [0,1],K;\mathbb{Q})$. Our main theorem is a…
Commutative complex numbers of the form u=x+\alpha y+\beta z+\gamma t in 4 dimensions are studied, the variables x, y, z and t being real numbers. Four distinct types of multiplication rules for the complex bases \alpha, \beta and \gamma…
Among the knots that are the connected sum of two torus knots with cobordism distance 1, we characterize those that have 4-dimensional clasp number at least 2, and we show that their n-fold connected self-sum has 4-dimensional clasp number…
We construct new knot polynomials. Let $V$ be the standard solid torus in 3-space and let $pr$ be its standard projection onto an annulus. Let $M$ be the space of all smooth oriented knots in $V$ such that the restriction of $pr$ is an…
Finding a systematic expansion of the spectrum of free superstrings on AdS${}_5\times $S${}^5$, or equivalently strongly coupled N = 4 SYM in the planar limit, remains an outstanding challenge. No first principle string theory methods are…
In a previous paper, V\'ertesi and the first author used grid-like Heegaard diagrams to define tangle Floer homology, which associates to a tangle $T$ a differential graded bimodule $\widetilde{\mathrm{CT}} (T)$. If $L$ is obtained by…
It is known that there are 21 ribbon knots with 10 crossings or fewer. We show that for every ribbon knot, there exists a tangle that satisfies two properties associated with the knot. First, under a specific closure, the closed tangle is…
The large size limit of matrix integrals with quartic potential may be used to count alternating links and tangles. The removal of redundancies amounts to renormalizations of the potential. This extends into two directions: higher genus and…
We extend the results of Deligne and Illusie on liftings modulo $p^2$ and decompositions of the de Rham complex in several ways. We show that for a smooth scheme $X$ over a perfect field $k$ of characteristic $p>0$, the truncations of the…
Let {T_n} be the bipolar filtration of the smooth concordance group of topologically slice knots, which was introduced by Cochran, Harvey, and Horn. It is known that for each n not equal to 1 the quotient group T_n/T_{n+1} has infinite rank…
We show that the knots $K\in\{4_1,5_1\}$ can be paired with a corresponding knot $K^\prime$ such that $u(K\#K^\prime)<u(K)+u(K^\prime)$. As a consequence unknotting number fails to be additive for these knots. We also provide a candidate…
We generalize the Guth--Katz joints theorem from lines to varieties. A special case says that $N$ planes (2-flats) in 6 dimensions (over any field) have $O(N^{3/2})$ joints, where a joint is a point contained in a triple of these planes not…
An ideal triangulation $\mathcal{T}$ of a hyperbolic 3-manifold $M$ with one cusp is non-peripheral if no edge of $\mathcal{T}$ is homotopic to a curve in the boundary torus of $M$. For such a triangulation, the gluing and completeness…
With a 4-ended tangle $T$, we associate a Heegaard Floer invariant $\operatorname{CFT^\partial}(T)$, the peculiar module of $T$. Based on Zarev's bordered sutured Heegaard Floer theory, we prove a glueing formula for this invariant which…
Solving tangle equations is deeply connected with studying enzyme action on DNA. The main goal of this paper is to solve the system of tangle equations $N(O+X_1)=b_1$ and $N(O+X_2)=b_2 \# b_3$, where $X_1$ and $X_2$ are rational tangles,…
We study the non-commutative instanton solution proposed in hep-th/0009142 and obtain the spectrum of small oscillations. The spectrum thus obtained is in exact agreement with the spectrum of stringy excitations in a configuration of point…
We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q:=H_q(N;Z)$. Our main result is a complete readily calculable classification of embeddings $N\to R^7$, up to the…