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We establish a criterion that ensures a bounded almost complex curve in a bounded almost complex 4-manifold minimizes genus amongst all smooth surfaces that share its homology class and the transverse link on its boundary. An immediate…

We study closed N=2 strings on orbifolds of the form T^4/Z_2 and C^2/Z_2. We compute the torus partition function and prove its modular invariance. We analyse the BRST cohomology of the theory, construct the vertex operators, and compute…

We examine the structure and dimensionality of the Jones polynomial using manifold learning techniques. Our data set consists of more than 10 million knots up to 17 crossings and two other special families up to 2001 crossings. We introduce…

This is a companion paper to earlier work of the authors, which proved an integral surgery formula for framed instanton homology. First, we present an enhancement of the large surgery formula, a rational surgery formula for null-homologous…

We prove that if two knots are concordant, their involutive knot Floer complexes satisfy a certain type of stable equivalence.

We consider an analogue of well-known Casson knot invariant for knotoids. We start with a direct analogue of the classical construction which gives two different integer-valued knotoid invariants and then focus on its homology extension.…

We explore nonrelativistic (NR) pulsating string configurations over torsion Newton-Cartan (TNC) geometry having topology $ R \times S^2 $ and check the corresponding analytic integrability criteria following Kovacic's algorithm. In the…

We extend knot contact homology to a theory over the ring $\mathbb{Z}[\lambda^{\pm 1},\mu^{\pm 1}]$, with the invariant given topologically and combinatorially. The improved invariant, which is defined for framed knots in $S^3$ and can be…

We discuss a matrix of periodic holomorphic functions in the upper and lower half-plane which can be obtained from a factorization of an Andersen-Kashaev state integral of a knot complement with remarkable analytic and asymptotic properties…

We sharpen the duality between open and closed topological string partition functions for topological gravity coupled to matter. The closed string partition function is a generalised Kontsevich matrix model in the large dimension limit. We…

Symmetry of geometrical figures is reflected in regularities of their algebraic invariants. Algebraic regularities are often preserved when the geometrical figure is topologically deformed. The most natural, intuitively simple but…

For a graph G embedded in an orientable surface \Sigma, we consider associated links L(G) in the thickened surface \Sigma \times I. We relate the HOMFLY polynomial of L(G) to the recently defined Bollobas-Riordan polynomial of a ribbon…

This paper, to be regularly updated, lists those prime knots with the fewest possible number of crossings for which values of basic knot invariants, such as the unknotting number or the smooth 4-genus, are unknown. This list is being…

We discuss some geometrical aspects of the semiclassical quantization of string solutions in Type IIB Green-Schwarz action on $\ads_5\times \sphere^5$. We concentrate on quadratic fluctuations around classical configurations, expressing the…

The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of rational orbifold conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms. Under a…

It is known that the fundamental group homomorphism $\pi_1(T^2) \to \pi_1(S^3\setminus K)$ induced by the inclusion of the boundary torus into the complement of a knot $K$ in $S^3$ is a complete knot invariant. Many classical invariants of…

We automate the process of machine learning correlations between knot invariants. For nearly 200,000 distinct sets of input knot invariants together with an output invariant, we attempt to learn the output invariant by training a neural…

We give a non-perturbative completion of a class of closed topological string theories in terms of building blocks of dual open strings. In the specific case where the open string is given by a matrix model these blocks correspond to a…

We prove a formula for the involutive concordance invariants of the cabled knots in terms of that of the companion knot and the pattern knot. As a consequence, we show that any iterated cable of a knot with parameters of the form (odd,1) is…

We classify all convex polyomino whose coordinate rings are Gorenstein. We also compute the Castelnuovo-Mumford regularity of the coordinate ring of any stack polyomino in terms of the smallest interval which contains its vertices. We give…