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We introduce a triple coproduct for knots on surfaces, providing a commutative framework that decomposes a single-component diagram into three components (Section 2). This construction is motivated by the interplay between intersection…
This paper is part expository and part presentation of calculational results. The target space of the Kontsevich integral for knots is a space of diagrams; this space has various algebraic structures which are described here. These are…
We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. We…
We started a program to study the open string integrality invariants (LMOV invariants) for toric Calabi-Yau 3-folds with Aganagic-Vafa brane (AV-brane) several years ago. This paper is devoted to the case of resolved conifold with one out…
Conformally invariant functionals on the space of knots are introduced via extrinsic conformal geometry of the knot and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form which can…
Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…
We study topological and integrable aspects of $\hat{c}=1$ strings. We consider the circle line theories 0A and 0B at particular radii, and the super affine theories at their self-dual radii. We construct their ground rings, identify them…
In this paper I consider all possible properties from commutative algebra for polynomial composites and monoid domains. The aim is full characterization of these structures. I start with the examination of group, ring, modules properties,…
To a smooth, compact, oriented, properly-embedded surface in the $4$-ball, we define an invariant of its boundary-preserving isotopy class from the Khovanov homology of its boundary link. Previous work showed that when the boundary link is…
We study the properties of glued knots, a sub-class of real rational knots, that can be constructed by gluing ellipses. We define an invariant called the gluing degree and relate it to various classical properties of knots and classify all…
Topological string theory with twistor space as the target makes visible some otherwise difficult to see properties of perturbative Yang-Mills theory. But left-right symmetry, which is obvious in the standard formalism, is highly unclear…
Freedman and Krushkal showed that if the surgery conjecture and the $s$-cobordism conjecture hold for all topological 4-manifolds, then every link with pairwise zero linking numbers is topologically round handle slice. Kim, Powell, and…
We prove an explicit formula for the tail of the colored Jones polynomial for a class of arborescent links in terms of a product of theta functions and/or false theta functions. We also provide numerical evidence towards a classification of…
The main goal of the present paper is to construct new invariants of knots with additional structure by adding new gradings to the Khovanov complex. The ideas given below work in the case of virtual knots, closed braids and some other cases…
This review summarizes the recent developments in topological string theory from the author's perspective, mostly focused on aspects of research in which the author is involved. After a brief overview of the theory, we discuss two aspects…
We study series invariants for plumbed 3-manifolds and knot complements twisted by a root lattice. Our series recover recent results of Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park, and Ri and apply more generally to 3-manifolds which are…
We use Ozsv\'ath, Stipsicz, and Szab\'o's Upsilon-invariant to provide bounds on cobordisms between knots that `contain full-twists'. In particular, we recover and generalize a classical consequence of the Morton-Franks-Williams inequality…
We study intrinsically linked graphs where we require that every embedding of the graph contains not just a non-split link, but a link that satisfies some additional property. Examples of properties we address in this paper are: a two…
We extend knot Floer homology to string links in D^{2} \times I and to d-based links in arbitrary three manifolds, without any hypothesis on the null-homology of the components. As for knot Floer homology we obtain a description of the…
In this paper we discuss the applications of knotoids to modelling knots in open curves and produce new knotoid invariants. We show how invariants of knotoids generally give rise to well-behaved measures of how much an open curve is…