Related papers: Cubes, cacti, and framed long knots
The main theorem characterizes all Legendrian negative torus knots in universally tight lens space in the sense of coarse equivalence. Together with Onaran's results on Legendrian positive torus knots, all Legendrian torus knots in…
McClure and Smith constructed a functor that sends a topological multiplicative operad O to an E_2 algebra TotO. They define in fact an operad D_2 (acting on the totalization TotO) weakly equivalent to the little 2-disks operad. On the…
This paper presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative…
It is well-known that self-linking is the only Z valued Vassiliev invariant of framed knots in S^3. However for most 3-manifolds, in particular for the total spaces of S^1-bundles over an orientable surface F not S^2, the space of Z-valued…
This paper contains the strongest and at the same time most calculable knot invariant ever. Let $\Theta$ be the topological moduli space of all ordered oriented tangles in 3-space. We construct a non-trivial combinatorial 1-cocycle…
We generalize Turaev's definition of torsion invariants of pairs $(M,\xi)$, where $M$ is a 3-dimensional manifold and $\xi$ is an Euler structure on $M$ (a non-singular vector field up to homotopy relative to the boundary of $M$ and local…
We study positive braid knots (the knots in the three-sphere realized as positive braid closures) through the lens of the L-space conjecture. This conjecture predicts that if $K$ is a non-trivial positive braid knot, then for all $r <…
Using spinning we analyze in a geometric way Haefliger's smoothly knotted (4k-1)-spheres in the 6k-sphere. Consider the 2-torus standardly embedded in the 3-sphere, which is further standardly embedded in the 6-sphere. At each point of the…
Starting from an extension of the Poisson bracket structure and Kubo-Martin-Schwinger-property of classical statistical mechanics of continuous systems to spin systems, defined on a lattice, we derive a series of, as we think, new and…
We give a general treatment of the somewhat unfamiliar operation on manifolds called Connected Sum at Infinity, or CSI for short. A driving ambition has been to make the geometry behind the well definition and basic properties of CSI as…
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 introduce a new approach to the description of multi-sorted clones (sets of $k$-tuples of operations of the same arity, closed under coordinatewise composition and containing all projection tuples) on a two-element domain. Leveraging the…
We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible…
We define a nontrivial mod 2 valued additive concordance invariant defined on the torsion subgroup of the knot concordance group using involutive knot Floer package. For knots not contained in its kernel, we prove that their iterated…
There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of…
We construct an infinite commutative lattice of groups whose dual spaces give Kauffman finite-type invariants of long virtual knots. The lattice is based "horizontally" upon the Polyak algebra and extended "vertically" using Manturov's…
Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the space of ordered pairs of distinct points…
We consider Milnor's "tower algorithm" in the space of piecewise monotone maps, an iterative algorithm on the space of metrics which unifies, on the one hand, Thurston's iterative scheme which converges to holomorphic models, and, on the…
We investigate the representation of lattices as sublattices of the lattice of all convex subsets (intervals) of a linearly ordered set $(X,\le)$. We introduce the purely lattice-theoretic notion of a \textit{loc-lattice} and prove that…
In this article we study the Poisson algebra structure on the homology of the totalization of a fibrant cosimplicial space associated with an operad with multiplication. This structure is given as the Browder operation induced by the action…