Related papers: A number theoretic result for Berge's conjecture

In 1999, Khovanov showed that a link invariant known as the Jones polynomial is the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups, and to explain their…

For pattern knots admitting genus-one bordered Heegaard diagrams, we show the knot Floer chain complexes of the corresponding satellite knots can be computed using immersed curves. This, in particular, gives a convenient way to compute the…

The Steenrod problem for closed orientable manifolds was solved completely by Thom. Following this approach, we solve the Steenrod problem for closed orientable orbifolds, proving that the rational homology groups of a closed orientable…

We prove that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL(2,C). For hyperbolic integer homology spheres this comes with the definition,…

We study Heegaard Floer homology and various related invariants (such as the $h$-function) for two-component L-space links with linking number zero. For such links, we explicitly describe the relationship between the $h$-function, the…

We consider the long-standing question of whether every regular LB-space is complete. This problem has been open since the 1950s and originates in Grothendieck's early work in functional analysis. Rather than seeking a direct proof or…

For any knot $K$ which bounds non-orientable and null-homologous surfaces $F$ in punctured $n\mathbb{C}P^2$, we construct a lower bound of the first Betti number of $F$ which consists of the signature of $K$ and the Heegaard Floer…

We show that special cycles generate a large part of the cohomology of locally symmetric spaces associated to orthogonal groups. We prove in particular that classes of totally geodesic submanifolds generate the cohomology groups of degree…

We prove that the (p,q)-cable of a knot K in S^3 admits a positive L-space surgery if and only if K admits a positive L-space surgery and q/p \geq 2g(K)-1, where g(K) is the Seifert genus of K. The "if" direction is due to Hedden.

We classify all the lattices realized as the intersection form of a positive definite four manifold with boundary $S_n^3(K)$ for a knot $K$ in the three sphere and a positive integer $n$ greater than $4g_4(K)+3$. We then use this result to…

We study the homology concordance group of knots in integer homology three-spheres which bound integer homology four-balls. Using knot Floer homology, we construct an infinite number of $\mathbb{Z}$-valued, linearly independent homology…

In the article The third homology of $SL_{2}(\mathbb{Q})$, Hutchinson determined the structure of $H_{3}\left(\mathrm{SL}_{2}(\mathbb{Q}),\mathbb{Z}\left[\frac{1}{2}\right]\right)$ by expressing it in terms of…

We prove a general result that relates certain pushouts of $E_k$-algebras to relative tensors over $E_{k+1}$-algebras. Specializations include a number of established results on classifying spaces, resolutions of modules, and (co)homology…

We define several concordance invariants using knot Floer homology which give improvements over known slice genus and clasp number bounds from Heegaard Floer homology. We also prove that the involutive correction terms of Hendricks and…

This is a contribution to the classification program of pointed Hopf algebras. We give a generalization of the quantum Serre relations and propose a generalization of the Frobenius-Lusztig kernels in order to compute Nichols algebras of…

Starting with a k-linear or DG category admitting a (homotopy) Serre functor, we construct a k-linear or DG 2-category categorifying the Heisenberg algebra of the numerical K-group of the original category. We also define a 2-categorical…

In 2001, \"Ostlund formulated the question: are Reidemeister moves of types 1 and 3 sufficient to describe a homotopy from any generic immersion of a circle in a two-dimensional plane to an embedding of the circle? The positive answer to…

We give an alternative proof of a recent theorem of Tange using the technology of changemaker lattices. Specifically, for $K\subset S^3$ a non-trivial knot with a lens space surgery, we give constraints on the Alexander polynomial of $K$…

Consider the space of `long knots' in R^n, K_{n,1}. This is the space of knots as studied by V. Vassiliev. Based on previous work of the authors, it follows that the rational homology of K_{3,1} is free Gerstenhaber-Poisson algebra. A…

We prove that twisted correction terms in Heegaard Floer homology provide lower bounds on the Thurston norm of certain cohomology classes determined by the strong concordance class of a 2-component link $L$ in $S^3$. We then specialise this…