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Cut-diagrams are diagrammatic objects, defined in dimensions 1 and 2, that generalize links in 3-space and surface-links in 4-space; in dimension 1, this coincides with the theory of welded links. Using cut-diagrams, we introduce an…
By using superisolated surface singularities whose link is a rational homology sphere we give counterexamples to some of the most important conjetures concernig invariants of normal surface singularities.
We prove that any diffeomorphism of a compact manifold can be C^1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially…
We give the first known topological model for the HOMFLY-PT polynomial constructed directly from link diagrams. More precisely, we prove that this invariant is given by graded intersections between explicit Lagrangian submanifolds in a…
We consider the double twist link $J(2m+1, 2n+1)$ which is the two-bridge link corresponding to the continued fraction $(2m+1)-1/(2n+1)$. It is known that $J(2m+1, 2n+1)$ has reducible nonabelian $SL_2(\mathbb{C})$-character variety if and…
We compute the tree-level $H^2R^3$ couplings of type II strings and provide some basic tests of the couplings by considering both K3 and Calabi-Yau threefold compactifications. Curiously, additional kinematical structures show up at tree…
This paper verifies a conjecture of Edelman and Reiner regarding the homology of the $h$-complex of a Boolean algebra. A discrete Morse function with no low-dimensional critical cells is constructed, implying a lower bound on connectivity.…
In the first part of the work (Sections 2-6) a special attention is given to relative separation axioms and relative connectedness, in particular, many relative versions of p-T_0, p-T_1, p-T_2, (i,j)- and p-regularities, (i,j)- and…
We give a construction for the d-dimensional simplices with all distances in {1,2} from the set of partitions of d+1.
We show that questions concerning the topological B-model on a Calabi-Yau manifold in the Landau-Ginzburg phase can be rephrased in the language of commutative algebra. This yields interesting and very practical methods for analyzing the…
The theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from sl(2) to sl(N) at arbitrary N -- but…
We conclude our analysis of bubble divergences in the flat spinfoam model. In [arXiv:1008.1476] we showed that the divergence degree of an arbitrary two-complex Gamma can be evaluated exactly by means of twisted cohomology. Here, we…
We provide a general method computing the mod $p$ Quillen homology of algebras over a monad that parametrizes the structure of mod $p$ homology of spectral Lie algebras. This is the $E^2$-page of the bar spectral sequence converging to the…
We study the spaces of string links and homotopy string links in an arbitrary manifold using multivariable manifold calculus of functors. We construct multi-cosimplicial models for both spaces and deduce certain convergence properties of…
We present new techniques to show hyperbolicity of links based on geometric/combinatorial topology. Our techniques are applicable to links that have at least one unknotted component. In particular, they are applicable to Brunnian links. We…
Let $p$ be a prime number. We develop a theory of $p$-adic Mahler measure of polynomials and apply it to the study of $\mathbb{Z}$-covers of rational homology 3-spheres branched over links. We obtain a $p$-adic analogue of the asymptotic…
For piecewise-linear maps, the phenomenon that a branch of a one-dimensional unstable manifold of a periodic solution is completely contained in its stable manifold is codimension-two. Unlike codimension-one homoclinic corners, such…
The recent paper "Linear Connectivity Forces Large Complete Bipartite Minors" by Boehme et al. relies on a structure theorem for graphs with no H-minor. The sketch provided of how to deduce this theorem from the work of Robertson and…
Suppose we concatenate two directed graphs, each isomorphic to a $d$ dimensional butterfly (but not necessarily identical to each other). Select any set of $2^k$ input and $2^k$ output nodes on the resulting graph. Then there exist node…
We derive a $U$-duality invariant formula for the degeneracies of BPS multiplets in a D1-D5 system for toroidal compactification of the type II string. The elliptic genus for this system vanishes, but it is found that BPS states can…