Related papers: Circle-valued Morse theory for frame spun knots an…
We introduce the notion of ascent sliceness of virtual knots. A representative of a virtual knot is an embedding $ S^1 \hookrightarrow \Sigma_{g} \times I $, for $ \Sigma_g $ a closed connected oriented surface of genus $ g $; the virtual…
The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the form of a "source-sink", that means identifying an attractor-repeller pair, which are attractor-repellent sets for all other…
For a knot $K$ in a homology $3$-sphere $\Sigma$, let $M$ be the result of $2/q$-surgery on $K$, and let $X$ be the universal abelian covering of $M$. Our first theorem is that if the first homology of $X$ is finite cyclic and $M$ is a…
Fintushel and Stern have proved that if S \subset X is a symplectic surface in a symplectic 4-manifold such that S has simply-connected complement and nonnegative self-intersection, then there are infinitely many topologically equivalent…
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 define a notion of complexity for shake-slice knots which is analogous to the definition of complexity for h-cobordisms studied by Morgan-Szab\'o. We prove that for each framing $n \ne 0$ and complexity $c \ge 0$, there is an…
We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants. The curve is obtained by exploiting the…
Bott and Taubes constructed knot invariants by integrating differential forms along the fiber of a bundle over the space of knots, generalizing the Gauss linking integral. Their techniques were later used to construct real cohomology…
This paper contains linear systems of equations which can distinguish knots without knot invariants. Let $M_n$ be the topological moduli space of all n-component string links and such that a fixed projection into the plane is an immersion.…
In recent work the author investigates perfect matchings of a bipartite graph obtained from a knot diagram and demonstrates that these correspond to discrete Morse functions on a 2-complex for the 2-sphere. This relationship is expounded…
We consider unbranched Willmore surfaces in the Euclidean space that arise as inverted complete minimal surfaces with embedded planar ends. Several statements are proven about upper and lower bounds on the Morse Index - the number of…
In this paper, we consider the non-singular extension problem for circle-valued Morse functions on closed orientable surfaces. The problem asks, given a circle-valued Morse function $f\colon M\to S^{1}$ on a closed orientable surface $M$,…
A celebrated theorem of Kirby identifies the set of closed oriented connected 3-manifolds with the set of framed links in $S^3$ modulo two moves. We give a similar description for the set of knots (and more generally, boundary links) in…
A knot K in the 3-sphere is said to have Property nR if, whenever K is a component of an n-component link L and some integral surgery on L produces the connected sum of n copies of S^1 x S^2, there is a sequence of handle slides on L that…
A singular knot is an immersed circle in $\mathbb R^{3}$ with finitely many transverse double points. The study of singular knots was initially motivated by the study of Vassiliev invariants. Namely, singular knots give rise to a decreasing…
This is a PhD thesis about low dimensional topology, in particular knot thory in 3-manifolds also different from the 3-sphere, topological applications of quantum invariants, and Turaev's shadows. There is an introduction and a survey for…
Let $K$ be a knot in the 3-sphere, viewed as the ideal boundary of hyperbolic 4-space $\mathbb{H}^4$. We prove that the number of minimal discs in $\mathbb{H}^4$ with ideal boundary $K$ is a knot invariant. I.e.\ the number is finite and…
Let $\mathrm S^3$ be the unit sphere of $\mathbb C^2$ with its standard Cauchy-Riemann (CR) structure. This paper investigates the CR geometry of curves in $\mathrm S^3$ which are transversal to the contact distribution, using the local CR…
The refined Chern-Simons theory is a one-parameter deformation of the ordinary Chern-Simons theory on Seifert manifolds. It is defined via an index of the theory on N M5 branes, where the corresponding one-parameter deformation is a natural…
In this paper we study Morse homology and cohomology with local coefficients, i.e. "twisted" Morse homology and cohomology, on closed finite dimensional smooth manifolds. We prove a Morse theoretic version of Eilenberg's Theorem, and we…