Related papers: Detecting essential surfaces as intersections in t…
We give families of knots and links with pairs of Seifert surfaces that are topologically non-isotopic in $D^4$. This generalizes the main example of Hayden-Kim-Miller-Park-Sundberg and the proof is similarly based on the double branched…
This paper is a subsequent paper of math.RT/0607673. Here we consider the irreducible components of Springer fibres (or orbital varieties) for two-column case in GL}_n. We describe the intersection of two irreducible components, and…
We study connections between the topology of generic character varieties of fundamental groups of punctured Riemann surfaces, Macdonald polynomials, quiver representations, Hilbert schemes on surfaces, modular forms and multiplicities in…
A marked strongly invertible knot is a triple $(K,h,\delta)$ of a knot $K$ in $S^3$, a strong inversion $h$ of $K$, and a subarc $\delta \subset \operatorname{Fix}(h)\cong S^1$ bounded by $\operatorname{Fix}(h)\cap K\cong S^0$. An invariant…
We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda's method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number…
We study character varieties of symmetric knots and their reductions mod p. We observe that the varieties present a different behaviour according to whether the knots admit a free or periodic symmetry.
By using double branched covers, we prove that there is a 1-1 correspondence between the set of knotoids in the 2-sphere, up to orientation reversion and rotation, and knots with a strong inversion, up to conjugacy. This correspondence…
For a given smooth $2$-knot in $S^4$, we relate the existence of a smooth Seifert hypersurface of a certain class to the existence of irreducible $ SU(2)$-representations of its knot group. For example, we see that any smooth $2$-knot…
Berge introduced knots that are primitive/primitive with respect to the genus 2 Heegaard surface, $F$, in $S^3$; surgery on such knots at the surface slope yields a lens space. Later Dean described a similar class of knots that are…
The first examples of totally geodesic Seifert surfaces are constructed for hyperbolic knots and links, including both free and totally knotted surfaces. Then it is proved that two bridge knot complements cannot contain totally geodesic…
A Kauffman bracket on a surface is an invariant for framed links in the thickened surface, satisfying the Kauffman skein relation and multiplicative under superposition. This includes representations of the skein algebra of the surface. We…
This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses that arise naturally in the study…
Closed essential surfaces in a three-manifold can be detected by ideal points of the character variety or by algebraic non-integral representations. We give examples of closed essential surfaces not detected in either of these ways. For…
We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds…
For every finite collection of curves on a surface, we define an associated (semi-)norm on the first homology group of the surface. The unit ball of the dual norm is the convex hull of its integer points. We give an interpretation of these…
Exceptional points play a pivotal role in the topology of non-Hermitian systems, and significant advances have been made in classifying exceptional points and exploring the associated phenomena. Exceptional surfaces, which are hypersurfaces…
We exhibit the first example of a knot in the three-sphere with a pair of minimal genus Seifert surfaces that can be distinguished using the sutured Floer homology of their complementary manifolds together with the Spin^c-grading. This…
Let M be a compact, orientable, irreducible, atoroidal 3-manifold with boundary an incompressible torus. Techniques based on the characteristic submanifold theory are used to bound the intersection number of two slopes \alpha and \beta on…
We show that for certain hyperbolic 3-manifolds, all boundary slopes are slopes of immersed incompressible surfaces, covered by incompressible embeddings in some finite cover. The manifolds include hyperbolic punctured torus bundles and…
Edmonds famously proved that every periodic knot of genus g possesses an equivariant Seifert surface of genus g. We show that this is not true if one instead considers nonorientable spanning surfaces of a periodic knot. We demonstrate by…