Related papers: Flips and Flops Constructed by GIT Quotient
Many quantum invariants of knots and 3-manifolds (e.g. Jones polynomials) are special cases of the Witten-Reshetikhin-Turaev 3D TQFT. The latter is in turn a part of a larger theory - the Crane-Yetter 4D TQFT. In this work, we compute the…
Plane perfect matchings of $2n$ points in convex position are in bijection with triangulations of convex polygons of size $n+2$. Edge flips are a classic operation to perform local changes both structures have in common. In this work, we…
Flip graphs are a ubiquitous class of graphs, which encode relations induced on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in…
In this paper we investigate the structure of flip graphs on non-crossing perfect matchings in the plane. Specifically, consider all non-crossing straight-line perfect matchings on a set of $2n$ points that are placed equidistantly on the…
We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include convex subsets of…
A general net of quadric surfaces, together with a choice of a base point, defines a net of plane cubics via the Gale transformation of the remaining seven base points. To both nets, one can also naturally associate the same smooth plane…
We adapt Seifert's algorithm for classical knots and links to the setting of tri-plane diagrams for bridge trisected surfaces in the 4-sphere. Our approach allows for the construction of a Seifert solid that is described by a Heegaard…
We provide topological obstructions to the existence of orbit cylinders of symmetric orbits, for mechanical systems preserved by antisymplectic involutions (e.g. the restricted three-body problem). Such cylinders induce continuous paths…
We construct new compactifications with good properties of moduli spaces of maps from nonsingular marked curves to a large class of GIT quotients. This generalizes from a unified perspective many particular examples considered earlier in…
Our earlier twisted-face-pairing construction showed how to modify an arbitrary orientation-reversing face-pairing on a faceted 3-ball in a mechanical way so that the quotient is automatically a closed, orientable 3-manifold. The…
We apply the specialization technique based on the decomposition of the diagonal to find an explicit example over $\mathbb{Q}$ of a quadric and cubic hypersurface in $\mathbb{P}^6$ such that their intersection is a smooth stably irrational…
A deep result of Voisin asserts that the Griffiths group of a general non-rigid Calabi-Yau (CY) 3-fold is infinitely generated. This theorem builds on an earlier method of hers which was implemented by Albano and Collino to prove the same…
A triangulation of a polygon is a subdivision of it into triangles, using diagonals between its vertices. Two different triangulations of a polygon can be related by a sequence of flips: a flip replaces a diagonal by the unique other…
We prove that every smoothly embedded surface in a 4--manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4--manifold; that is, after isotopy, the surface meets components of the trisection in…
By use of a variety of techniques (most based on constructions of quasipositive knots and links, some old and others new), many smooth 3-manifolds are realized as transverse intersections of complex surfaces in complex 3-space with strictly…
We introduce bridge trisections of knotted surfaces in the four-sphere. This description is inspired by the work of Gay and Kirby on trisections of four-manifolds and extends the classical concept of bridge splittings of links in the…
We introduce defects, with internal gauge symmetries, on a knot and Seifert surface to a knot into the combinatorial construction of finite gauge-group Dijkgraaf-Witten theory. The appropriate initial data for the construction are certain…
We describe explicit birational maps from some rational complete intersections of three quadrics in $\mathbb{P}^7$ to some prime Fano manifolds together with their Sarkisov decomposition via a single Secant Flop, allowing us to recover the…
Flips of diagonals in colored triangle-free triangulations of a convex polygon are interpreted as moves between two adjacent chambers in a certain graphic hyperplane arrangement. Properties of geodesics in the associated flip graph are…
We study how Gromov-Witten invariants of projective 3-folds transform under a standard flop and a small extremal transition in the algebro-geometric setting from the recent development of algebraic relative Gromov-Witten theory and its…