arXiv subjects
Publications and source records attributed to Juanita Pinzón-Caicedo.
We prove that the Conway knot has infinite order in the concordance group, and in particular that it is not slice. Ours is an alternative, independent proof with respect to those of Piccirillo and of Donatone, Kegel, Lewark, and Truöl.
We show that two 3-manifolds that admit Sol geometry and have first homology group of order 16 are integer homology cobordant if and only if they are homeomorphic.
We study the existence of irreducible $SU(2)$-representations for cyclic branched covers of knots in $S^3$. Our main result establishes that if $K$ is a non-trivial prime knot and $d$ is an integer such that $d \geq 2$ and $Σ_d(K)$ is an integer homology sphere, then $π_1(Σ_d(K))$ admits an irreducible $SU(2)$-representation, whenever $K$ satisfies one of two conditions: either $K$ is $2$-periodic, or $K$ can be represented as the closure of a tangle adapted to a $d\times d$ SICUP matrix. The first condition leverages a commuting trick for covering spaces to realize higher-degree branched covers as 2-fold covers, allowing us to apply recent results of Kronheimer-Mrowka and others. The second condition uses equivariant surgery descriptions and the $ν^\sharp$ invariant from instanton Floer homology. As applications, we provide new infinite families of hyperbolic integer homology spheres admitting irreducible representations, including examples where previously known criteria fail.
These notes are an introduction to knot theory from the perspective of surfaces. The notes cover fundamental concepts such as isotopies, Reidemeister moves, torus knots, and (orientable, connected) surfaces with one boundary component. They also present knot invariants defined through Seifert surfaces and their associated matrices, including the 3-genus, the Alexander polynomial, and the signature. Finally, starting from the notion of concordance and the connected sum operation, a group structure on the set of knots is introduced.
We prove that satellite operations that satisfy a certain positivity condition and have winding number other than one are not homomorphisms. The argument uses the $d$-invariants of branched covers. In the process, we prove a technical result relating $d$-invariants and the Torelli group which may be of independent interest.
We prove that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)-representations. Our methods use instanton Floer homology, and in particular the surgery exact triangle, holonomy perturbations, and a non-vanishing result due to Kronheimer-Mrowka, as well as results about surgeries on cables due to Gordon.
Given a handle decomposition of a 4-manifold with boundary, and an open book decomposition of the boundary, we show how to produce a trisection diagram of a trisection of the 4-manifold inducing the given open book. We do this by making the original proof of the existence of relative trisections more explicit, in terms of handles. Furthermore, we extend this existence result to the case of 4-manifolds with multiple boundary components, and show how trisected 4-manifolds with multiple boundary components glue together.
A theorem of Furuta and Fintushel-Stern provides a criterion for a collection of Seifert fibred homology spheres to be independent in the homology cobordism group of oriented homology 3-spheres. In this article we use these results and some 4-dimensional constructions to produce infinite families of positive torus knots whose iterated Whitehead doubles are independent in the smooth concordance group.
We establish a correspondence between trisections of smooth, compact, oriented $4$--manifolds with connected boundary and diagrams describing these trisected $4$--manifolds. Such a diagram comes in the form of a compact, oriented surface with boundary together with three tuples of simple closed curves, with possibly fewer curves than the genus of the surface, satisfying a pairwise standardness condition. This should be thought of as the $4$--dimensional analog of a sutured Heegaard diagram for a sutured $3$--manifold. We also give many foundational examples.
The main goal of this article is to obtain a condition under which an infinite collection $\mathscr{F}$ of satellite knots (with companion a positive torus knot and pattern similar to the Whitehead link) freely generates a subgroup of infinite rank in the smooth concordance group. This goal is attained by examining both the instanton moduli space over a 4-manifold with tubular ends and the corresponding Chern-Simons invariant of the adequate 3-dimensional portion of the 4-manifold. More specifically, the result is derived from Furuta's criterion for the independence of Seifert fibred homology spheres in the homology cobordism group of oriented homology 3-spheres. Indeed, we first associate to $\mathscr{F}$ the corresponding collection of 2-fold covers of the 3-sphere branched over the elements of $\mathscr{F}$ and then introduce definite cobordisms from the aforementioned covers of the satellites to a number of Seifert fibered homology spheres. This allows us to apply Furuta's criterion and thus obtain a condition that guarantees the independence of the family $\mathscr{F}$ in the smooth concordance group.
Given a 2-stranded tangle in a $\ZZ/2$ homology ball, $T\subset Y$, we investigate the character variety $R(Y,T)$ of conjugacy classes of traceless SU(2) representations of $π_1(Y\setminus T)$. In particular we completely determine the subspace of binary dihedral representations, and identify all of $R(Y,T)$ for many tangles naturally associated to knots in S^3. Moreover, we determine the image of the restriction map from $R(T,Y)$ to the traceless SU(2) character variety of the 4-punctured 2-sphere (the {\it pillowcase}). We give examples to show this image can be non-linear in general, and show it is linear for tangles associated to pretzel knots.