Margaret Doig | Creighton University (original) (raw)

Papers by Margaret Doig

arXiv (Cornell University), May 26, 2015

It follows implicitly from recent work in Heegaard Floer theory that lens spaces are homology cob... more It follows implicitly from recent work in Heegaard Floer theory that lens spaces are homology cobordant exactly when they are oriented homeomorphic. We provide a new combinatorial proof using the Heegaard Floer d-invariants, which themselves may be defined combinatorially for lens spaces.

arXiv (Cornell University), Feb 25, 2013

For a fixed p, there are only finitely many elliptic 3-manifolds given by p/q-surgery on a knot i... more For a fixed p, there are only finitely many elliptic 3-manifolds given by p/q-surgery on a knot in S 3. We prove this result by using the Heegaard Floer correction terms (d-invariants) to obstruct elliptic manifolds from arising as knot surgery.

In this paper, we exhibit an explicit one-dimensional deformation retract Dk(Sn) of the unordered... more In this paper, we exhibit an explicit one-dimensional deformation retract Dk(Sn) of the unordered k-point configuration space U top k (Sn) for any star Sn. These spaces have recently been studied by Abrams and Ghrist. We use this retract to compute an explicit set of free generators βk of the corresponding braid group Bk(Sn, ck). In particular, we show that the natural map ik∗ : Bk−1(Sn, ck−1) →֒ Bk(Sn, ck) sends βk−1 to βk injectively.

arXiv: Geometric Topology, 2020

We model the typical behavior of knots and links using grid diagrams. Links are ubiquitous in the... more We model the typical behavior of knots and links using grid diagrams. Links are ubiquitous in the sciences, and their normal or typical behavior is of significant importance in understanding situations such as the topological state of DNA or the statistical mechanics of ring polymers. We examine three invariants: the expected size of a random knot; the expected number of components of a random link; and the expected writhe of a random knot. We investigate the first two numerically and produce generating functions which codify the observed patterns, we perform an exploratory data analysis for the third. We continue this project in a future work, where we investigate genus and the effects of crossing change on it.

We study the Randić index for cactus graphs. It is conjectured to be bounded below by radius (for... more We study the Randić index for cactus graphs. It is conjectured to be bounded below by radius (for other than an even path), and it is known to obey several bounds based on diameter. We study radius and diameter for cacti then verify the radius bound and strengthen two diameter bounds for cacti. Along the way, we produce several other bounds for the Randić index in terms of graph size, order, and valency for several special classes of graphs, including chemical nontrivial cacti and cacti with starlike BC-trees.

Proceedings of the American Mathematical Society, 2015

We study finite, non-cyclic knot surgeries, that is, surgeries which give manifolds of finite but... more We study finite, non-cyclic knot surgeries, that is, surgeries which give manifolds of finite but not cyclic fundamental group. These manifolds are known to be knot surgeries except for the dihedral manifolds. We show that, for a fixed p, there are finitely many dihedral manifolds that are p/q-surgery, and we place a bound on which manifolds they may be. In the process, we calculate a recursive relationship among the Heegaard Floer d-invariants of dihedral manifolds with a given first homology and calculate a bound on which d-invariants would occur if such a manifold were surgery on a knot in S 3 .

Algebraic & Geometric Topology, 2015

It is well known that any 3-manifold can be obtained by Dehn surgery on a link, but not which one... more It is well known that any 3-manifold can be obtained by Dehn surgery on a link, but not which ones can be obtained from a knot or which knots can produce them. We investigate these two questions for elliptic Seifert fibered spaces (other than lens spaces) using the Heegaard Floer correction terms or d-invariants associated to a 3-manifold Y and its torsion Spin c structures. For 1 .Y / finite and jH 1 .Y /j Ä 4, we classify the manifolds which are knot surgery and the knot surgeries which give them; for jH 1 .Y /j Ä 32, we classify the manifolds which are surgery and place restrictions on the surgeries which may give them. 57M25; 57R65

Transactions of the American Mathematical Society, 2016

We calculate the intersection ring of three-dimensional graph manifolds with rational coefficient... more We calculate the intersection ring of three-dimensional graph manifolds with rational coefficients and give an algebraic characterization of these rings when the manifold's underlying graph is a tree. We are able to use this characterization to show that the intersection ring obstructs arbitrary three-manifolds from being homology cobordant to certain graph manifolds.

A grid graph is a Cartesian product G = γ 1 × γ 2 × • • • × γ k where the γ i are cycles or paths... more A grid graph is a Cartesian product G = γ 1 × γ 2 × • • • × γ k where the γ i are cycles or paths. The run length of a Hamiltonian cycle in a grid graph is defined to be the maximum number r such that any r consecutive edges include no more than one edge in any dimension. In this paper, we present several methods for producing cycles of high run length from cycles of lower run length; sample applications include showing that the maximum run length of G is (i) less than k if each γ i is directed; (ii) at least ⌊k/3⌋ + 1 if each |γ i | has the form p ri for some fixed prime p; and (iii) at least ⌊k/2⌋ + 1 if every |γ i | is of the form 2 qi p ri for fixed prime p.

Geometriae Dedicata, 2014