Maggy Tomova

Alternate Heegaard genus bounds distance

Suppose M is a compact orientable irreducible 3-manifold with Heegaard splitting surfaces P and Q. Then either Q is isotopic to a possibly stabilized or boundary-stabilized copy of P or the distance d(P)<= 2genus (Q). More generally, if P and Q are bicompressible but weakly incompressible connected closed separating surfaces in M then either P and Q can be well-separated or P and Q are isotopic or d(P)<= 2genus (Q).

Uniqueness of bridge surfaces for 2-bridge knots

Any 2-bridge knot in S3 has a bridge sphere from which any other bridge surface can be obtained by stabilization, meridional stabilization, perturbation and proper isotopy.

Flipping bridge surfaces and bounds on the stable bridge number

We show that if K is a knot in S 3 and U is a bridge sphere for K with high distance and 2n punctures, the number of perturbations of K required to interchange the two balls bounded by U via an isotopy is n. We also construct a knot with two different bridge spheres with 2n and 2n 1 bridges respectively for which any common perturbation has at least 3n 4 bridges. We generalize both of these results to bridge surfaces for knots in any 3‐manifold. 57M25, 57M27, 57M50

Radio numbers for generalized prism graphs

A radio labeling is an assignment c : V (G) ! N such that every distinct pair of vertices u,v satisfies the inequality d(u,v )+ |c(u) c(v) |diam(G) + 1. The span of a radio labeling is the maximum value of c. The radio number of G, rn(G), is the minimum span over all radio labelings of G. Generalized prism graphs, denoted Zn,s, s 1, n s, have vertex set {(i,j)|i =1 ,2 and j =1 ,...,n} and edge set

COMPANIONS OF THE UNKNOT AND WIDTH ADDITIVITY

It has been conjectured that for knots K and K′ in S3, w(K # K′) = w(K) + w(K′) - 2. In [7], Scharlemann and Thompson proposed potential counterexamples to this conjecture. For every n, they proposed a family of knots

Note: The N-queens Problem on a symmetric Toeplitz matrix

\{K^n_i\}

Additive invariants for knots, links and graphs in

for which they conjectured that

3

w(B^n\, \#\, K^n_i) = w(K^n_i)

–manifolds

where Bn is a bridge number n knot. We show that for n > 2 none of the knots in

Neighbors of Knots in the Gordian Graph

\{K^n_i\}