Yongwu Rong

Degree one maps between geometric 3-manifolds

Let M and N be two compact orientable 3-manifolds, we say that M ≥ N, if there is a degree one map from M to N. This gives a way to measure the complexity of 3-manifolds. The main purpose of this paper is to give a positive answer to the following conjecture: if there is an infinite sequence of degree one maps between Haken manifolds, then eventually all the manifolds are homeomorphic to each other. More generally, we prove a theorem which says that any infinite sequence of degree one maps between the so-called «geometric 3-manifolds» must eventually become homotopy equivalences

Degree one maps of Seifert manifolds and a note on Seifert volume

Abstract The existence of degree one maps defines an interesting partial order in the set of geometric 3-manifolds, modulo homotopy equivalences. Here we classify this partial order among aspherical Seifert fibered spaces. We also consider the Seifert volume and show that it has some properties similar to that of Gromovs norm under a degree d map.

EXAMPLES OF KNOTS WITH THE SAME POLYNOMIALS

This work is motivated by the question of whether the Jones polynomial can detect knottedness. Following the previous works of Eliahou–Kauffman–Thistlethwaite, Kanenobu, and Watson, we construct specific new examples of knots which cannot be distinguished by the Jones polynomial.

On derivatives of link polynomials

Abstract The higher order link polynomials are a class of link invariants related to both Homfly polynomial and Vassiliev invariants. Here we study their partial derivatives. We prove that each partial derivative of an n th order Homfly polynomial is an ( n + 1)th order Homfly polynomial. In particular, each d th partial derivative of the Homfly polynomial is a d th order Homfly polynomial. For d = 1, we show that these derivatives span all the first order Homfly polynomials. Similar constructions are made for other link polynomials. Questions on linear span and computational complexities are discussed.

A CATEGORIFICATION FOR THE PENROSE POLYNOMIAL

Given a graph, we construct homology groups whose Euler characteristic is the Penrose polynomial of the graph, evaluated at an integer. This work is motivated by Khovanovs work on the categorification of the Jones polynomial for knots, and the subsequent categorifications of the chromatic and Tutte polynomials for graphs.