Thomas W. Mattman

THE CULLER-SHALEN SEMINORMS OF THE (-2, 3, n) PRETZEL KNOT

We show that the

Many, many more intrinsically knotted graphs

{\rm SL}_2 (\mathbb C)

Seifert-fibered surgeries which do not arise from primitive/Seifert-fibered constructions

-character variety of the (-2, 3, n) pretzel knot consists of two (respectively three) algebraic curves when 3 ∤ n (respectively 3 | n) and given an explicit calculation of the Culler-Shalem seminorms of these curves. Using this calculation, we describe the fundamental polygon and Newton polygon for these knots and give a list of Dehn surgerise yielding a manifold with finite or cyclic fundamental group. This constitutes a new proof of property P for these knots.

A proof of the Kauffman–Harary Conjecture

We list more than 200 new examples of minor minimal intrinsically knotted graphs and describe many more that are intrinsically knotted and likely minor minimal.

Commensurability classes of (−2,3,n) pretzel knot complements

We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Deans primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert-fibered surgeries arise from Deans primitive/Seifert-fibered construction. The (-3, 3,5)-pretzel knot belongs to both of the infinite families.

Graphs on 21 edges that are not 2-apex

We prove the Kauffman-Harary Conjecture, posed in 1999: given a reduced, alternating diagram D of a knot with prime determinant p, every non-trivial Fox p-coloring of D will assign different colors to different arcs.

RIBBONLENGTH OF TORUS KNOTS

Let K be a hyperbolic (−2,3, n) pretzel knot and M = S 3 \ K its complement. For these knots, we verify a conjecture of Reid and Walsh: there are at most three knot complements in the commensurability class of M. Indeed, if n 6 7, we show that M is the unique knot complement in its class. We include examples to illustrate how our methods apply to a broad class of Montesinos knots.

BOUNDS ON THE CROSSCAP NUMBER OF TORUS KNOTS

We show that the 20 graph Heawood family, obtained by a combination of triangle-Y and Y-triangle moves on

Six variations on a theme: almost planar graphs

K_7

Forbidden Minors: Finding the Finite Few

, is precisely the set of graphs of at most 21 edges that are minor minimal for the property not