Every Two-Generator Knot is Prime

Every Complete Binary Tree is Prime

A graph with a vertex set V is said to have a prime labeling if its vertices can be labeled with distinct integers 1; 2; ; jV j such that for every edge fx; yg, the labels assigned to x and y are relatively prime. A tree is prime if it has at least one prime labeling. Around 1980, Entringer conjectured that every tree is prime. After three decades, this conjecture remains open. Nevertheless, a ...

Every Reidemeister Move Is Needed for Each Knot Type

We show that every knot type admits a pair of diagrams that cannot be made identical without using Reidemeister Ω2-moves. We also show that our proof is compatible with known results for the other move types, in the sense that every knot type admits a pair of diagrams that cannot be made identical without using all of the move types.

Knotty: Knot Generator

We present a tool for the automatic procedural generation of a single-string Chinese-style knot that resembles any arbitrary input 3D model. We call this tool Knotty.

A Prime Strongly Positive Amphicheiral Knot Which Is Not Slice

When every $P$-flat ideal is flat

In this paper‎, ‎we study the class of rings in which every $P$-flat‎ ‎ideal is flat and which will be called $PFF$-rings‎. ‎In particular‎, ‎Von Neumann regular rings‎, ‎hereditary rings‎, ‎semi-hereditary ring‎, ‎PID and arithmetical rings are examples of $PFF$-rings‎. ‎In the context domain‎, ‎this notion coincide with‎ ‎Pr"{u}fer domain‎. ‎We provide necessary and sufficient conditions for‎...