arXiv subjects

Chad Musick

Publications and source records attributed to Chad Musick.

4 recordsLinked to original sources

arXiv2026

Locally Minimal Bridge Presentations of Knots

For classical knots, many characteristics are difficult to determine from arbitrary diagrams of the knot. Bridge number is of this type. It is possible to have a bridge presentation of a knot in which the number of bridges is only locally minimal, as shown by Ozawa and Takao. Here, we give a method for beginning with an arbitrary classical knot diagram having $n$ crossings and producing a locally minimal bridge presentation of the knot with size $O(n^2)$. This method requires only polynomial time and space. For some classes of knots, any local minimum is also the global minimum, which was shown for the trivial knot by Otal. Because the unknot is the only knot with bridge number $1$, unknot recognition is in $P$.

math.GT↗

arXiv2012

Minimal bridge projections for 11-crossing prime knots

We give the bridge indices for 11-crossing prime knots and give a minimal bridge projection for each of these knots. The results on the indices may be easily summarized: all of these knots that are not rational knots or Montesinos knots have bridge index three.

math.GT↗

arXiv2012

A method of encoding generalized link diagrams

We describe a method of encoding various types of link diagrams, including those with classical, flat, rigid, welded, and virtual crossings. We show that this method may be used to encode link diagrams, up to equivalence, in a notation whose length is a cubic function of the number of 'riser marks'. For classical knots, the minimal number of such marks is twice the bridge index, and a classical knot diagram in minimal bridge form with bridge index $b$ may be encoded in space $\mathcal{O}(b^2)$. A set of moves on the notation is defined. As a demonstration of the utility of the notation we give another proof that the Kishino virtual knot is non-classical.

math.GT↗

arXiv2011

Recognizing trivial links in polynomial time

Trivial links are unique up to number of link components, but they can be hard to recognize from arbitrary diagrams. We define a new measure of the complexity of a link embedding, the crumple, and show how this may be used to measure progress toward a trivial embedding. In conjunction with a modified form of arc presentations of links, we obtain a strictly monotonic, deterministic algorithm that recognizes triviality in links within polynomial time and space.

math.GT↗