arXiv subjects

Frank Swenton

Publications and source records attributed to Frank Swenton.

3 recordsLinked to original sources

arXiv2023

An algorithm to find ribbon disks for alternating knots

We describe an algorithm to find ribbon disks for alternating knots, and the results of a computer implementation of this algorithm. The algorithm is underlain by a slice link obstruction coming from Donaldson's diagonalisation theorem. It successfully finds ribbon disks for slice two-bridge knots and for the connected sum of any alternating knot with its reverse mirror, as well as for 662,903 prime alternating knots of 21 or fewer crossings. We also identify some examples of ribbon alternating knots for which the algorithm fails to find ribbon disks, though a related search identifies all such examples known. Combining these searches with known obstructions, we resolve the sliceness of all but 3,276 of the over 1.2 billion prime alternating knots with 21 or fewer crossings.

math.GT↗

arXiv2021

Determining the doubly slice genera of prime knots with up to 12 crossings

For a knot $K$, the doubly slice genus $g_{ds}(K)$ is the minimal $g$ such that $K$ divides a closed, orientable, and unknotted surface of genus $g$ embedded in $S^4$. In this paper, we identify the doubly slice genera of 2909 of the 2977 prime knots which have a crossing number of 12 or fewer.

math.GT↗

arXiv2004

Limits and the system of near-numbers

In this paper, we present a comprehensive system for the treatment of the topic of limits--conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts to force limit discussion into the language of individual real numbers and equality. The system of near-numbers properly fit the context of limits, allowing precise and unambiguous notation in limit computation. More importantly, the near-numbers are a set of concrete mathematical objects that provide a scaffold both for building an accurate intuitive understanding of the limit concept and for connecting that understanding to the formal definition of the limit.

math.GM↗