TL;DR
This paper explores the quadruple crossing number of knots and links, demonstrating how the span of the bracket polynomial can be used to determine this number for various knots and links.
Contribution
It introduces a method to compute quadruple crossing numbers using the span of the bracket polynomial, expanding understanding of knot projections.
Findings
Every knot and link has a quadruple crossing projection.
The span of the bracket polynomial can determine quadruple crossing numbers.
The paper provides specific calculations for various knots and links.
Abstract
A quadruple crossing is a crossing in a projection of a knot or link that has four strands of the knot passing straight through it. A quadruple crossing projection is a projection such that all of the crossings are quadruple crossings. In a previous paper, it was proved that every knot and link has a quadruple crossing projection and hence, every knot has a minimal quadruple crossing number. In this paper, we investigate quadruple crossing number, and in particular, use the span of the bracket polynomial to determine quadruple crossing number for a variety of knots and links.
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