2 problems

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Non-foil free knot diagrams realize the figure-eight knot
Let $K$ be a minimal free knot diagram with at least four crossings that is not a foil or a connected sum of solely foil knots. An assignment of crossings resolves every free cross…
Conjectureopenfigure-eight-knots free-knots knot-diagramsarXiv posted 2 days ago
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Foil diagrams attain the upper bound on figure-eight descendants
A free foil diagram with $n$ crossings is a free knot diagram, and an algebraic free diagram is a free knot diagram arising from an algebraic tangle. Foil figure-eight upper-bound…
Conjectureopenalgebraic-tangles figure-eight-knots free-knotsarXiv posted 2 days ago

Non-foil free knot diagrams realize the figure-eight knot