15 problems

Free knots initially appeared as isotopy classes of Gauss words, namely equivalence classes of Gauss words under the relevant Reidemeister moves. Turaev's conjecture. Every free kn…

Let a free knot diagram be given, and let a nontrivial knot be a knot other than the unknot. The resultant knot probability is the proportion of crossing assignments of the free kn…

A free tangle is a tangle whose crossings are free crossings, and its four anchor points remain fixed on the boundary. The resultant knot probability is the proportion of crossing…

Let $K_1$ be a nontrivial prime knot with resultant unknot probability $\alpha$, and let $K_2$ be a nontrivial prime knot formed by recursive connected sums with $K_1$. Write…

A free $21n$ knot is the free knot diagram formed from the $21$-tangle and a free $n$-tangle, as used in the source. Resultants are knots obtained by assigning all free crossings.…

A minimal prime algebraic free knot diagram is a minimal free knot diagram representing a prime knot and arising from an algebraic tangle. A free $2n$ knot is the family considered…

A free $2n$ knot is the free knot diagram referred to in the source, with $n+2$ crossings. A minimal free knot diagram is one with the minimum crossing number for its knot. $2n$-un…

A resultant knot probability is the proportion of crossing assignments of a free knot diagram that produce a specified knot. An algebraic tangle is a tangle obtained through the al…

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…

Let $n$ be odd with $n\geq 3$. A minimal prime free knot diagram is a minimal free knot diagram representing a prime knot, and a trefoil descendant is a resultant trefoil obtained…

Let a free $2m$ diagram denote the free closure of the $2m$-tangle referred to in the source. A free diagram with $m$ crossings is considered sufficiently complicated as in the sou…

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…

Let $K$ be a framed four-valent graph, viewed as a free knot diagram, and let its girth be the length of its shortest cycle. If … then the unique minimal diagram conjecture. The di…

A free knot is a one-component free link: an equivalence class of framed $4$-valent graphs under the three Reidemeister moves, where at each vertex the four incident half-edges are…

A free knot is a homotopy class of Gauss words, equivalently a free knot diagram considered up to the corresponding free-knot moves. Turaev's conjecture. Every free knot is trivial…