4 problems

Matched-diagram conjecture. (i) Not every link has a matched diagram. (ii) Every oriented link is $t_3$-move equivalent to a link with a matched diagram.

Tait's conjecture. A reduced alternating diagram of $L$ has minimal crossing number, and if $L$ is prime then no non-alternating diagram has minimal crossing number.

Let $L$ be a link in $S^3$ with a diagram that is prime, twist-reduced, and $3$-highly twisted, and that has at least two twist regions. Hyperbolicity conjecture. The link $L$ is h…

Let $D(K)$ be a semi-adequate diagram of a satellite link $K$. Visible satellite torus conjecture. Either $D(K)$ or its mirror image can be obtained using the construction describe…