5 problems

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Harikae–Nakanishi–Uchida conjecture on $(2,2)$ -move equivalence
Harikae–Nakanishi–Uchida conjecture. Every link is $(2,2)$ -move equivalent to a trivial link.
openknot-theory link-equivalence tangle-moves
04
Przytycki's 4-move conjecture
Przytycki's 4-move conjecture. Any knot is 4-equivalent to the unknot.
openknot-theory link-equivalence unknotting
05
The 11-coloring-preserving $(2,5)$ and $(4,-3)$ move conjecture
A rational $(m,q)$ -move is the local move corresponding to the indicated rational tangle, and its inverse and mirror-image moves are also allowed. The Fox $11$ -coloring space of a…
openknot-invariants knot-theory link-equivalence
04
Harikae–Nakanishi $(2,2)$ -move conjecture for links
A $(2,2)$ -move is the local link modification shown in Figure 1.10; two links are $(2,2)$ -move equivalent if they are related by such moves, their inverses, and Reidemeister moves.…
openknot-theory link-equivalence
04
Nakanishi's 4-move conjecture for knots
A 4-move is the local operation on a link diagram that replaces four consecutive half-twists between two strands by the opposite four half-twists. Nakanishi's 4-move conjecture. Ev…
openknot-theory link-equivalence
012

Harikae–Nakanishi–Uchida conjecture on (2,2)(2,2)(2,2)-move equivalence