Minimal generating sets of directed oriented Reidemeister moves (1601.00559v2)

Abstract: Polyak proved that the set ${\Omega1a,\Omega1b,\Omega2a,\Omega3a}$ is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining $32$ different types of moves, which we call directed oriented Reidemeister moves. In this article we prove that the set of $8$ directed Polyak moves ${ \Omega{1a}^\uparrow, \Omega{1a}^\downarrow, \Omega{1b}^\uparrow, \Omega{1b}^\downarrow, \Omega{2a}^\uparrow, \Omega{2a}^\downarrow, \Omega{3a}^\uparrow, \Omega{3a}^\downarrow }$ is a minimal generating set of directed oriented Reidemeister moves. We also specialize the problem, introducing the notion of a $L$-generating set for a link $L$. The same set is proven to be a minimal $L$-generating set for any link $L$ with at least $2$ components. Finally, we discuss knot diagram invariants arising in the study of $K$-generating sets for an arbitrary knot $K$, emphasizing the distinction between ascending and descending moves of type $\Omega3$.