Involutions of alternating links

Equivariant 4‐genera of strongly invertible and periodic knots

“…This was also shown in[34, Appendix II] and[7, Theorem 1.11] 2. The lower bound g4 ⩾ 3 is obtained by Proposition 3.10, and the upper bound g4 ⩽ g 3 = 3 is obtained by Edmonds' theorem[11] 3.…”

“…† An alternating knot with a periodic symmetry of order > 2 always has an alternating diagram in which the symmetry is visible; see [5] or [1]. For symmetries of order 2, the situation is more complicated [2]. ‡ We follow the convention for Montesinos knots notation from [19].…”

Khovanov homology of strongly invertible knots and their quotients

“…Like Stoffregen-Zhang's and Borodzik-Politarczyk-Silvero's spectral sequence for periodic knots, the spectral sequence we construct for strongly invertible knots relates the Khovanov homology of K to the annular Khovanov homology of its quotient K. If τ denotes the 180 • rotation around the axis then by a slight τ -equivariant perturbation of the knot K we may assume that the projection of K to the plane perpendicular to the axis is a knot diagram. Following the literature [Boy21,BI], we call such a diagram intravergent. The quotient knot K may be viewed as an annular knot in two natural ways, K 0 and K 1 , corresponding to taking the quotients of the 0-resolution or the 1-resolution of the fixed crossing of the intravergent diagram K; see Figure 5.2.…”