The $(2,1)$-cable of the figure-eight knot is not smoothly slice

Abstract

We prove that the (2, 1)-cable of the figure-eight knot is not smoothly slice by showing that its branched double cover bounds no equivariant homology ball.

Equivariant knots and knot Floer homology

“…The authors do not believe that knot Floer homology detects these examples; it is still unclear whether 𝔎 𝜏,𝜄 is abelian. (2) Building on the Floer-theoretic formalism of the present work (in particular, Theorem 4.3), the authors of this paper (in joint work with Kang and Park) have recently shown that the (2,1)-cable of the figure-eight is not slice [9]. This was previously an open question, and as such may provide some motivation for the extensive framework we establish here.…”

Ribbonness on Classical Link

“…However, in I. Dai, S. Kang, A. Mallick, J. Park and M. Stoffregen showed that it is not a slice knot [3]. In this paper, the author comes back to elementary research beginning point on the difference between a slice knot and a ribbon knot [4].…”

“…The upper-half 4-space R 4 + is denoted by R 3 [0,+∞). Let k be a link in the 3-space R 3 , and F a proper oriented surface in the upper-half 4-space R 4 + with ∂F = k. Let b j (j = 1,2,...,m) be finitely many disjoint oriented bands spanning the link k in R 3 , which are regarded as framed arcs spanning k in R 3 . Let k′ be a link in R 3 obtained from k by surgery along these bands.…”

“…for disk systems d, d′ in R 3 with ∂d = o and ∂d′ = o′. A modified semi-closed realizing surface scl(F u s ) + of the band surgery sequence Every band surgery operation k → k ′ along a band system b is realized a surface F u s in R 3 [s, u] for any interval [s, u], as follows (see [10]):…”

Ribbonness of a stable-ribbon surface-link, I. A stably trivial surface-link

“…For a long time, the author has considered the (2,1)-cable of the figure-eight knot, which is not ribbon but rationally slice, as a candidate for a non-ribbon knot which might be slice (see [4,5]). However, in [1], I. Dai, S. Kang, A. Mallick, J. Park and M. Stoffregen showed that it is not a slice knot.…”