On amphicheiral knots

Abstract

Amphicheiral knots with up to 12 crossings are discussed from the perspective of their symmetry properties. By use of an algorithm that involves the development of appropriate vertexbicolored knot graphs, rigidly achiral presentations have been found for all amphicheiral invertible prime knots with up to 10 crossings and for a selected number of such knots with 12 crossings, including 121994, the first example of an amphicheiral prime knot whose S2n diagram is also a reduced diagram. Characteristic properties of wire models of these presentations have been examined. The adjacency matrices of the vertex-bicolored graphs of amphicheiral knots exhibit twofold antisymmetry, and, with the sole exception of 12427, all such knots are capable of rigidly antisymmetric presentations.

A left-right classification of topologically chiral knots

“…3 illustrates this principle for a few selected prime knots. This equivalence is also expressed in the twofold antisymmetry of the corresponding adjacency matrix and in the condition that the derived polynomial satisfies e(t) = e(t -1) [12].…”

“…twofold rotation combined with a transposition of colors [12]). The positive and negative component ei(p)'s of all Wp'S in the knot's standard diagram cancel exactly, so the Wp'S are zero for all values ofp.…”

“…A knot is amphicheiral if and only if the vertex-bicolored graph [11] of at least one of its reduced diagrams is composed of equivalent black and white subgraphs, inclusive of connectivities [12]; we refer to this as the standard diagram. Note that, according to this criterion, the diagrams of PI#P~ or P2#P~, but not of P1 #P~ or P~#P2, qualify as "standard".…”

Chiral: A confusing etymology

“…Because Kelvin probably had the Greek word in mind, it is also curious that without explanation he dropped the letter e from the spelling of cheiral. As Liang and Mislow 9 and Mislow 10 have noted, there is a further connection with Tait. Earlier work by Kelvin on vortex atoms had stimulated Tait to work on knots leading to his use of the word, ''amphicheiral'' to describe a ''twofold asymmetry.…”

Braids for Twist Knots

“…The amphicheiral [5,14,18,19] knot 63 is not a twist knot, it has Conway notation label [13] of a form distinct from that of the amphicheiral knot 83, and has a distinct form of braid word [2,3]:…”