Null-homologous unknottings

Abstract

Every knot can be unknotted with two generalized twists; this was first proved by Ohyama.Here we prove that any knot of genus g can be unknotted with 2g null-homologous twists and that there exist genus g knots that cannot be unknotted with fewer than 2g null-homologous twists.

Slicing knots in definite 4-manifolds

“…The minimum number of generalized crossings one must add to the unknot to construct a given knot is called its untwisting number, and was first defined in [MD88]. The untwisting number of knots was investigated in [Liv02,CT14,İnc16,İnc17,McC19,McC21,Liv19]. It is straightforward to see that if a knot is obtained from a slice knot by adding m positive generalized crossings, then it is slice in # m CP 2 [CT14, Lemma 2.8].…”

“…In each case, the knot can be changed to a ribbon knot by a negative to positive crossing change, and is therefore H-slice in CP 2 . Now apply Corollary 5.6 and the result of [Liv02,Liv19], that every knot of genus one can be converted to the unknot by two generalized crossing changes (one positive to negative and one negative to positive).…”

“…According to [Liv02,Liv19], every knot with Seifert genus one can be converted to the unknot via two generalized crossing changes (one positive to negative and one negative to positive), and hence is H-slice in CP 2 #CP 2 . We give some new examples of knots K, specifically 3strand pretzel knots, for which we determine the set X(K) precisely, in Corollary 5.7.…”

Null-homologous twisting and the algebraic genus

“…In order to complete the proof we need to show that there are always pairs of null-homologous twisting operations that decrease the algebraic genus. This can be done by adapting the argument used by Livingston to prove that any knot can be converted to the unknot using at most 2g null-homologous twists [Liv19].…”

Untwisting 3‐strand torus knots

“…We will not see any of these surfaces. Rather, we will find a precise upper bound for the topological 4-genus via an operation called null-homologous twisting, which has recently received some attention [6,8,10,11]. A null-homologous twist is an operation on oriented links that inserts a full twist into an even number 2m of parallel strands, m of which point upwards, and m of which point downwards (see e.g.…”