# Gaps between consecutive untwisting numbers.

###### Abstract.

For one can define a generalization of the unknotting number called the th untwisting number which counts the number of null-homologous twists on at most strands required to convert the knot to the unknot. We show that for any the difference between the consecutive untwisting numbers and can be arbitrarily large. We also show that torus knots exhibit arbitrarily large gaps between and .

## 1. Introduction

Given a knot in , we perform a null-homologous twist by taking an unknotted curve disjoint from with and performing -surgery or -surgery on . If bounds an embedded disk intersecting transversely in points, then we call this a null-homologous twist on strands. Such a twist can always be performed locally by adding a full twist on parallel strands with appropriate orientations. An example of a null-homologous twist on four strands is shown in Figure 1.

Ince used null-homologous twisting operations to define an infinite sequence of generalizations to the unknotting number [Inc16]. For a knot the th untwisting number, denoted , is the minimum number of null-homologous twists on at most strands required to convert to the unknot. Since a null-homologous twist on two strands is equivalent to a standard crossing change, coincides with the classical unknotting number. One may also define the untwisting number by . Clearly the untwisting numbers form a decreasing sequence:

The main purpose of this article is to show that the difference between consecutive pairs of untwisting numbers can be arbitrarily large.

###### Theorem 1.

For any pair of positive integers and , there is a knot such that

The gaps between untwisting numbers have previously been studied by Ince, who showed that the gap between and can be arbitrarily large [Inc16]. Ince also considered the separation between higher untwisting numbers, showing for example that for any the gap between and can be arbitrarily large (cf. [Inc17, Example 6.5]). Our examples are similar to those studied by Ince, however we are able to establish stronger results through better lower bounds on . These lower bounds are provided by relating and the smooth slice genus :

(1) |

Here denotes smooth slice genus of . For fixed , the lower bound in (1) turns out to be optimal as the knots used to prove Theorem 1 will be knots attaining equality in (1).

Whilst (1) shows that the admit lower bounds based on the smooth slice genus, these lower bounds do not yield any information about . It turns out that one can obtain lower bounds on using the topological slice genus:

(2) |

This can be seen from results of Ince [Inc16], who used the work of Borodzik and Friedl [BF14, BF15] to show that . Alternatively one can establish (2) using the concept of algebraic genus [McC19].

Given that the unknotting numbers of torus knots were notoriously hard to compute, it is natural to wonder what one can say about the behaviour of untwisting numbers for torus knots. For torus knots with braid index at least four the untwisting number, is strictly smaller than the unknotting number.

###### Theorem 2.

If , then . Furthermore, for any we have

(3) |

Since the unknotting number satisfies , it follows that for torus knots the difference between and grows arbitrarily large as the braid index increases. For torus knots with braid index two, i.e those of the form we have . This follows from the fact that the classical knot signature provides a lower bound for and hence for . The same reasoning shows that and . However, for the remaining torus knots of braid index three understanding their untwisting numbers seems much more challenging.

## 2. Unbounded gaps

First we prove the following proposition, which implies (1).

###### Proposition 3.

If and are knots related related by a null-homologous twist on strands, then

###### Proof.

We observe that a null-homologous twist on strands can be accomplished by oriented band moves. This can be proven by induction on . Consider a full twist on strands with strands oriented up and strands oriented down. Such a twist can be arranged as a full twist on strands with two more strands, one oriented up and the other down, “wrapping around” the full twist as in the left hand side of Figure 2. As illustrated in Figure 2 one can perform two oriented band moves and isotopies to produce a full twist on strands with two parallel strands alongside. Thus, proceeding inductively, we see that the full twist on strands can be converted to parallel strands by oriented band moves.

Thus if and are related by a null-homologous twist on strands, then there is a sequence of oriented band moves and isotopies that convert into . These moves allow one to construct a smoothly embedded surface of genus properly embedded in so that . Thus

as required.

[width=0.95]band_addition