# 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 can be unknotted with null-homologous twists and that there exist genus knots that cannot be unknotted with fewer than null-homologous twists.

## 1. Introduction

Figure 1 illustrates an operation that can be performed on a knot, twisting a set of parallel strands. In this example, orientations are shown and the linking number of the twist is one in absolute value. In 1994, Ohyama [MR1297516] proved the unexpected result that every knot can be unknotted with two twists. Here we will give an alternative perspective on the proof of this theorem and use this to show that every knot of three-genus can be unknotted using null-homologous twists, that is, with all twists having linking number 0. It will also be shown that is the best possible bound.

There is extensive literature concerning the general problem of unknotting. Some of this research concerns the question of which knots can be unknotted with a single operation that consists of performing many full twists with no constraint on the linking number; a small sample of papers on this topic includes [MR1177414, MR2233011, MR1282760, MR1194998]. More generally, one can consider an equivalence relation on knots generated by twisting, with constraints on the linking number and number of twists; the starting points of research on this topic is a paper by Fox [MR0131871], and a few later papers include [MR3544442, MR881755, MR1049833].

This work is closely related to the study of classes of unknotting operations; a good initial reference is [MR1075165]. Our approach is based on simplifying a Seifert surface for the knot; one can also consider the problem of simplifying closed surfaces and compact three-manifolds embedded in using twisting operations. Research along these lines includes [MR3676048, MR3807598], in which similar techniques to those used here appear. Geometric operations on a knot correspond to algebraic operations on its Seifert matrix, and this in turn leads to the notion of algebraic unknotting. See, for instance, [MR1043226, MR1727885, 2017arXiv170910269B].

Our focus on linking number 0 arose because of its relationship to problems in four-dimensional aspects of knot theory: if a knot can be unknotted with positive and negative null-homologous full twists, then bounds a null-homologous disk embedded in the punctured connected sum . This is a topic that has received considerable attention; a few early references include [MR603768, MR0246309, MR1195083, MR0248851, MR852974, MR561245]. From the four-dimensional perspective, this in turn leads to the question of converting a knot into a knot with Alexander polynomial 1 (which would then be topologically slice [MR679066]); this is explored, for instance, in [2017arXiv170910269B].

Acknowledgement In [MR1936979], the main result of this paper was stated without proof. Recently, Maciej Borodzik and Lukas Lewark, who needed the result, noted that a follow-up paper promised in [MR1936979] never appeared. Maciej and Lukas also had outlined their own proof of the desired result. Here we present the argument. Thanks are also due to Makoto Ozawa and Akira Yasuhara for valuable comments and for identifying important references.

## 2. Surgery descriptions of knots and unknotting

Throughout this paper, we will use surgery descriptions of knots, links, and their cyclic covering spaces. A basic reference is the text by Rolfsen [MR0515288, Chapter 9H]. More details can be found in [MR1707327] and original sources such as [MR0467753].

An –component framed link in a three-manifold is the isotopy class of an oriented embedded copy of the disjoint union of copies of in along with the choice of an isotopy class of a nowhere vanishing section of the normal bundle to . For oriented links in , linking numbers define a natural correspondence between framings on each component and integers.

In the introduction we described the general twisting operation. We now make this formal in a way that facilitates the proof of the unknotting theorem.

A surgery diagram for a framed –component link in a three-manifold consists of an oriented link in for which each component has an integer framing; it should have the property that is a surgery description of and represents the framed link . The framing of is denoted .

For our work, the key result concerning modifications of surgery diagrams states that two such diagrams represent isotopic links in if they are related by a sequence of three types of moves along with the inverse of the second move: (1) diagram isotopy; (2) removing an unknotted component with framing and adding a full twist to the remaining strands of that pass through , twisting left or right depending on whether or , respectively; (3) sliding a component of over an (other than itself), adjusting framings appropriately. If , the second move, blowing down , decreases the framing of each remaining component by the square of the linking number; if , then it increases framings by the square of the linking number. Details are presented in the references; in the work below we will summarize how framings can be tracked using associated linking matrices.

###### Definition 2.1.

A link can be unlinked with twists if there is a surgery diagram of , in which both and are unlinks. The unlinking is called null-homologous if each is null-homologous in the complement of .

Note that the choice of framing for is not relevant in this definition. Also note that in a surgery diagram for for which is an unlink, all framings are necessarily . Thus, this definition of unlinking corresponds to the definition that involves applying twists to a standard diagram of .

## 3. The Light Bulb Trick

A well-known result, called The Light Bulb Trick, states that any two (unoriented) knots in , each of which meets a nonseparating two-sphere in exactly one point, are isotopic. There is a simple generalization for connected sums of which quickly yields the following result. To set up notation for the proof, in the statement of the theorem we make precise the notion that a subset of forms a set of meridians for some of the .

###### Theorem 3.1.

Let be a link with surgery diagram . Suppose that for some , the link has all framings 0 and bounds a set of disjoint disks in such that consists of exactly one point, and that intersection point is on . Then has a surgery diagram

for which is an unlink bounding a set of disjoint disks with interiors in the complement of . Furthermore, it can be arranged that if is an arbitrary set of integers satisfying , then

###### Proof.

The isotopy is constructed by repeatedly sliding elements of over elements of . On the left in Figure 2, a schematic diagram of a portion of the diagram for is presented. In this portion of the diagram, it is possible that . In the full diagram, it is also possible that some of the for intersect the disks nontrivially. What is essential is that full intersection of with consist of only the one point, .

The schematic on the right in Figure 2 illustrates the effect of sliding over ; recall that has framing 0. The effect is to change one crossing between and . If , then the framing of is changed by . (It is also possible that the linking of with some for is changed, but the resulting link diagram continues to satisfy the initial conditions of the theorem.) It follows that arbitrary crossing changes between components of and components of can be performed so that in the new link, which we denote , we have that is an unlink separated from . Note that since no crossing changes between components of were performed, .

Once the link is constructed, for a Reidemeister move I can be preformed to to put a small left-handed kink in the diagram. That crossing can be changed to be right-handed by sliding over . This does not change the link type of , but it changes the framing of by 2. Thus, this move and its inverse can change all framings by arbitrary multiples of 2.

∎

## 4. Unknotting knots with a single pair of twists

We begin by considering unknotting a single knot, offering a new perspective on Ohyama’s theorem. Our approach keeps track of framings.

###### Theorem 4.1 (Unknotting Theorem).

Any knot in can be unknotted with two twists of opposite sign. If has framing , then in the diagram in which is unknotted, it can be arranged that has framing for any integer satisfying . The absolute values of the linking numbers of the two unknotting curves with differ by 1.

###### Proof.

Figure 3 is a schematic framed link diagram of a knot in . Sliding the framed component over the framed component, and then sliding over the framed component yields a diagram as shown on the right in Figure 3.

The framing/linking matrix is given by

If we denote this surgery diagram for by , with denoting the 0–framed surgery component, then it satisfies the conditions of Theorem 3.1, and thus we can unknot . If the number of slides of over is (algebraically) , the linking matrix for the resulting link is given by

Sliding the –framed curve over the –framed surgery curve converts into an unlink with framings and . The resulting framing/linking matrix becomes

As in the proof of Theorem 3.1, by placing kinks in and then sliding it over , the choice of is seen to be arbitrary.

∎

## 5. Null-homologous unknotting

###### Theorem 5.1.

If , then can be unknotted with null-homologous twists.

###### Proof.

Let be a Seifert surface for built from a disk by adding pairs of bands. Then can be illustrated as in Figure 4. Each of the bands is drawn with a small gap in it, indicating that the bands are perhaps knotted, linked together, and twisted. There is a symplectic basis of represented by simple closed curves built from the cores of those bands. We denote these curves by , as shown.

Let denote the negative push-offs of the from . Thus, in Figure 4 the visible portion of lies under . If the link forms an unlink with all Seifert framings 0, as shown in Figure 5, and that unlink lies everywhere over , then is an unknot: the Seifert surface can be ambiently surgered to form a disk. Figure 5 is drawn to shown the forming a trivial link along which can be surgered. The bands are still drawn broken, to indicate that they are perhaps knotted, twisted, and link, but it is assumed they pass everywhere under the bands.

Thus, to unknot , we can proceed as follows. Introduce pairs of two component unlinks in the complement of and perform and surgery on each. This is illustrated in Figure 6. As in the proof of the unknotting theorem, Theorem 4.1, we can slide each curve and an curve (and its corresponding band) over a surgery curve to arrange that each band has a small linking circle on which 0–surgery is performed. We are now in the setting of Theorem 3.1, with the set corresponding to the set . However, instead of sliding the , we work with the corresponding bands.

As in the earlier arguments, a sequence of slides (of the bands) over the –framed linking curves can ensure that: (1) each curve is unknotted; (2) the set of curves forms an unlink, and finally; (3) by sliding curves over the –framed surgery curves, that the link formed from the curves is split from the link formed from the curves, lying completely beneath it.

It remains to arrange that the curves all have framing 0. In Theorem 4.1, we saw that the framings of the can be changed by any odd integer. Thus, if the initial Seifert pairing of each curve were odd, then we would be done. Consider each pair . If either has framing odd, perhaps after switching the labels, we would be able to proceed. If both are even, we can consider a new symplectic basis . Letting denote the Seifert form, we have

By using a symplectic basis in which is one of the basis elements, we have arranged that it has odd framing. Thus, the argument is complete.

∎

## 6. Converse

###### Theorem 6.1.

For every , there exists a genus knot that cannot be unknotted with fewer than null-homologous twists.

###### Proof.

If can be unknotted with null-homologous twists, the first homology of the infinite cyclic cover of , , has rank at most as an –module for any field . (See [MR0515288, Chapter 7C].) In general, this homology group is presented by , where is a Seifert matrix for . If is a genus one knot with Seifert matrix

then letting be the field with three elements,

as an –module. This is of rank 2.

The knot is now seen to have of rank as an –module, and thus it provides the necessary example to conclude the proof. ∎

## 7. Concluding remarks

There is an interesting point of overlap between our null-homologous unknotting result and Ohyama’s original theorem. That is in the case of . For such a knot, Ohyama says it can be unknotted with a positive and negative twist with linking numbers for some integer . Our result says that it can be unknotted with such twists of linking numbers . It is not difficult to modify a proof of Ohyama’s theorem to show that for every knot and for every value of , there is a unknotting of .

One question is to determine for each knot , for which pairs there is an unknotting with these linking numbers. Notice that for any integers and , the connected sum of torus knots can be unknotted with a pair of twists of linking numbers , with the signs of the twists determined by the signs of and . On the other hand, for any given knot, there are potential obstructions to unknotting with linking numbers any given pair . If , it seems possible that the condition is . These issues will be the subject of further research.

Such questions can be considered from a four-dimensional perspective. According to [MR0246309, MR0248851], every knot bounds a smoothly embedded disk in , and by Ohyama’s result, this can be improved: for every integer , any knot bounds a disk in that represents . If , then it also bounds a disk representing . Again, what other possibilities can occur?