2·1. Background

A 1-link is a link in $ S^3 $ , and 1-link of one component is a 1-knot. A 2-knot is an embedding $ S^2 \hookrightarrow S^4 $ , and a surface-knot is an embedding $ F \hookrightarrow S^4 $ for F a closed orientable surface. A surface-knot is trivial if it bounds an embedded handlebody in $ S^4 $ . All of the above objects are considered up to ambient isotopy.

Henceforth we denote by $ S^3_0 $ an equator in $ S^4 $ , and by $ B^4_+ $ , $ B^4_- $ the associated hemispheres (so that $ S^4 = B^4_+ \cup_{S^3_0} B^4_- $ ).

Every 1-knot divides a surface-knot, and this relationship has been of great interest to low-dimensional topologists for almost a century. Questions on this relationship are broadly of two kinds. First, given a fixed 1-knot how complex are the surface-knots that it divides? Obversely, for surface-knots of a given complexity what are the 1-knots that divide them? The focus of this paper is a question of the second type: studying the 1-knots that divide ribbon surface-knots.

Before presenting formal definitions of our main objects of interest we fix some further terminology.

This Morse-theoretic definition of a ribbon surface is equivalent to the definition via surfaces immersed in $ S^3 $ with ribbon singularities (see, for example, [ Reference Kamada11 , Lemma 11·9]). Note that slice surface and surface-knot are distinct concepts, likewise ribbon surface and ribbon surface-knot.

Given a slice surface F for a 1-knot K we may form a surface-knot divided by K as follows. Regard K as lying in an equator $ S^3_0 $ and F in $ B^4_+ $ . Denote by $ \overline{F} $ the surface obtained by reflecting F through $ S^3_0 $ . The surface-knot $ F \cup_K \overline{F} $ is known as the double of F, and K divides it by construction.

An embedded 2-sphere in $ S^4 $ is standard if it is in Morse position with exactly two critical points (of index 0 and 2 necessarily). An isotopy representative of a surface-knot is a sphere-tube presentation if it is obtained by attaching 3-dimensional 1-handles to a disjoint union of standard 2-spheres. A surface-knot is ribbon if and only if it possesses a sphere-tube presentation [ Reference Kamada12 , Section 5·6].

We frequently make use of the fact that the double of a ribbon surface for K is a ribbon surface-knot divided by K. This is described in [ Reference McDonald19 , Figure 2] and the related discussion (for full details see, for example, [ Reference Kamada12 , Section 5]). We suffice ourselves by observing, as in Figure 1, that a ribbon surface is made up of discs and bands; upon doubling discs become trivial 2-spheres and bands become 1-handles in a sphere-tube presentation of the double.

Fig. 1.

On the left: a ribbon surface F formed of discs and bands. On the right: the induced sphere-tube presentation of the double of F.

2·2. Dividing spheres

We say that a 1-knot K is slice if it divides a 2-knot [ Reference Artin2, Reference Fox and Milnor7 ], and that K is doubly slice if it divides the trivial 2-knot [ Reference Fox6, Reference Sumners24 ].

Using Definition 3 we introduce a notion that lies between slice and doubly slice.

Notice that as the trivial 2-knot is ribbon it follows that a doubly slice 1-knot is half ribbon (the converse fails, for example, on the 1-knot $ 6_1 $ ). Half ribbon knots are of course slice, but the status of the converse is an open question (as outlined in Question 1).

Recall that a 1-knot K is ribbon if it bounds a ribbon surface of genus 0; such a surface is known as a ribbon disc for K.

Thus doubly slice implies slice but the converse is false, and ribbon implies half ribbon but the converse may be false. In other words, ribbon is to the property of dividing a ribbon 2-knot as doubly slice is to slice, whence the name half ribbon.

Proposition 6 shows that the slice-ribbon conjecture splits into the following questions.

2·3. Dividing surfaces

The slice genus of a 1-knot K, $ g_4 ( K ) $ , is the minimum genus of a slice surface for K. The ribbon genus of K, $ g_r ( K ) $ , is the minimum genus of a ribbon surface for K.

Notice that taking the double of a slice surface for K yields a surface-knot divided by K. Generically this surface-knot will be nontrivial. Restricting to trivial surface-knots (not necessarily doubles of slice surfaces for 1-knots) yields the double slice genus of K, $ g_{ds} ( K ) $ , the minimum genus of a trivial surface-knot divided by K [ Reference Livingston and Meier17 ].

Just as in Section 2·2 we can use Definition 3 to define a quantity intermediate to the slice and double slice genera.

Of course, K is of half ribbon genus zero if and only if it is half ribbon. The half ribbon genus is finite for all 1-knots as it is bounded above by the double slice genus and twice the ribbon genus.

Proposition 8. Let K be a 1-knot, then

\begin{equation*} {}2 g_4 ( K ) \leq g_{hr} ( K ) \leq 2g_{r} ( K ) \leq g_{ds} ( K ). {}\end{equation*}

Proof. Let $ K_1 $ , $ K_2 $ be 1-knots and $ S_1 $ , $ S_2 $ ribbon surface-knots such that $ K_i $ divides $ S_i $ . Denote by S the split union of $ S_1 $ and $ S_2 $ . That is, S is a disjoint union of $ S_1 $ and $ S_2 $ , and there exists a 4-ball B such that $ S_1 \cap B = S_1 $ , $ S_2 \cap B = \varnothing $ .

Let S’ be the result of adding a 1-handle between the components of S, chosen so that its core intersects $ \partial B $ in exactly one point and the connected sum of $ K_1 $ and $ K_2 $ appears as the equatorial cross-section.

As $ S_1 $ and $ S_2 $ are ribbon they possess sphere-tube presentations. Let $ S^{\prime\prime} $ be the surface-knot obtained by first isotoping $ S_1 $ and $ S_2 $ into such presentations and then adding a 1-handle between them, the core of which intersects $ \partial B $ in exactly one point. (Notice that the isotopy taking $ S_1 $ and $S_2$ to their sphere-tube presentations may be chosen to fix $ \partial B $ pointwise.) A schematic for the split union of $ S_1 $ and $S_2$ , together with S’ and $ S^{\prime\prime} $ , is given in Figure 2.

Fig. 2.

Schematic diagrams of the split union of $ S_1 $ and $S_2$ , and the surface-knots S’ and $ S^{\prime\prime} $ .

It follows that $ S^{\prime\prime} $ is a ribbon surface-knot as it possesses a sphere-tube presentation. By [ Reference Kamada12 , Proposition 1·2·11] the surface-knots S’ and $ S^{\prime\prime} $ are identical. Thus S’ is a ribbon surface-knot divided by $ K_1 \# K_2 $ of genus $ g(S_1) + g(S_2) $ , so that $ g_{hr} ( K_1 \# K_2 ) \leq g_{hr} ( K_1 ) + g_{hr} ( K_2 ) $ .

W. Chen constructed the first examples of 1-knots of arbitrarily large double slice genus [ Reference Chen4 ]. These 1-knots are ribbon and are therefore of half ribbon genus zero by Proposition 6.

Let $ \overline{K} $ denote the mirror image of a 1-knot K. Orson and Powell showed that the knot $ J = \left( \#^{\frac{M}{2}} \overline{5_2} \right) \# \left( \#^{N-M} 8_{20} \right) $ satisfies $ 2 g_4 ( J ) = M $ and $ g_{ds} ( J ) = N $ , for all integers $ 0 \leq M \leq N $ with M even [ Reference Orson and Powell21 ]. As $ 2 g_4 \left(\overline{5_2}\right) = g_{ds} \left(\overline{5_2}\right) = 2 $ and $ 8_{20} $ is ribbon we have that $ g_{hr} \left(\overline{5_2}\right) = 2 $ and $ g_{hr} ( 8_{20} ) = 0 $ by Proposition 8, and $ g_{hr} ( J ) = M $ by Proposition 9. It follows that the half ribbon and doubly slice genera differ by an arbitrarily large amount. The analogous question regarding the slice genus remains open.

Question B. Given integers $ 0 \leq M \leq N \leq P $ does there exist a 1-knot K such that

\begin{equation*} {}2 g_{4} ( K ) = M, \; g_{hr} ( K ) = N, \; g_{ds} ( K ) = P\text{?} {}\end{equation*}

Does there exist a prime 1-knot with this property?

Answering Question 2 for all M, N, P is at least as hard as finding a 1-knot with distinct slice and ribbon genera.

In Section 3 we determine the half ribbon genus of all 1-knots up to 12 crossings to be even.

Specialising further, establishing the existence of 1-knots of half ribbon genus one would resolve the slice-ribbon conjecture in the negative.

Satoh defined a surjective map from the category of welded knots to that of ribbon tori [ Reference Satoh23 ]. Can this map be used to address Question 4?

2·4. Fusion numbers

In Sections 2·2 and 2·3 we consider the problem of minimising the genus of a surface-knot divided by a given 1-knot. In this section we consider minimising the following alternative measure of complexity.

Let K be a ribbon 1-knot. The fusion number of K, f (K), is the minimum number of bands in a ribbon disc for K. Let S be a ribbon 2-knot. The fusion number of S, f (S), is the minimum number of 1-handles in a sphere-tube presentation for S.

For half ribbon 1-knots we define a new quantity in terms of the fusion number of the ribbon 2-knots they divide.

Note that $ f ( S ) = 0 $ if and only if S is a trivial 2-knot, so that $ f_h ( K ) = 0 $ if and only if $ g_{ds} ( K ) = 0 $ . The proof of Proposition 9 also establishes the subadditivity of the half fusion number with respect to the connected sum of 1-knots.

Orson and Powell showed that the Levine–Tristram signatures bound the double slice genus from below [ Reference Orson and Powell21 ]. The same is true of the half fusion number.

Theorem 11. Let K be a half ribbon 1-knot. Then

\begin{equation*} {}f_{h} ( K ) \geq \underset{{\omega \in S^1 \smallsetminus \lbrace 1 \rbrace}}{\text{max}} \left\lbrace |\sigma_{\omega} (K)| \right\rbrace. {}\end{equation*}

Proof. Let S be a ribbon 2-knot divided by K such that $ f ( S ) = f_h ( K ) $ . By adding f (S) 1-handles to S it can be converted into a trivial surface-knot S’ [ Reference Miyazaki20 ]. Moreover, as the result of adding a 1-handle depends only on its core [ Reference Kamada12 , Proposition 5·1·6] these handles may be chosen so that $ K \cup U $ divides S’, for $ K \cup U $ the split union of K and an unlink U. Generically the handles may intersect the equatorial $ S^3 $ so that we cannot avoid the appearance of this unlink, as per the schematic given in Figure 3.

Fig. 3.

Attaching 1-handles with cores given by the dashed arcs introduces an unlink to the equatorial cross-section.

The genus of S’ is equal to f (S) and bounds from above the weak double slice genus of $ K \cup U $ , denoted $ g^1_{ds} ( K \cup U ) $ [ Reference Conway and Orson5 , Equation 1]. Conway and Orson showed that the Levine–Tristram signatures bound this quantity from below [ Reference Conway and Orson5 , Corollary 1·3], so that

\begin{equation*} {}\left| \sigma_{\omega} ( K \cup U ) \right| \leq g^1_{ds} ( K \cup U ) \leq g ( S' ) = f_h ( K ). {}\end{equation*}

The proposition follows by the additivity of the signature under disjoint union.

In the proof above the surface-knot divided by K is trivialised by attaching $ f_h ( K ) $ 1-handles. However, this does not allow us to conclude that $ g_{ds} ( K ) $ bounds $ f_h ( K ) $ from below, as we can guarantee only that $ K \cup U $ divides this trivial surface-knot.

Let D be a ribbon disc for a 1-knot K. The bands of D yield 1-handles in the double of D, that is a ribbon 2-knot divided by K. It follows that $ f_h ( K ) \leq f ( K ) $ . There exist ribbon knots whose fusion and half fusion numbers are arbitrarily far apart.

Proof. Juhaśz, Miller and Zemke defined an invariant of 1-knots using knot Floer homology, denoted $ \text{Ord}_v ( K ) $ , and proved that it bounds the fusion number of ribbon 1-knots from below [ Reference Juhász, Miller and Zemke10 , Corollary 1·7]. Denote by $ T_{p,q} $ the positive (p,q)-torus knot and let $ C_{p,q} = T_{p,q} \# \overline{T_{p,q}} $ . It is established in [ Reference Juhász, Miller and Zemke10 , Equation 1·7] that

\begin{equation*} {}\text{Ord}_v ( C_{p,q} ) = f ( C_{p,q} ) = \text{min} \left\lbrace p, q \right\rbrace - 1. {}\end{equation*}

Notice that $ f_h ( C_{p,q} ) = 0 $ as $ C_{p,q} $ is doubly slice. Let $ K_M = C_{p,q} \# \left( \#^M 8_{20} \right) $ . The 1-knot $ 8_{20} $ is chosen as for $ \omega = e^{\pi i / 3} $ we have $ \sigma_{\omega} ( 8_{20} ) = f_h ( 8_{20} ) = f ( 8_{20} ) = 1 $ . Observe that $ M = \sigma_{ \omega } ( K_M ) \leq f_h ( K_M ) $ by Theorem 11 and the additivity of the signature with respect to connected sum. That the half fusion number is subadditive with respect to connected sum implies that $ f_h ( K_M ) = M $ , in fact.

Further, applying [ Reference Juhász, Miller and Zemke10 , Equation 1·4] yields

\begin{equation*} {}\begin{aligned} {}\text{Ord}_v ( K_M ) &= \text{max} \left\lbrace \text{Ord}_v ( C_{p,q} ), \text{Ord}_v ( 8_{20} ) \right\rbrace \\[5pt] {}&= \text{min} \left\lbrace p, q \right\rbrace - 1 {}\end{aligned} {}\end{equation*}

so that $ \text{min} \left\lbrace p, q \right\rbrace - 1 \leq f ( K_M ) $ . A suitably large choice of p and q completes the proof.

Proposition 12 establishes that the set of ribbon 2-knots divided by a 1-knot K is in some sense distinct to the set of ribbon discs for K.

Note that as $ g_{ds} ( 8_{20 }) = 1 $ and the double slice genus is subadditive the proof of Proposition 12 shows that the difference $ f ( K ) - g_{ds} ( K ) $ can also be made arbitrarily large.

3·1. Handle attachments defined by ribbon cobordisms

Recent calculations of the slice and double slice genera have employed upper bounds obtained from various sequences of band attachments. Our calculations rely on the observation that if K divides a surface-knot S, attaching bands to K may be realised by adding 1-handles to S.

The superslice genus, $ g^{s} ( K ) $ , of a 1-knot K is the minimum genus of a slice surface for K the double of which is a trivial surface-knot [ Reference Chen4 ].

Picking J as the unknot in Corollary 14 (i), (ii) recovers [ Reference McDonald19 , Theorem 3·1]. It is unknown if attaching a 1-handle to a ribbon surface-knot preserves the ribbon property. This causes Corollary 14 (iii) to be of a different form to Corollary 14 (i), (ii). The result of attaching a 1-handle h to a surface-knot S depends only on the homotopy class of the core, $ \gamma $ , of h in the complement of S [ Reference Kamada12 , Proposition 5·1·6]. If S is ribbon it may be isotoped into a sphere-tube presentation. The trace of this isotopy is of codimension 1 so that it and $ \gamma $ generically intersect in points. It is therefore unclear if the isotopy can be completed in the presence of h.

Let K and J be 1-knots and C a cobordism between them. We say that C is a ribbon cobordism from K to J if we do not encounter a birth of an unknotted and unlinked component when traversing C from K to J. A ribbon concordance is a ribbon cobordism of genus 0.

Theorem 13 is similar to the following description of the union of a ribbon concordance with its reverse given by Zemke [ Reference Zemke26 ]Footnote ¹ . If C is a ribbon concordance from K to J then $ C \cup_K \overline{C} $ is obtained from $ J \times [0,1] $ by taking a disjoint union with trivial 2-knots and attaching them to $ J \times [0,1] $ via 1-handles. This operation is known as taking a tube sum with trivial 2-knots.

This description can be combined in a straightforward manner with Theorem 13 to obtain the following results. We expect these more general results to be useful in further study of the double slice and half ribbon genera.

Proof. (i)Suppose that J divides a trivial surface-knot S with $ g ( S ) = g_{ds} ( J ) $ . By Theorem 15 K divides a surface-knot, S’, obtained from S by attaching $ (s - d) $ 1-handles to S and taking d tube sums with trivial 2-knots. Denote by $ S_1 $ the result of attaching the $ (s - d) $ 1-handles to S. As S is trivial $ S_1 $ is trivial, and $ g ( S_1 ) = g ( S ) + s - d $ .

Denote by T the set of trivial 2-knots that will be tube summed to $ S_1 $ to produce S’. A 1-handle is trivial if it bounds an embedded $ D^1 \times D^2 $ . As the d components of T are formed by doubling a ribbon cobordism they may be connected with $ (d - 1) $ trivial 1-handles to produce a single trivial 2-knot, T’, without altering the equatorial cross-section. We may add an additional trivial 1-handle between $ S_1 $ and T’ to produce a trivial surface-knot, $ S_2 $ , also without altering the cross-section. Notice that $ g ( S_2 ) = g ( S ) + s - d $ .

Fig. 4.

On the left: bands defining a ribbon cobordism C. On the right: a schematic of $ C \cup \overline{C} $ (some handles have been isotoped away from the equator for aesthetic purposes).

Finally, consider the set of d 1-handles defined by the tube sums that produce S’. We may add these 1-handles to $ S_2 $ (as it includes both S and T) to produce a trivial surface-knot, $ S_3 $ , of genus $ g ( S_2 ) + d = g_{ds} ( J ) + s $ . That the trivial 1-handles above are attached without altering the cross-section ensures that K divides $ S_3 $ .

(ii) Let F be a ribbon surface for J with $ g ( F ) = g_{r} ( J ) $ . Then J divides the genus $ 2g_{r} ( J ) $ surface-knot S with sphere-tube presentation given by the double of F. By Theorem 15 K divides a surface-knot S ^′ obtained from S by attaching 1-handles and taking tube sums with trivial 2-knots. Thus S’ is ribbon as it also has a sphere-tube presentation, and $ g ( S ' ) = 2g_{r} ( J ) + s - d $ (only the $ (s - d) $ 1-handles attached to S affect the genus of S’).

Picking J as the unknot in Corollary 16 (i) recovers [ Reference McDonald19 , Theorem 3·2].

3·1. Determining genera

As employed by Lewark–McCoy and Brittenham–Hermiller the operations of switching a crossing, switching a pair of crossings of zero writhe, and taking the oriented resolution at two crossings are all realizable by attaching two bands [ Reference Lewark and McCoy15 , Lemma 5].

To calculate the half ribbon genus we combine specific examples of these operations found by Lewark–McCoy and Brittenham–Hermiller, calculations of the double slice genus by Karageorghis–Swenton, and Corollary 14.

For the 1-knots that Lewark–McCoy and Brittenham–Hermiller do not provide a suitable operation we made an independent computer search for crossing changes to ribbon or doubly slice 1-knots.

Proof of Theorem 1. The slice-ribbon conjecture has been verified up to 12 crossings. Therefore $ 2 g_4 ( K ) = g_{hr} ( K ) = 0 $ for all slice 1-knots.

If K is not slice and $ g_{ds} ( K ) $ was determined prior to this work then $ 2 g_4 ( K ) = g_{ds} ( K ) $ [ Reference Livingston and Moore18 ], so that $ 2 g_4 ( K ) = g_{hr} ( K ) $ also by Theorem 8.

This leaves 58 cases of undetermined half ribbon genus, as described in Table I (7 of the 65 cases undetermined by Karageorghis and Swenton are slice). These 1-knots are obtainable from a ribbon 1-knot by attaching two bands as shown by Lewark–McCoy [ Reference Lewark and McCoy15 , Appendix A], Brittenham–Hermiller [ Reference Brittenham and Hermiller3 , Section 4], or our crossing change search. Therefore these 1-knots have half ribbon genus equal to 2 by Corollary 14 and Proposition 8 (recall that they are not slice). An example of this process in the case of $ 10_{74} $ is given in Figure 5.

Table 1.

The second and third columns list the result of attaching two bands, as given by [ Reference Lewark and McCoy15 , Appendix A] and [ Reference Brittenham and Hermiller3 , Section 4], respectively. The fourth column lists the result of a crossing change, found by our computer search. If we were able to calculate a previously unknown value of $ g_{ds} $ it is listed in the fifth column.

Fig. 5.

A diagram of the 1-knot $ 10_{74} $ , together with three bands. Attaching the bands labelled A and B realises a crossing change that yields a diagram of the 1-knot $ 9_{46} $ . Subsequently attaching the band labelled C defines a ribbon disc for $ 9_{46} $ .

Finally, as depicted in Table 1 the 1-knots $ 9_{37} $ , $ 10_{74} $ , 11n148, 12a554, 12a896, 12a921, 12a1050, 11n148, 12n554 are obtainable from a doubly slice 1-knot by attaching two bands, so that they have double slice genus 2 by Corollary 14 (they were shown to have double slice genus 2 or 3 by Karageorghis and Swenton).

This leaves 57 prime 1-knots of up to 12 crossings with unknown double slice genus.

Our methods extend fruitfully into 13-crossing 1-knots. Specifically, we restricted to 13-crossing 1-knots of signature 2 and searched for crossing changes to ribbon or doubly slice 1-knots. This allows us to show that 2156 such 1-knots have half ribbon genus 2, of which 247 have double slice genus 2. The full results of these calculations are provided on the webpage https://www.mas.ncl.ac.uk/william.rushworth/ccdata.html and on the arXiv listing https://arxiv.org/abs/2209.15577.

Just as this paper studies 1-knots that appear as cross-sections of ribbon 2-knots, one could study those that appear as cross-sections of homotopy ribbon 2-knots. In addition to being possibly distinct to half ribbon in the smooth category, such a definition extends to the topological category. The most basic question one might ask in this setting is as follows.

Finally, although we do not pursue it here, the notion of half ribbon genus can readily be extended to allow for knotted surfaces of more than one component, as is done for the double slice genus [ Reference Conway and Orson5 , Equation 1]. There are natural generalisations of Theorems 13 and 15 to this setting.