2. The Kakimizu complex

The proof of Theorem 1.2 hinges on understanding the space of incompressible Seifert surfaces for a non-split, alternating link. This space (for a general link) is called the Kakimizu complex. In the literature, it is common to restrict to minimal genus surfaces. For the sake of discussion, we give both definitions.

• • The vertices of $\mathrm{MS}(L)$ are in bijection with proper isotopy (rel. boundary) classes of minimal genus Seifert surfaces for L.

• • The vertices of $\mathrm{IS}(L)$ are in bijection with proper isotopy (rel. boundary) classes of incompressible Seifert surfaces for L.

• • In each of $\mathrm{MS}(L), \mathrm{IS}(L)$ , for $n>0$ , there is an n-simplex with vertices $v_0,\ldots, v_n$ exactly when the isotopy classes corresponding to $v_0,\ldots, v_1$ admit Seifert surface representatives with mutually disjoint interiors.

The notation ‘ $\mathrm{MS}(L)$ ’ and $`\mathrm{IS}(L)$ ’ chosen by Kakimizu is meant to remind the reader that these simplicial complexes are associated to minimal surfaces or incompressible surfaces bounded by the link L.

The methods of Kakimizu suggest the following strengthened version of Theorem 2.2.

Note that a Seifert surface for L restricts to a properly embedded, oriented surface in $S^3\setminus\nu(L)$ that has no closed components and has boundary a longitude on each component of $\partial S^3\setminus\nu(L)$ . For convenience, we will refer interchangeably to such properly embedded surfaces and Seifert surfaces for L.

We will make use of the following definition, used by Kakimizu in [Reference KakimizuKak92].

Kakimizu [Reference KakimizuKak92, Proposition 3.1(1)] proved that the covering distance of S and S’ is equal to the minimum length of a path between vertices corresponding to S,S’ in $\mathrm{IS}(L)$ . Even without this fact (which we will not directly employ in this paper, so as to keep our arguments relatively self-contained), we might make the following observation.

We now prove Theorem 2.4 via the following lemma, which mirrors that of [Reference KakimizuKak92, Theorem 2.1] and closely follows the same proof, taking care to consider the Euler characteristic of the produced surfaces.

The following proof of Lemma 2.7 is illustrated in Figure 1.

Figure 1.

A schematic of the infinite cyclic cover M of $E:=S^3\setminus\nu(L)$ . We indicate lifts $S_i$ of S and $S'_j$ of S’. The region between $S_i,S_{i+1}$ is $E_i$ and the region between $S'_j,S'_{j+1}$ is $E'_j$ . In this schematic, $d(S,S')=3$ as $E'_0$ intersects $E_i$ for the values $i\in\{0,1,2,3\}$ . In the proof of Lemma 2.7, we obtain surfaces $R\subset E'_0$ and $R'\subset E_0$ whose union is isotopic to the cut-and-paste sum of $S'_0$ and $S_1$ . It follows that one of R,R’ has Euler characteristic bounded below by $\min\{\chi(S),\chi(S')\}$ . Fully compressing that surface (in $E'_0$ or $E_0$ accordingly) yields an incompressible surface that projects to the desired Seifert surface S”.

Similarly, let $S'_j$ be the translate of $S'_0$ under the action of $j\in\mathbb{Z}$ and let $E'_j$ denote the closure of the region of M bounded by $-S'_j\cup S'_{j+1}$ . Then $E'_0\cap E_0\neq\emptyset$ and $E'_i\cap E_0=\emptyset$ whenever $i>0$ . Let X be a regular neighborhood of $S'_0\cup(E_0\cap E_0')$ in $E_0'$ . Since $S'_0\cap E_n=\emptyset$ , we also have $X\cap E_n=\emptyset$ . Let R be the closure of $\partial X\setminus\partial E'_0$ .

We simultaneously produce another surface, exchanging the roles of S and S’. That is, we note that $E_j\cap E'_0=\emptyset$ whenever $j<0$ . Let X’ be a regular neighborhood of $S_1\cup(E_0\cap E'_0)$ in $E_0$ . Since $S_1\cap E'_{-n}=\emptyset$ , we also have $X'\cap E'_{-n}=\emptyset$ . Let R’ be the closure of $\partial X'\setminus\partial E_0$ .

The union of R and R’ is isotopic to the cut-and-paste sum of $S_1$ and $S'_0$ . (That is, the union of R and R’ is isotopic to the embedded surface obtained by resolving $S_1\cup S'_0$ at intersections while respecting orientation. Note that if $S_1, S'_0$ are disjoint, as may be the case if $n=1$ , then R is a parallel copy of $S'_0$ while R’ is a parallel copy of $S_1$ .) We conclude $\chi(R)+\chi(R')=\chi(S)+\chi(S')$ , so at least one of $\chi(R),\chi(R')$ is bounded below by $\min\{\chi(S),\chi(S')\}$ . By potentially exchanging the roles of S and S’ (and hence also R and R’), assume without loss of generality that $\chi(R)\ge\min\{\chi(S),\chi(S')\}$ . Since S,S’ intersect in essential curves, any closed components of R are positive genus. Delete any closed components; we still have $\chi(R)\ge\min\{\chi(S),\chi(S')\}$ .

Since $R\subset X\subset E_0'$ , R does not intersect any $S_i'$ . Then $p(R)\cap S'=\emptyset$ , where p(R) is the projection of R to E (and hence a Seifert surface for L). Moreover, R does not intersect $E_j$ with $j<0$ or $j>n$ . We already remarked that X does not intersect $E_n$ , so R also does not intersect $E_n$ . We conclude that if R is incompressible, then the Seifert surface p(R) satisfies $d(S,p(R))<n$ , satisfying the claim (with $S'':=p(R)$ ).

Unfortunately, the surface R may fail to be incompressible. Since S, S’ intersect in essential circles and R is a subset of the cut-and-paste sum of $S_1$ and $S'_0$ , the surface R does not include any 2-sphere components. Since $\partial E'_0$ is incompressible in M, if a surface in the interior of $E'_0$ is incompressible in $E'_0$ then it is also incompressible in M. Suppose R is compressible in M and hence in $E'_0$ . Let D be a compressing disk for R in $E'_0$ . Since $S_n$ is incompressible, again by a standard innermost circle argument we may assume D does not intersect $S_n$ . Then compressing R along D yields a new surface in $E'_0$ that has no 2-sphere components and does not intersect $E_n$ . Repeat until obtaining a surface that is incompressible in $E'_0$ and hence also incompressible in M. Delete any closed components and call the result $\tilde{R}$ . Since $\tilde{R}$ is obtained from R by compression and deletion of positive-genus components, $\chi(\tilde{R})\ge\chi(R)\ge\min\{\chi(S),\chi(S')\}$ . Let S” denote the incompressible Seifert surface $p(\tilde{R})$ . Since $\tilde{R}\subset E'_0\setminus E_n$ , we find that the interiors of S” and S’ are disjoint and $d(S,S'')<n$ as desired.

Theorem 2.4 follows easily from Lemma 2.7.

If $d(F_1,S)=0$ , then by Observation 2.6 the surfaces $F_1, S$ are isotopic rel. boundary and we conclude there is a path $(v',v_1,v)$ in $\mathrm{IS}_k(L)$ . Otherwise, repeat the above argument replacing S’ with $F_i$ to obtain a surface $F_{i+1}$ representing a vertex $v_{i+1}\in\mathrm{IS}(L)$ such that $\chi(F_{i+1})\ge\min\{\chi(S),\chi(F_i)\}\ge \min\{\chi(S),\chi(S')\}\ge k$ (so in particular $v_{i+1}\in \mathrm{IS}_k(L)$ ), the vertices $v_{i+1},v_i$ are adjacent in $\mathrm{IS}(L)$ , and $d(F_{i+1},S)<d(F_i,S)$ . Repeat until obtaining a surface $F_a$ with $d(F_a,S)=0$ and hence (again by Observation 2.6) the surface $F_a$ is isotopic rel. boundary to S. We conclude there is a path $(v',v_1,v_2,\ldots, v_a,v)$ in $\mathrm{IS}_k(L)$ .

The following corollary follows from Kakimizu [Reference KakimizuKak92, Proposition 3.1(1)], but we prove it here for completeness.

On the other hand, if $d(S,S')=1$ , then it follows from Lemma 2.7 that there is a third Seifert surface S” such that S” is disjoint from S’ in its interior and $d(S,S'')<1$ . Then $d(S,S'')=0$ and Observation 2.6 shows that S,S” are isotopic rel. boundary in $S^3$ . We conclude that, up to isotopy rel. boundary in $S^3$ , the surfaces S,S’ have disjoint interior and thus represent adjacent vertices in $\mathrm{IS}(L)$ .

In the case that L is an alternating link, there is even more known about surfaces bounded by or containing L. It is a well-known theorem of [Reference CrowellCro59, Reference MurasugiMur58] that Seifert’s algorithm applied to an alternating diagram yields a minimal genus Seifert surface. However, a minimal genus Seifert surface for an alternating link need not arise from Seifert’s algorithm applied to an alternating diagram [Reference Hirasawa and SakumaHS97, Theorem 1.2]Footnote ¹ , and in general it is not clear what, if any, relationship is held between distinct minimal genus Seifert surfaces for an alternating link.

We will make use of the following key lemma proved by Kindred [Reference KindredKin18].

Lemma 2.11 (Kindred [Reference KindredKin18] (Crossing Tube Lemma)). Given a nontrivial, reduced alternating diagram $\mathcal{D}$ of a non-split link L and a closed, essential surface F containing L, there exists an isotopy of $S^3$ fixing L after which F has a standard tube above some crossing c of $\mathcal{D}$ as in Figure 2. That is, F admits a compressing disk lying above a small neighborhood of c and that intersects L in the two points of L above c.

Figure 2.

Left: a generic essential surface F containing a link L. We feature the portion of F lying above a neighborhood of some crossing in the diagram $\mathcal{D}$ (shown at the bottom of the figure). This subset of F includes two disks that contain two arcs of L and potentially other disks that do not meet L. Right: above this crossing, the surface F includes a standard tube containing the two strands that project to the crossing.

3. Proof of Theorem 1.2

We will reduce to the setting of incompressible surfaces using the following lemma, which is based on a commonly used fact in the study of surfaces in 4-manifolds. See [Reference BoyleBoy88, Corollary 4] for a general statement and proof. Here, when we say a Seifert surface $S_i$ is obtained by compressing another Seifert surface $\Sigma_i$ , we mean that we compress $\Sigma_i$ and delete any resulting closed components.

Proof. If the compression takes place along a non-separating curve, then $\Sigma$ is obtained from S by attaching one tube, i.e. surgery along a framed arc $\gamma$ . Because S is boundary parallel in $B^4$ , we have $\pi_1(B^4\setminus S)\cong\mathbb{Z}$ with generator represented by a meridian of S. Therefore, any two arcs with endpoints on S and interior in $B^4\setminus S$ are isotopic keeping ends on S and interior disjoint from S. In ambient dimension four, the normal bundle $N(\gamma)$ of the arc $\gamma$ rel. boundary admits framings in bijection with $\pi_1(SO(3))\cong\mathbb{Z}/2\mathbb{Z}$ . Such a framing corresponds with a choice of how to attach a tube to S along $\gamma$ : we choose a trivialization $\langle v_1, v_2, v_3\rangle$ of $N(\gamma)$ with $\langle v_1, v_2\rangle$ tangent to S at the ends of $\gamma$ and then surger S along the three-dimensional 1-handle $\gamma\times\langle v_1,v_2\rangle$ . Exactly one of the two potential framings of $N(\gamma)$ yields a 1-handle over which the orientation of S extends, while the other framing yields a 1-handle over which the orientation does not extend. Since $\Sigma$ is oriented, the framing of $\gamma$ is uniquely determined. We conclude that there is a unique way of attaching an oriented tube to S in $B^4$ , one of which would yield $S\# T$ , and hence that $\Sigma$ is isotopic rel. boundary to $S\# T$ .

On the other hand, suppose that S is obtained by compressing $\Sigma$ along a separating curve and then deleting a closed component S’ disjoint from S. Recall that a closed surface in $S^3$ always bounds a handlebody in $B^4$ (see e.g. the discussion in the beginning of [Reference HatcherHat07, §1.2]: every positive-genus surface in $S^3$ is compressible and every smooth $S^2$ in $S^3$ bounds a 3-ball). Let V be a handlebody bounded by S’ into $B^4$ . Let $\gamma$ be a framed arc connecting S, S’ such that surgery along $\gamma$ yields $\Sigma$ . Push $S', V, \gamma$ into the interior of $B^4$ . Since $\gamma$ is one-dimensional and disjoint in its interior from V and $\pi_1(B^4\setminus V)=1$ , as the nonseparating case we conclude that $\gamma$ is isotopic to any other arc connecting S to $S'=gT$ that is disjoint from V and also that the framing on $\gamma$ is determined by the orientation of $\Sigma$ . Therefore, $\Sigma$ is isotopic rel. boundary in $B^4$ to $S\# (g(S')T)$ .

We will make use of the following subcase of Theorem 1.2 in the proof of the main theorem.

In the proof of Lemma 3.2, we use the following easy fact.

Proof. Suppose D is a compressing disk for $\Sigma_1$ . Let C be a curve of intersection in $D\cap\Sigma_2$ that is innermost on D, so C bounds a disk $D'\subset D$ whose interior is disjoint from $\Sigma_2$ . Compress $\Sigma_2$ along D’ and delete any closed components formed. Repeat until the interior of D is disjoint from $\Sigma_2$ , and finally perform the compression along $\Sigma_1$ . Repeat until obtaining an incompressible Seifert surface $S_1$ . Now any compressing disk for $\Sigma_2$ intersects $S_1$ in inessential curves that can similarly be removed by an innermost circle argument, so compress $\Sigma_2$ along disks disjoint from $S_1$ to obtain an incompressible surface $S_2$ whose interior is disjoint from $S_1$ .

Proof of Lemma 3.2. If $d(S_1,S_2)=0$ , then by Observation 2.6 the surfaces $S_1$ and $S_2$ are isotopic rel. boundary. Then up to isotopy rel. boundary in $S^3$ , the surfaces $\Sigma_1,\Sigma_2$ are each obtained from $S_1$ by surgery along n arcs for some n. By Lemma 3.1, $\Sigma_1$ and $\Sigma_2$ are isotopic rel. boundary in $B^4$ .

Now suppose $d(S_1,S_2)\le 1$ . We will prove that $\Sigma_1,\Sigma_2$ are isotopic rel. boundary in $B^4$ by inducting on the genus g of $\Sigma_1, \Sigma_2$ (although for convenience, we will phrase this as a condition on the Euler characteristic, inducting on the Euler characteristic decreasing from the maximum possible value of 1).

If $\chi(\Sigma_i)=1$ , then it is a well-known consequence of the three-dimensional Schoenflies theorem (see e.g. the discussion in [Reference HatcherHat07, Section 1.1]) that the disks $\Sigma_1,\Sigma_2$ are isotopic rel. boundary in $S^3$ and hence in $B^4$ .

Now suppose that Lemma 3.2 holds for Seifert surfaces of Euler characteristic strictly greater than $\chi(\Sigma_i)$ . If $d(S_1,S_2)=0$ then $\Sigma_1$ and $\Sigma_2$ are isotopic rel. boundary in $B^4$ , so assume that $d(S_1,S_2)=1$ . By Corollary 2.8, we may apply an isotopy rel. boundary in $S^3$ to $S_1,S_2$ so that they have disjoint interior. Then $S_1\cup S_2$ is an embedded surface F that is essential with respect to L.

Choose a reduced, alternating diagram $\mathcal{D}$ for L. Lemma 2.11 implies that F includes a standard tube over some crossing c of $\mathcal{D}$ as in Figure 3 (left). Fixing L, isotope $S_1$ and $S_2$ so that their interiors agree precisely at a band b above c as in Figure 3 (center). Let $\mathcal{D}'$ be the diagram obtained by smoothing $\mathcal{D}$ at c, respecting orientations, and let L’ be the link described by $\mathcal{D}'$ . Since $\mathcal{D}$ is alternating, non-split, and reduced, the smoothed diagram $\mathcal{D}'$ is alternating and non-split.

Figure 3.

Left: incompressible surfaces $S_1,S_2$ bounded by L. By Lemma 2.11, the essential surface $S_1\cup S_2$ includes a standard tube over some crossing c in a reduced, alternating diagram $\mathcal{D}$ for L. Center: we isotope $S_1,S_2$ near the standard tube so that they agree in a band b. Elsewhere, we add $g(S_1)-g(S_2)$ tubes to $S_2$ to obtain a surface $\tilde{S}_2$ of the same genus as $S_1$ . Right: deleting the band b from $S_1,\tilde{S}_2$ yields surfaces $S'_1,S'_2$ bounded by a link L’.

Without loss of generality, assume $g(S_1)\ge g(S_2)$ . Attach tubes to $S_2$ (away from b) to obtain a surface $\tilde{S}_2$ of the same genus as $S_1$ . By Lemma 3.1, $\tilde{S}_2$ is isotopic rel. boundary in $B^4$ to $S_2\#(g(S_1)-g(S_2))T$ , where T is a standard torus in $S^4$ .

Consider the surfaces $S'_1:=S_1\setminus b$ , $S'_2:=\tilde{S}_2\setminus b$ bounded by the link L’. We have $\chi(S'_1)=\chi(S'_2)=\chi(S_1)+1>\chi(\Sigma_i)$ . By Proposition 3.3, $S'_1, S'_2$ compress to surfaces with disjoint interior and hence covering distance at most 1. Then by inductive hypothesis, we find that $S'_1$ and $S'_2$ are isotopic rel. boundary in $B^4$ . Since $\Sigma_1=S'_1\cup b$ and $\tilde{S}_2=S'_2\cup b$ , we find also that $S_1$ and $\tilde{S}_2$ are isotopic rel. boundary in $B^4$ .

By Lemma 3.1, the surface $\Sigma_1$ is isotopic rel. boundary in $B^4$ to

$S_1\#(g(\Sigma_1)-g(S_1)) T.$

Similarly, $\Sigma_2$ is isotopic rel. boundary in $B^4$ to

Since $S_1$ and $\tilde{S}_2$ are isotopic rel. boundary in $B^4$ , we conclude that $\Sigma_1$ and $\Sigma_2$ are isotopic rel. boundary in $B^4$ as claimed.

We can now prove Theorem 1.2.

Proof. Let $\chi:=\chi(\Sigma_i)$ . Let $S_i$ be an incompressible surface obtained from $\Sigma_i$ by $k_i$ compressions. Assume $k_1\le k_2$ , so $\chi\le \chi(S_1)\le \chi(S_2)$ .

By Theorem 2.4, there exist incompressible Seifert surfaces $S_1=F_0,F_1,F_2,\ldots, F_n,F_{n+1}=S_2$ such that $\chi(F_j)\ge \chi(S_2)\ge\chi$ and the surfaces $F_j, F_{j+1}$ have disjoint interior for each $j\in\{0,\ldots, n\}$ . For each $j=1,\ldots, n$ , let $\Sigma'_j$ be a Seifert surface obtained from $F_j$ by attaching tubes to $F_j$ such that $\chi(\Sigma'_j)=\chi$ for all j. Let $\Sigma'_0:=\Sigma_1, \Sigma'_{n+1}:=\Sigma_2$ .

Now for each $j=0,\ldots, n$ , the surfaces $\Sigma'_j, \Sigma'_{j+1}$ are same-genus Seifert surfaces for L that compress to incompressible Seifert surfaces $F_j, F_{j+1}$ . Note that $d(F_j, F_{j+1})\le 1$ since the interiors of $F_j, F_{j+1}$ are disjoint. Then by Lemma 3.2, $\Sigma'_j, \Sigma'_{j+1}$ are isotopic rel. boundary in $B^4$ . Since $\Sigma_1=\Sigma'_0, \Sigma_2=\Sigma'_{n+1}$ , we conclude that $\Sigma_1,\Sigma_2$ are isotopic rel. boundary in $B^4$ .

Corollaries 1.3 and 1.4 now follow easily.

Corollary 1.3. Up to isotopy rel. boundary in the 4-ball, a non-split, alternating link L bounds a unique minimal genus Seifert surface S. Any other Seifert surface for L is isotopic rel. boundary in $B^4$ to a connected sum of S and some number of standard tori in $S^4$ .

Proof. Let S be a minimal genus Seifert surface for L. By Theorem 1.2, any other minimal genus Seifert surface for L is isotopic to S rel. boundary in $B^4$ . Let $\Sigma$ be a Seifert surface for L, and let S’ be a genus- $g(\Sigma)$ Seifert surface obtained by attaching n tubes to S. By Theorem 1.2, $\Sigma$ is isotopic rel. boundary in $B^4$ to S’, which by Lemma 3.1 is isotopic rel. boundary in $B^4$ to $S\#_n T$ .

Corollary 1.4. Let $S_1, S_2$ be Seifert surfaces for a non-split, alternating link L in $S^3$ . Viewing $S^3$ as an equator of $S^4$ , push the interiors of $S_1$ and $S_2$ off $S^3$ toward opposite normal directions of $S^3$ to obtain surface $S_1^+, S_2^-$ in $S^4$ with the same boundary but disjoint interior. The closed surface $S_1^+\cup S_2^-$ bounds a handlebody smoothly embedded in $S^4$ , i.e. is smoothly unknotted.

Proof. Let S be a minimal genus Seifert surface for L. Obtain a closed surface F in $S^4$ by gluing two copies of S, with the interiors of each copy pushed off $S^3$ toward opposite sides. Then F is isotopic to the boundary of the handlebody $S\times I$ .

By Theorem 1.2, $S_1^+\cup S_2^-$ is isotopic to $F\#_n T$ , where $n=(g(S_1)-g(S))+(g(S_2)-g(S))$ . We conclude $S_1^+\cup S_2^-$ also bounds a handlebody.

4. Open questions

We end with some natural open questions for other families of links. We first suggest one consider almost-alternating links, i.e. links that admit a diagram which becomes alternating after applying one crossing change (note the distinction between being almost alternating vs. admitting a crossing change to some alternating link).

An affirmative answer to (a) implies an affirmative answer to (b).

Arguably, it would be more interesting if the answer to Question 4.1(a) is ‘no,’ as this would provide a sharpness result for Theorem 1.2 is sharp.

Theorem 1.2 is the first theorem proving uniqueness of minimal genus Seifert surfaces for a family of knots up to isotopy rel. boundary in $B^4$ without having uniqueness up to isotopy in $S^3$ .

Hyperbolic knots may bound more than one minimal genus Seifert surface up to isotopy rel. boundary in $B^4$ [Reference Hayden, Kim, Miller, Park and SundbergHKM⁺23]. However, a hyperbolic knot bounds a finite number of minimal genus Seifert surfaces in $S^3$ (and in fact, a finite number of Seifert surfaces of any fixed genus) according to constraints from normal surface theory [Reference Schubert and SoltsienSS64] (see also [Reference Johnson, Pelayo and WilsonJPW14, Theorem 4]). Perhaps a similar analysis could be used to give a nontrivial explicit upper bound on the number of Seifert surfaces for a hyperbolic knot up to isotopy rel. boundary in $B^4$ .