1.1 Existence of $\mathbb {Z}$ -disks in $(\mathbb {C}P^2)^\circ$

Necessary and sufficient criteria for deciding whether a knot $K$ is $\mathbb {Z}$ -slice in $(\mathbb {C}P^2)^\circ$ can be deduced from the work of Fogel [Reference FogelFog94]. To state this result, use $E_K$ to denote the exterior of $K$ , and recall that the Blanchfield form of $K$ is a nonsingular, sesquilinear, Hermitian form

\begin{equation*} {\textrm {Bl}}_K \colon H_1(E_K^\infty ) \times H_1(E_K^\infty ) \to \mathbb {Q}(t)/\mathbb {Z}[t^{\pm 1}],\end{equation*}

which, roughly speaking, measures equivariant linking in the infinite cyclic cover $E_K^\infty$ of $E_K$ . We say that ${\textrm {Bl}}_K$ is represented by a size $n$ Hermitian matrix $A$ with $\det (A) \neq 0$ and coefficients in $\mathbb {Z}[t^{\pm 1}]$ if ${\textrm {Bl}}_K$ is isometric to the following linking form:

(1.1) \begin{align}\begin{array}{rl} \mathbb {Z}[t^{\pm 1}]^n/A^T\mathbb {Z}[t^{\pm 1}]^n \times \mathbb {Z}[t^{\pm 1}]^n/A^T\mathbb {Z}[t^{\pm 1}]^n &\to \mathbb {Q}(t)/\mathbb {Z}[t^{\pm 1}]\\[4pt] ([x],[y]) &\mapsto -x^TA^{-1}\overline {y}. \end{array} \end{align}

If such a matrix has size $1$ , then it is a symmetric polynomial $p(t)$ and, in this case, we say that ${\textrm {Bl}}_K$ is represented by $p(t)$ . Here, and in what follows, we write $\Delta _K$ for the unique symmetric representative of the Alexander polynomial of $K$ that evaluates to $-1$ at $t=1$ ; the reason for this convention will be elucidated in § 1.3. Recalling that we are working in the topological category, $\mathbb {Z}$ -sliceness in $(\mathbb {C}P^2)^\circ$ can be characterized as follows; see § 3 for the proof.

Here, given a curve $\gamma \subset S^3 \setminus K$ with $\ell k(K,\gamma )=0$ that is unknotted in $S^3$ , we say that a knot $K'$ in $S^3$ is obtained from $K$ by a positive generalized crossing change along $\gamma$ if $(S^3_{-1}(\gamma ),K)\cong (S^3,K')$ .As illustrated in Figure 1, when only two strands of $K$ are involved, we omit the word ‘generalized’. We note that we are also considering $K \subset S^3$ as a knot in $S^3\setminus \nu (\gamma )$ and hence in $S^3_{-1}(\gamma )$ . We emphasize that the last three items of Theorem1.1 were already known to be equivalent by work of Fogel [Reference FogelFog94]; see also [Reference Borodzik and FriedlBF14] for the case of multiple crossing changes.

Figure 1 Top: a positive crossing change from $K$ to $K'$ realized as a (generalized) crossing change about a curve $\gamma$ (bold). Bottom: a generalized positive crossing change about $\gamma$ transforms $K$ into $K'$ . In this example, $\gamma$ links $K$ geometrically more than two times (although as always links $K$ zero times algebraically).

In summary, Theorem1.1 describes conditions for obstructing a knot from bounding a $\mathbb {Z}$ -disk in $(\mathbb {C}P^2)^\circ$ as well as ways to construct $\mathbb {Z}$ -slice knots in $(\mathbb {C}P^2)^\circ$ . In § 1.6 we list related work on the existence question for $\mathbb {Z}$ -surfaces in $4$ -manifolds with boundary $S^3$ such as [Reference Feller and LewarkFL18, Reference Feller and LewarkFL24, Reference Kjuchukova, Miller, Ray and SakalliKMRS24, Reference Conway, Piccirillo and PowellCPP23] but first we discuss the classification of $\mathbb {Z}$ -disks in $(\mathbb {C}P^2)^\circ$ .

1.2 Classification of $\mathbb {Z}$ -disks in $(\mathbb {C}P^2)^\circ$

Assume now that a generalized crossing change along the surgery curve $\gamma$ results in an Alexander polynomial one knot. As we recall in more detail in § 4, one can associate to $K$ and $\gamma$ a $\mathbb {Z}$ -disk in $(\mathbb {C}P^2)^\circ$ , which we refer to as a generalized crossing change $\mathbb {Z}$ -disk.

From now on, we fix a basepoint $z \in \partial E_K$ and a lift $\widetilde {z} \in \partial E_K^\infty$ of $z$ to the infinite cyclic cover. When we refer to loops and lifts, it is with respect to these basepoints. We note in Proposition 4.10 below that the homology class of the lift to $E_K^\infty$ of the surgery curve $\gamma \subset E_K$ belongs to

(1.2) \begin{equation} \mathcal {G}_K:= \left \lbrace x \in H_1(E_K^\infty ) \ \Big | \ x \text { is a generator and } {\textrm {Bl}}_K(x,x) =\frac {1}{\Delta _K} \right \rbrace . \end{equation}

Observe that the set $\lbrace t^k \rbrace _{k \in \mathbb {Z}}$ acts on $ \mathcal {G}_K$ by multiplication. The following theorem (which, recall, takes place in the topological category) gives a complete characterization of $\mathbb {Z}$ -disks in $(\mathbb {C}P^2)^\circ$ as well as a complete invariant; see § 5 for the proof.

Note that $\mathcal {D}_{\mathbb {Z}}(K,(\mathbb {C}P^2)^\circ )$ is non-empty if and only if $K$ is $\mathbb {Z}$ -slice in $(\mathbb {C}P^2)^\circ$ , whereas $\mathcal {G}_K$ can be non-empty as soon as $K$ has cyclic Alexander module; the bijection only holds if $K$ is $\mathbb {Z}$ -slice.

In summary, Theorem1.2 shows that all $\mathbb {Z}$ -disks in $(\mathbb {C}P^2)^\circ$ arise as crossing change $\mathbb {Z}$ -disks and that the lift of the surgery curve leads to a complete invariant. In § 1.6 we compare Theorem1.2 to the classification of $\mathbb {Z}$ -surfaces obtained in [Reference Conway, Piccirillo and PowellCPP23] but first we describe how the bijection with $\mathcal {G}_K/\lbrace t^k \rbrace _{k \in \mathbb {Z}}$ allows us to explicitly enumerate $\mathbb {Z}$ -disks when $K$ has trivial or quadratic Alexander polynomial.

Figure 2. Left: the twist knot $K_n$ where the box denotes $n$ full twists. Middle: the figure eight knot $K_1$ . Right: the right-handed trefoil $K_{-1}$ .

1.3 Knots with quadratic Alexander polynomial: an explicit enumeration

Alexander polynomial one knots bound a unique $\mathbb {Z}$ -disk in $(\mathbb {C}P^2)^\circ$ up to isotopy rel. boundary; this is a consequence of Theorem1.2 (and should be compared to there being a unique $\mathbb {Z}$ -sphere in ${\mathbb {C}} P^2$ [Reference Conway and OrsonCO25, Proposition A.1]) but also follows from the combination of [Reference Conway and PowellCP23, Theorem1.2] and [Reference Orson and PowellOP22, Theorem C]. When the polynomial is nontrivial, some number theory is needed to understand the cardinality of $\mathcal {G}_K/\lbrace t^k \rbrace _{k \in \mathbb {Z}}$ , as we now explain. If the Alexander polynomial of a knot is trivial or quadratic (e.g. if $K$ is a genus one knot), then up to multiplication by $\pm t^k$ , it is of the form

\begin{equation*}\Delta _n:=nt-(2n+1)+nt^{-1},\end{equation*}

for some $n \in \mathbb {Z}$ . Note that $\Delta _0=1, \Delta _{-1}=-t+1-t^{-1}$ is the Alexander polynomial of the trefoil, $\Delta _1$ is the Alexander polynomial of the figure eight knot, and more generally, $\Delta _n$ is the Alexander polynomial of the twist knot $K_n$ with $n$ full twists illustrated in Figure 2. Twists knots are genus one knots that are $\mathbb {Z}$ -slice in $(\mathbb {C}P^2)^\circ$ because they can be unknotted by switching a single positive crossing change; recall Theorem1.1. We find it more convenient to index the $\Delta _n$ using their leading coefficient which is why we work with a symmetric representative of the Alexander polynomial that satisfies $\Delta _K(1)=-1$ : this way $\Delta _{K_n}=\Delta _n$ .

Our next result enumerates $\mathbb {Z}$ -disks in $(\mathbb {C}P^2)^\circ$ for knots with trivial or quadratic Alexander polynomial; its proof follows by combining Theorem1.2 with the number-theoretic Theorem1.7.

Theorem 1.3. If a knot $K$ with trivial or quadratic Alexander polynomial is $\mathbb {Z}$ -slice in $(\mathbb {C}P^2)^\circ$ , then it bounds $1,2,4$ or infinitely many $\mathbb {Z}$ -disks in $(\mathbb {C}P^2)^\circ$ up to isotopy rel. boundary. More precisely, up to isotopy rel. boundary,

\begin{equation*} K \text{ bounds } \begin{cases} 1 \text { disk} & \text {if $\Delta _K=\Delta _0,\Delta _{-1}$}, \\ 2 \text { disks} & \text {if $\Delta _K=\Delta _1,\Delta _2$, or $\Delta _K=\Delta _n$ for $n=-p^k$ with $p$ prime and $k$ odd}, \\ 4 \text { disks} & \text {if $\Delta _K=\Delta _n$ for $n=-p^k$ with $p$ prime and $k$ even}, \\ \infty \text { disks} & \text {otherwise}.\\ \end{cases} \end{equation*}

We emphasize the difference with $\mathbb {Z}$ -disks in $D^4$ where $|\mathcal {D}_{\mathbb {Z}}(K,D^4)| \in \lbrace 0,1\rbrace$ ; in contrast, when $\Delta _K$ is quadratic, Theorem1.3 shows that, generically, $\mathcal {D}_{\mathbb {Z}}(K,(\mathbb {C}P^2)^\circ )$ is infinite.

Remark 1.4. Theorems 1.2 and 1.3 admit analogues where the word ‘isotopy’ is replaced by ‘equivalence’. To form the analogue of Theorem 1.2, one substitutes multiplication by $t^k$ for multiplication by $\pm t^k$ , whereas for Theorem 1.3, the outcome is that if $K$ is a knot with trivial or quadratic Alexander polynomial that is $\mathbb {Z}$ -slice in $(\mathbb {C}P^2)^\circ$ , then, up to equivalence rel. boundary,

\begin{equation*} K \text { bounds } \begin{cases} 1 \text { disk} & \text {if $\Delta _K=\Delta _0,\Delta _{-1},\Delta _1,\Delta _2$}, \\ & \text {or $\Delta _K=\Delta _n$ for $n=-p^k$ with $p$ prime and $k$ odd}, \\ 2 \text { disks} & \text {if $\Delta _K=\Delta _n$ for $n=-p^k$ with $p$ prime and $k$ even}, \\ \infty \text { disks} & \text {otherwise}.\\ \end{cases} \end{equation*}

We refer to Remarks 1.8 and 5.5 for further details.

We describe some further context surrounding Theorem1.3 in § 1.6 but presently we discuss examples of Theorem1.3 with an emphasis on realizing the disks smoothly.

As we explain in § 4, the construction of the $\mathbb {Z}$ -disk $D_\gamma$ associated to a generalized crossing change curve $\gamma$ depends on a choice of orientation for $\gamma$ . Reversing the orientation on $\gamma$ results in a disk $D_{-\gamma }$ which is equivalent, but not necessarily isotopic, to the original rel. boundary. Each generalized crossing change curve thus gives two $\mathbb {Z}$ -disks which are equivalent rel. boundary, but usually distinct in the sense of Theorem1.3. For knots with trivial or quadratic Alexander polynomial, we note in Remark 8.3 that $D_{\pm \gamma }$ are isotopic rel. boundary if and only if $\Delta _K = \Delta _0$ or $\Delta _{-1}$ .

We further illustrate Theorems1.1 and 1.3 by calculating the number $d_K$ of $\mathbb {Z}$ -disks in $(\mathbb {C}P^2)^\circ$ bounded by a knot $K$ . Thanks to Theorem1.1, we know that if a knot or its mirror is $\mathbb {Z}$ -slice in $(\mathbb {C}P^2)^\circ$ , then its Blanchfield form is presented by a size one nondegenerate Hermitian matrix. Work of Fogel [Reference FogelFog94, Theorem] therefore implies that if a knot $K$ or its mirror is $\mathbb {Z}$ -slice in $(\mathbb {C}P^2)^\circ$ , then $u_a(K)=1$ , where $u_a(K)$ denotes the algebraic unknotting number of $K$ . The algebraic unknotting number is listed on KnotInfo [Reference Livingston and MooreLM24].

1.4 The underlying number theory

Next we state the number-theoretic result that underlies Theorem1.3. First we recast $\mathcal {G}_K$ in a more algebraic way. Assume that $K$ is a knot that is $\mathbb {Z}$ -slice in $(\mathbb {C}P^2)^\circ$ so that by Theorem1.1 its Alexander module is cyclic (i.e. $H_1(E_K^\infty )\cong \mathbb {Z}[t^{\pm 1}]/(\Delta _K)$ ) and ${\textrm {Bl}}_K(x,y)={x\overline {y}}/({\Delta _K})$ for every $x,y \in H_1(E_K^\infty )$ . Here $x \mapsto \overline {x}$ denotes the involution $p(t) \mapsto p(t^{-1}).$ This implies that polynomial multiplication induces a free and transitive action of

\begin{equation*} U\left (\frac {\mathbb {Z}[t^{\pm 1}]}{(\Delta _K)}\right ):= \left \lbrace u \in \frac {\mathbb {Z}[t^{\pm 1}]}{(\Delta _K)} \ \Big | \ u\overline {u}=1 \right \rbrace, \end{equation*}

on $\mathcal {G}_K$ . In particular, there is a noncanonical bijection

\begin{equation*} \mathcal {G}_K \xrightarrow {\approx } U\left (\frac {\mathbb {Z}[t^{\pm 1}]}{(\Delta _K)}\right ).\end{equation*}

The general setting is as follows. Given a ring $R$ with involution $x \mapsto \overline {x}$ , the group of unitary units $U(R)$ refers to those $x \in R$ such that $x \overline {x}=1$ . For example, when $R=\mathbb {Z}[t^{\pm 1}]$ with the involution $p(t) \mapsto p(t^{-1})$ , all units are unitary and are of the form $\pm t^{k}$ with $k \in \mathbb {Z}$ . More generally, if $R=\mathbb {Z}[t^{\pm 1}]/p(t)$ for some $p(t) \in \mathbb {Z}[t^{\pm 1}]$ , then the group $U(R)$ always contains the image of the set $\lbrace \pm t^k \rbrace _{k \in \mathbb {Z}} \subset \mathbb {Z}[t^{\pm 1}]$ under the projection map $\mathbb {Z}[t^{\pm 1}] \to \mathbb {Z}[t^{\pm 1}]/p(t).$

Theorem1.3 therefore follows from Theorem1.2 and the study of $U(\mathbb {Z}[t^{\pm 1}]/p(t))/\lbrace t^k\rbrace _{k \in \mathbb {Z}}$ for quadratic Laurent polynomials $p(t)$ that arise as Alexander polynomials, i.e. for the polynomials $\Delta _n=nt-(2n+1)+nt^{-1}$ .

We write

\begin{equation*}U_n:=U\left (\frac {\mathbb {Z}[t^{\pm 1}]}{(\Delta _n)}\right ),\end{equation*}

as a shorthand for the set of unitary units of $\mathbb {Z}[t^{\pm 1}]/(\Delta _n)$ . In order to prove Theorem1.3, it suffices to determine for what values of $n$ the set $U_n/\lbrace t^k \rbrace _{k \in \mathbb {Z}}$ is infinite and to determine its cardinality when it is finite. The next theorem (which is proved in § 7) achieves slightly more as it also determines the rank of $U_n/\lbrace t^k \rbrace _{k \in \mathbb {Z}}$ in the infinite case in terms of the number $\Omega (n)$ of positive primes dividing $n$ .

We use the convention that strictly negative rank should be interpreted as zero rank; we write this for simplicity of the theorem statement. The cases in which $(2)$ gives a rank less than or equal to zero correspond precisely to the cases in $(1)$ ; see Remark 7.1.

1.5 Challenges in the smooth category

We now briefly discuss the results of Theorem1.3 in the context of the smooth category. The most immediate question in this regard is to determine which of the isotopy classes in Theorem1.3 are realized by smooth disks. As we have seen in Example 1.5, there are knots illustrating the first through third cases of Theorem1.3 for which all isotopy classes are realized smoothly.

Recall that two smoothly embedded disks in a $4$ -manifold with boundary $S^3$ are exotic if they are topologically but not smoothly isotopic rel. boundary. Exotic disks are known to exist in $D^4$ [Reference AkbulutAkb91, Reference HaydenHay20] (see also [Reference Hayden and SundbergHS24, Reference Dai, Mallick and StoffregenDMS23]) as well as in many $2$ -handlebodies with $S^3$ boundary, including $(\mathbb {C}P^2)^\circ$ [Reference Conway, Piccirillo and PowellCPP23, Theorem 1.13]. On the other hand, given a knot $K$ that is smoothly slice in a $4$ -manifold $N$ , it remains challenging to decide whether or not $K$ bounds any exotic slice disks in $N$ . Theorem1.2 implies that two smooth crossing change disks $D_{\gamma _1}$ and $D_{\gamma _2}$ with boundary a knot $K$ are exotic if they are smoothly distinct and the classes $[\widetilde {\gamma }_1]$ and $[\widetilde {\gamma }_2]$ agree in $\mathcal {G}_K/\lbrace t^k \rbrace _{k \in \mathbb {Z}}$ .

One broad approach towards generating pairs of exotic disks would thus be to choose a knot $K$ falling into the first three cases of Theorem1.3 and find several distinct unknotting curves that turn $K$ into a smoothly $\mathbb {Z}$ -slice knot. If sufficiently many curves are found, then certainly $[\widetilde {\gamma }_1]=[\widetilde {\gamma }_2] \in \mathcal {G}_K/\lbrace t^k \rbrace _{k \in \mathbb {Z}}$ for some pair $(\gamma _1, \gamma _2)$ . One could then attempt to leverage smooth invariants such as knot Floer homology to distinguish the corresponding pair of disks (see for example [Reference Dai, Mallick and StoffregenDMS23]).

For instance, the trefoil bounds a unique crossing change $\mathbb {Z}$ -disk in $(\mathbb {C}P^2)^\circ$ up to topological isotopy rel. boundary. Thus, any pair of smooth $\mathbb {Z}$ -disks for the trefoil which can be distinguished in the smooth category would constitute an exotic pair. Unfortunately, the authors have not been able to find any such candidates for exotic pairs of disks, although a systematic investigation of this strategy is beyond the scope of this paper.

1.6 Further context and related work

Let $N$ be a simply-connected $4$ -manifold with boundary $S^3$ , and let $K \subset S^3$ be a knot. $\mathbb {Z}$ -surfaces with boundary $K$ considered up to equivalence rel. boundary were classified in [Reference Conway and PowellCP23, Reference Conway, Piccirillo and PowellCPP23]. Complete invariants are given by the equivariant intersection form and automorphism invariant of their exterior. Prior work on the existence question includes [Reference Feller and LewarkFL18, Reference Feller and LewarkFL24] (for $\mathbb {Z}$ -sliceness in $N=D^4 \#_m S^2 \times S^2 \#_n {\mathbb {C}} P^2 \# \overline {{\mathbb {C}} P}^2)$ and [Reference Kjuchukova, Miller, Ray and SakalliKMRS24] (for $\mathbb {Z}$ -sliceness in $N=(\#_\ell {\mathbb {C}} P^2)^\circ$ ). Concerning this latter result, to the best of our knowledge, [Reference Kjuchukova, Miller, Ray and SakalliKMRS24, Theorem1.3] does not recover Theorem1.1. We emphasize again that Theorem1.1 follows fairly promptly from [Reference FogelFog94]. We also note that since $\mathbb {Z}$ -disks are nullhomologous (see for example [Reference Conway and PowellCP23, Lemma 5.1]), Theorem1.1 also yields criteria for a knot to be $H$ -slice, a topic that has attracted some attention recently.

We now focus on $\mathbb {Z}$ -disks. In what follows, the exterior of a disk $D \subset N$ is denoted by $N_D$ . Given a Hermitian form $\lambda$ over $\mathbb {Z}[t^{\pm 1}]$ , we write $\mathcal {D}_\lambda (K,N)^{\textrm {equiv}}$ for the set of rel. boundary equivalence classes of $\mathbb {Z}$ -disks in $N$ with boundary $K$ and equivariant intersection form $\lambda _{N_D}\cong \lambda$ ; we refer to § 2.3 for a brief review of equivariant intersection forms. The automorphism invariant from [Reference Conway, Piccirillo and PowellCPP23] gives rise to a (noncanonical) bijection

\begin{equation*}b \colon \mathcal {D}_\lambda (K,N)^{\textrm {equiv}} \xrightarrow {\approx } {\textrm {Aut}}({\textrm {Bl}}_K)/{\textrm {Aut}}(\lambda ). \end{equation*}

The target of $b$ is the orbit set of the left action of the automorphism group ${\textrm {Aut}}(\lambda )$ on the automorphism group ${\textrm {Aut}}({\textrm {Bl}}_K)$ by $F \cdot h=h \circ \partial F^{-1}$ ; we refer to the introductions of [Reference Conway and PowellCP23, Reference Conway, Piccirillo and PowellCPP23] for more details.

A statement involving isotopy classes of disks instead of equivalence classes of disks has not yet appeared in the literature but can be obtained by tracing through the proofs of [Reference Conway, Piccirillo and PowellCPP23] and applying [Reference Orson and PowellOP22, Theorem C] to upgrade equivalences to isotopies.

We now restrict to $N=(\mathbb {C}P^2)^\circ$ . We observe in Proposition 3.2 that the exterior of a $\mathbb {Z}$ -disk $D \subset (\mathbb {C}P^2)^\circ$ with boundary $K$ necessarily has equivariant intersection form $\lambda _{N_D}(x,y)=x\Delta _K\overline {y}$ . This implies that ${\textrm {Aut}}({\textrm {Bl}}_K)/{\textrm {Aut}}(\lambda ) \approx U(\mathbb {Z}[t^{\pm 1}]/(\Delta _K))/\lbrace \pm t^k \rbrace _{k \in \mathbb {Z}}$ and so $b$ induces a bijection

\begin{equation*}b \colon \mathcal {D}(K,(\mathbb {C}P^2)^\circ )^{\textrm {equiv}} \xrightarrow {\approx } {\textrm {Aut}}({\textrm {Bl}}_K)/{\textrm {Aut}}(\lambda ) \xrightarrow {\approx } U(\mathbb {Z}[t^{\pm 1}]/(\Delta _K))/\lbrace \pm t^k \rbrace _{k \in \mathbb {Z}}. \end{equation*}

Although this has not appeared in print, if one traces through [Reference Conway, Piccirillo and PowellCPP23] and applies [Reference Orson and PowellOP22, Theorem C] when necessary, one can verify that $b$ induces a bijection

(1.3) \begin{equation} b \colon \mathcal {D}(K,(\mathbb {C}P^2)^\circ ) \xrightarrow {\approx } U(\mathbb {Z}[t^{\pm 1}]/(\Delta _K))/\lbrace t^k \rbrace _{k \in \mathbb {Z}}. \end{equation}

In practice, the nature of the automorphism invariant $b_D$ can make it difficult to distinguish concrete $\mathbb {Z}$ -disks in $(\mathbb {C}P^2)^\circ$ with boundary $K$ .

Thus the main novelty of Theorem1.2 resides less in the existence of a bijection as in (1.3) than in the explicit nature of the invariant as well as in the fact that every $\mathbb {Z}$ -disk in $(\mathbb {C}P^2)^\circ$ is isotopic to a crossing change $\mathbb {Z}$ -disk. Another feature of Theorem1.2 is that while the bijection in (1.3) can be derived by modifying the lengthy argument of [Reference Conway, Piccirillo and PowellCPP23, § 5], our argument is more self-contained and instead relies on [Reference FogelFog94]. In summary, our main insight is that, for generalized crossing change $\mathbb {Z}$ -disks, the automorphism invariant boils down to the homology class of the lift of the surgery curve in the Alexander module.

Finally, we describe some additional context surrounding Theorem1.3. In [Reference BoyerBoy86, Reference StongSto93, Reference BoyerBoy93, Reference Crowley and SixtCS11, Reference Conway, Crowley, Powell and SixtCCPS21, Reference Conway, Piccirillo and PowellCPP23, Reference Conway, Crowley and PowellCCP23], isometries of linking forms have been used to study surfaces in $4$ -manifolds as well as the (stable) classification of $4k$ -manifolds. A subset of the applications in these papers rely on being able to find examples of Hermitian forms $(H,\lambda )$ such that the automorphism set ${\textrm {Aut}}(\partial \lambda )/{\textrm {Aut}}(\lambda )$ is large; here $\partial \lambda$ denotes the boundary linking form of $\lambda$ , see for example [Reference Conway and PowellCP23, § 2].

In [Reference Conway, Piccirillo and PowellCPP23], examples were given of $3$ -manifolds $Y$ that have arbitrarily large ${\textrm {Aut}}({\textrm {Bl}}_Y)/{\textrm {Aut}}(\lambda )$ , and this was later improved to examples with infinite ${\textrm {Aut}}({\textrm {Bl}}_Y)/{\textrm {Aut}}(\lambda )$ [Reference Conway, Crowley and PowellCCP23]; we refer to [Reference Conway, Crowley, Powell and SixtCCPS21] for applications of this sort of algebra to the stable classification of $4k$ -manifolds with $k\gt 1$ . Prior to this article, however, few examples of knots $K$ had been produced for which ${\textrm {Aut}}({\textrm {Bl}}_K)/{\textrm {Aut}}(\lambda )$ is nontrivial (i.e. for which the equivariant intersection form does not determine the isotopy type of the disk) and no example for which it is infinite. Surprisingly, Theorem1.3 suggests that the orbit set ${\textrm {Aut}}({\textrm {Bl}}_K)/{\textrm {Aut}}(\lambda )$ might in fact be, generically, infinite.