2·1. Maximal rank Coxeter quotients

Our technique throughout will be the following: given $L_\mathfrak{t}$ a loose fishnet (see Definition 1·2) of width m, we will construct a quotient $\varphi\;:\; \pi_1(S^3\setminus L_\mathfrak{t}) \rightarrow G_\mathfrak{t},$ where $G_\mathfrak{t}$ is a Coxeter group of Coxeter rank ${m}/{2}$ . As the notation suggests, the group $G_\mathfrak{t}$ will be determined by the parameter vector $\mathfrak{t}$ which defines the fishnet. Note that $\varphi$ will be a maximal rank Coxeter quotient in the sense of [ Reference Blair, A. and MorrisonBKM24 ]; in particular, $\varphi$ maps meridians of $L_\mathfrak{t}$ to reflections of $G_\mathfrak{t}$ . Then, the Coxeter rank of $G_\mathfrak{t}$ is equal to the number of local maxima of $L_\mathfrak{t}$ depicted in Figure 1. Such a quotient gives a tight lower bound on the meridional rank of $\pi_1(S^3 \backslash L_\mathfrak{t})$ , as in [ Reference Baader, Blair and KjuchukovaBBK21, Reference Baader, Blair, Kjuchukova and MisevBBKM23 ], proving that $\beta(L)=\mu(L)={m}/{2}$ as claimed.

We will represent quotients $\varphi\;:\; \pi_1(S^3\backslash L_\mathfrak{t})\to G_\mathfrak{t}$ in a diagram D of $L_\mathfrak{t}$ in the usual way, going back to Fox: by labelling strands in D with the images under $\varphi$ of the corresponding Wirtinger meridians in $G_\mathfrak{t}$ . This labelling defines a quotient map iff at each crossing the image of the Wirtinger relation holds in $G_\mathfrak{t}$ . Given a twist region in D, to determine the image of all strands in this region under $\varphi,$ it suffices to label the two strands incoming into the twist region from the top (resp. the bottom).

The following observations will be used repeatedly in the paper. Given any twist region $\sigma_{k}^{t_{j,k}}$ in the fishnet $L_\mathfrak{t}$ , let a and b denote the labels of the two incoming strands from the top and let r denote the order of (ab) in $G_\mathfrak{t}$ . If r divides $t_{j,k},$ then the Wirtinger meridians of the two outgoing strands are also labelled by a and b; moreover, a and b appear in the same order, left to right, at the top and the bottom of the twist region. If $r=2$ (equivalently, a and b commute), then, regardless of whether r divides $t_{j, k}$ , the outgoing strands are again labelled a and b; the order in which the labels a, b appear is determined by the parity of $t_{j, k}$ in the obvious way. These diagrammatic constructions were inspired by Brunner [ Reference BrunnerBru92 ], who studied Artin quotients of link groups.

Proof of Proposition 1·4. Consider the Coxeter group

\[G_\mathfrak{t}=\left\langle a_1, a_2, \dots, a_{\frac{m}{2}}\ \big\lvert \ a_1^2=a_2^2\dots = a_{\frac{m}{2}}^2=1 \text{ and } (a_ja_{j+1})^{d_{2j}}=1 \text{ for all } j \in\{1,...,\tfrac{m}{2}{-}1\} \right\rangle.\]

Since $d_{2j}\gt1$ for all j, it follows that $G_\mathfrak{t}$ has Coxeter rank equal to ${m}/{2}$ [ Reference Felikson and TumarkinFT10 , lemma 2·1]. Let $L_\mathfrak{t}$ be represented in the fishnet diagram determined by the parameter vector $\mathfrak{t}$ in the sense of Definition 1·2. We begin by labelling the strands containing the ${m}/{2}$ local maxima in the diagram by $a_1, a_2, \dots, a_{\frac{m}{2}},$ in this order. Recall that, by assumption, for each j, $d_{2j}$ divides $t_{i,2j} \ \forall i$ . Therefore, by the discussion above, for each twist region in the diagram, the labels of the incoming strands are the same as those of the outgoing strands. This implies that the induced labels at the two endpoints of a strand containing a local minimum agree. We have therefore arrived at a coherent colouring of the diagram by elements of the group $G_\mathfrak{t}$ . This defines the desired quotient $\varphi\;:\; \pi_1(S^3\backslash L_\mathfrak{t})\to G_\mathfrak{t}$ of Coxeter rank ${m}/{2}$ , proving the claim that $\beta(L_\mathfrak{t})=\mu(L_\mathfrak{t})={m}/{2}$ .

To prove the final claim of the proposition, assume that the parameter vector $\mathfrak{t}$ satisfies $|t_{i,j}|\geq 3 \ \forall i,j$ . Then, $L_\mathfrak{t}$ fulfills the hypotheses of Theorem 4·4. Therefore, the distance of the induced bridge sphere in the fishnet diagram is $\lceil n /(m - 4) \rceil$ .

Proof of Corollary 1·5. Fix $r\in \mathbb{R}_{\gt0}$ and $b\in \mathbb{N}_{\geq 3}$ . Let $L_\mathfrak{t}$ be the fishnet determined by the parameter vector

\[ \mathfrak{t}= (t_{1,2},t_{1,4}, \dots, t_{1,m-2}; \dots, t_{n,m{-}2}), \]

where ${m}/{2}=b\geq 3$ and where we choose n such that $n\geq {rm^2}/{2}.$ In this construction, we may select the integer parameters freely, so we assume that $\forall i,j$ , we have $t_{i,j}\gt3$ and $d_{2j}\gt1$ . By Proposition 1·4, it follows that $\beta(L_\mathfrak{t})={m}/{2}$ and that $d(L_\mathfrak{t})\geq {n}/({m-4})$ . Thus, ${d}/{b}\geq ({n}/({m-4}))\cdot({2}/{m}) \gt r,$ as claimed. To conclude the proof, note that by Lemma 3·1 we may furthermore ensure that $L_\mathfrak{t}$ is a knot.

2·2. Other fishnets which satisfy MRC

In what follows, we will work in the setting of loose fishnets. We will see in this section many new families of fishnets which admit maximal rank Coxeter quotients and thus satisfy MRC. We encourage the reader to understand these families as extensions of representative examples.

2·2·1. Decomposing fishnets

The key observation is that the existence of a quotient for a fishnet $L_\mathfrak{t}$ can be detected by decomposing $L_\mathfrak{t}$ into a union of sublinks, $L_\mathfrak{t}=\bigcup_i L_i$ . In particular, the $L_i$ are not necessarily knots.

Consider the fishnet M given by Figure 2. With the majority of boxes at most determined up to parity, M does not necessarily satisfy the gcd condition of Proposition 1·4. However, M admits a maximal rank Coxeter quotient. For fishnets with more than one component, it is common for the existence of a quotient to be dependent on relatively few parameters, as is the case for M.

Figure 2.

A fishnet M which satisfies MRC without fulfilling the hypotheses of Proposition 1·4. M is the link of a dotted unknot, a solid and dashed sublink in 5-bridge position, and a (2, 11)-torus knot, faintly pictured. Unnumbered twist regions which involve two patterns are determined mod 2 as dictated by the exiting strands. Omitted parameters in one-patterned twist regions are arbitrary, subject to the condition that the dashed strands do not form a trivial tangle.

The patterns in Figure 2 are included to guide the construction of the desired quotient. First, the faintly pictured sublink First, (pictured so for diagrammatic clarity) is a 2-bridge link and therefore has a maximal rank Coxeter quotient given by a dihedral group. In this case, the quotient group is $D_{11},$ the dihedral group of order 22. Next, consider the solid/dashed sublink. Because the solid black meets the dashed in a single twist region, we regard each of the differently patterned halves as a summand. If both summands admit maximal rank Coxeter quotients, then the solid/dashed sublink does as well. By assumption, the dashed is a 2-bridge link summand, hence has some dihedral quotient, $D_b$ . The solid black depicts a 3-bridge link summand, but due to the twist region parameterized by $t_{4,2}=1$ , it fails the gcd condition given in Proposition 1·4 and does not have an obvious quotient. Instead, we consider the solid summand along with the dotted unknot. Together these have 4-bridges, and in fact satisfy the gcd condition needed for the solid/dotted to admit a maximal rank Coxeter quotient, $C_{r,g}$ . Moreover, the dotted does not meet the dashed in any twist region, so we can regard the solid/dotted and the dashed as two summands of the solid/dotted/dashed link, summands which both admit maximal rank quotients. By reconciling these two quotient groups, we construct a maximal rank Coxeter quotient, $C_{r,g,b}$ , for the solid/dotted/dashed sublink. Acting by conjugation on $C_{r,g}$ and $D_b$ ensures that two Coxeter generators, r and b, from $C_{r,g}$ and $D_b$ respectively, label the strands entering the top of twist region $t_{5,12}=13$ . Then, a minimal Coxeter generating set is given by the union of minimal Coxeter generating sets of $C_{r,g}$ and $D_{b}$ ; the relations are given by the union of relations for these two groups, together with the additional relation $(rb)^{13}=1$ . At last, this results in the maximal rank Coxeter quotient for M, $C_M = D_{11} \oplus C_{r,g,b}$ .

Note that none of the components of the link in the above example is presented in a diagram of a regular fishnet (not even the dashed summand). Moreover, the example illustrates the following more general phenomena:

1. 1. If k disjoint sublinks of a link $L= L_1 \cup \dots \cup L_k$ independently admit maximal rankFootnote ¹ Coxeter quotients, $G_1, \dots, G_k$ , then L admits a maximal rank quotient onto the direct sum $G\;:\!=\;G_1\oplus \dots \oplus G_k$ . The existence of a quotient $\pi_1(S^3\backslash L)\twoheadrightarrow G$ is not affected by the way these sublinks link each other. In particular, they do not need to form a regular fishnet.

2. 2. Our ability to detect the existence of a quotient in a fixed diagram may depend on the way we subdivide a link into a union of sublinks. For instance, if some component $K_s$ of a link L does not admit a maximal rank Coxeter quotient, it may still be the case that the link $K_s \cup K_t$ admits one, where $K_t$ is some other component of L which meets $K_s$ in at least one twist region.

3. 3. Let $L_1$ and $L_2$ form a split link, presented in a diagram such that $L_1$ and $L_2$ are contained in disjoint balls in the plane. Leaving all but two strands unchanged in the diagrams of $L_1$ and $L_2$ , we can obtain a diagram of a link of the form $L_1\#T_{(2,t)}\#L_2 \,=\!:\,L$ , where $T_{(2,t)}$ denotes the (2, t)-torus link. If the initial diagrams $D_1,D_2$ of $L_1,L_2$ are (possibly loose) fishnets, it is easy to obtain a (necessarily loose) fishnet diagram for L: simply juxtapose $D_1$ and $D_2$ and insert a new column between them, in which one parameter is equal to t and the remaining are equal to zero. (When $L_1$ and $L_2$ have different height, rows of zeroes are added to either $D_1$ or $D_2$ .) In particular, if $L_1$ and $L_2$ admit quotients onto the Coxeter groups $G_1$ and $G_2$ , then L admits a quotient onto a Coxeter group determined by $G_1$ , $G_2$ , and t in the natural way.

Using the above remarks, one readily constructs new families of links which admit maximal rank Coxeter quotients. In particular, one can build regular fishnets of arbitrary dimensions which admit “nonobvious” Coxeter quotients and satisfy MRC.

2·2·2. Combining fishnets

In Figure 3 we introduce the key construction used in Section 3.

Figure 3.

Interweaving fishnets: a fishnet L made up of black sublink $L_1$ and gray sublink $L_2$ , with shaded boxes only determined mod 2. If $L_1$ and $L_2$ each admit maximal rank quotients (determined by the unshaded boxes), so does L. Independently, if L satisfies the gcd condition of Proposition 1·4 (determined by a subset of the shaded boxes), then L admits a different maximal rank quotient.

Figure 3 Long description

A diagram of a fishnet link structure. The diagram represents a fishnet link L made up of a black sublink L1 and a gray sublink L2. The shaded boxes are determined modulo 2. If L1 and L2 each admit maximal rank quotients, determined by the unshaded boxes, then L also admits maximal rank quotients. Additionally, if L satisfies the greatest common divisor condition of Proposition 14, determined by a subset of the shaded boxes, then L admits a different maximal rank quotient. The diagram visually illustrates the relationships and conditions between the sublinks and the overall link structure.

We start with two fishnets, $L_1$ and $L_2$ , in bridge position with b and $b-1$ bridges, respectively; they may have any number of rowsFootnote ² . The two links are depicted, superimposed, in Figure 3. $L_1$ is black, $L_2$ is grey. When the diagrams are simply placed on top of each other, the shaded boxes contain single crossings. We use this setup to construct new families of fishnets in $(2b-1)$ -bridge position by changing the values of the parameters in the shaded boxes. We study the existence of Coxeter quotients of Coxeter rank $2b-1$ for these fishnets.

As in observation (1) above, if sublinks $L_1$ and $L_2$ admit Coxeter quotients of rank b and $b-1$ , then L admits a maximal rank Coxeter quotient given by the direct sum of these two. Unique to this arrangement is that L may be constructed so that it admits a maximal rank quotient regardless of whether sublinks $L_1$ and $L_2$ do. Indeed, the gcd condition of Proposition 1·4 — namely $d_j\gt1$ for all even j — depends only on parameters of L which are in shaded boxes in Figure 3. These parameters are fixed mod 2 in order to maintain the sublink decomposition of L into $L_1$ and $L_2$ , but can always be chosen to satisfy this gcd condition in a way that preserves parity. Therefore, there is always a choice in the construction of L which ensures $L_1 \cup L_2$ admits a maximal rank Coxeter quotient.

3·1. Component number of fishnets

We begin with a discussion of the component number of a fishnet. This will be used to demonstrate that the links which satisfy the hypotheses of Proposition 1·4 can have any number of components between 1 and the bridge number.

Let $L_\mathfrak{t}$ denote a loose fishnet which is the plat closure of an m-stranded braid $\zeta_\mathfrak{t}$ . Consider the homomorphism $\phi\;:\; B_m \twoheadrightarrow S_m$ from the braid group to the symmetric group given by $\phi\;:\; \sigma_i \mapsto (i\;i{+}1)$ . Using this map, we associate a permutation $\phi(\zeta_\mathfrak{t})$ to $L_\mathfrak{t}$ which can be used to obtain the component number of $L_\mathfrak{t}$ .

We will show that any component number between 1 and ${m}/{2}$ can be achieved by a fishnet of width m satisfying a prescribed gcd condition on the parameters in even columns. Precisely, fix any even integer $m\geq 4$ and any set of odd integers $\delta_2, \delta_4 \dots, \delta_{m-2}$ . The following lemma shows that there exists a transitive permutation in $S_m$ which is in the image of a fishnet $L_\mathfrak{t}$ , whose parameter set $\mathfrak{t}$ has the property that $d_{2j}$ is divisible by $\delta_{2j}$ for all $j=1, 2,\dots, {m}/{2}$ .

Lemma 3·1. [ Reference PfaffPfa24 ]. Fix $m \in 2\mathbb N$ , and fix a set $\Delta = \{\delta_2,\delta_4, \dots, \delta_{m{-}2}\}$ such that $\delta_{2j} \geq 3$ and $\delta_{2j} \equiv 1 \mod\;2$ for all $\delta_{2j} \in \Delta$ . Let $\mathcal L_\Delta$ be the set of strong fishnets $L_\mathfrak{t}$ defined by

\[ \mathcal L_\Delta \;:\!=\; \left\{L_{\mathfrak{t}=(t_{1,2},t_{1,4}, \dots, t_{1,m{-}2}; \dots, t_{n,m-2})}\ \big\lvert\ |t_{i,j}|\geq 3 \ \forall i,j \text{ and $\delta_{2j}\mid t_{i,2j} \ \forall i \in \{1,3,\dots,n\}$}, n \in 2\mathbb N+1 \right\}. \]

Then for any $k \in \{1,2, \dots, {m}/{2}\}$ , there exists a strong fishnet $L_{\mathfrak{t}_k} \in \mathcal L_\Delta$ which has k components.

Proof. For $L_\mathfrak{t} \in \mathcal L_\Delta$ , denote by $\zeta_\mathfrak{t}$ its corresponding braid, given explicitly in Definition 1·2. Let $\zeta_\Delta \;:\!=\; \{\zeta_\mathfrak{t} \mid L_\mathfrak{t}\in\mathcal L_\Delta\} \subset B_m$ . Since the component number of $L_\mathfrak{t}$ is determined by its image under map $\phi\;:\; B_m \twoheadrightarrow S_m$ where $\phi(\sigma_i) = (i\ i{+}1)$ as above, it suffices to show that the image of $\zeta_\Delta$ under $\phi$ surjects onto $S_m$ . To do this, we will find fishnets in $\mathcal L_\Delta$ which map to a generating set for $S_m$ , and then realise multiplication in $S_m$ by combining fishnets in a concrete way, defined below.

We will define $L_{\mathfrak{t}_1}, L_{\mathfrak{t}_2} \in \mathcal L_\Delta$ which correspond to an n-cycle and a transposition in $S_m$ , respectivelyFootnote ³ . Let $\mathfrak{t}_1$ be a parameter vector which defines a particular fishnet $L_{\mathfrak{t}_1}$ of width m and height $m{+}1$ with odd entries along a diagonal and even entries elsewhere. Specifically, we set:

\begin{align*} \text{for } i=j+1,\;t_{i,j} =\begin{cases} \delta_j, & i \text{ odd} \\ 3, & i \text{ even} \end{cases}; \text{ and for } i\neq j+1,\;t_{i,j} =\begin{cases} 2\delta_j, & i \text{ odd} \\ 4, & i \text{ even} \end{cases}. \end{align*}

Let $\mathfrak{t}_2 = (\delta_2, 2\delta_4, \dots, 2\delta_{m{-}2})$ , defining a fishnet with a single row. The parameter vectors $\mathfrak{t}_1$ and $\mathfrak{t}_2$ satisfy the necessary conditions to ensure that $L_{\mathfrak{t}_1},L_{\mathfrak{t}_2} \in \mathcal L_\Delta$ . Hence, $\zeta_{\mathfrak{t}_1}, \zeta_{\mathfrak{t}_2} \in \zeta_\Delta$ . Additionally, $\phi(\zeta_{\mathfrak{t}_1}) = (m\ m{-}1\ \dots\ 1)$ and $\phi(\zeta_{\mathfrak{t}_2}) = (2\ 3)$ , so $\phi(\zeta_{\mathfrak{t}_1})$ and $\phi(\zeta_{\mathfrak{t}_2})$ generate $S_m$ .

We now describe how to realise multiplication in $S_m$ by “composing” fishnet diagrams. It is not possible to directly stack one fishnet of width m on top of another due to the parity constraint on rows. Instead, we do the following. Let $\mathfrak{r} = (r_{1,2},r_{1,4}, \dots, r_{1,m{-}2}; \dots, r_{n,m-2})$ and $\mathfrak{s} = (s_{1,2},s_{1,4}, \dots, s_{1,m{-}2}; \dots, s_{n,m-2})$ be parameter vectors which determine fishnets $L_\mathfrak{r}$ and $L_\mathfrak{s}$ of width m. Their associated braids are denoted $\zeta_\mathfrak{r}, \zeta_\mathfrak{s}$ . Then $\phi(\zeta_\mathfrak{r})\phi(\zeta_\mathfrak{s})=\phi(\zeta_\mathfrak{r} \ast \zeta_\mathfrak{s})$ in $S_m$ , where $\ast$ denotes composition in $B_m$ . In order to compose the fishnets $L_\mathfrak{r}$ and $L_\mathfrak{s}$ , we will stack the fishnet diagram of $L_\mathfrak{r}$ above the fishnet diagram of $L_\mathfrak{s}$ with an additional row of even parameters between them, resulting in the fishnet $L_{\mathfrak{r}\ast \mathfrak{s}}$ , where

$$\mathfrak{r}\ast \mathfrak{s} \;:\!=\; (s_{1,2},s_{1,4}, \dots, s_{1,m{-}2}; \dots, s_{n,m-2};\;4,4,\dots,4;\;r_{1,2},r_{1,4}, \dots, r_{1,m{-}2}; \dots, r_{n,m-2}).$$

Then $\phi(\zeta_{\mathfrak{r} \ast \mathfrak{s}}) = \phi(\zeta_\mathfrak{r})(1)\phi(\zeta_\mathfrak{s}) = \phi(\zeta_\mathfrak{r})\phi(\zeta_\mathfrak{s}) \in S_m$ . By construction, all integer parameters in $L_{\mathfrak{r}\ast \mathfrak{s}}$ are at least 3, so $L_{\mathfrak{r}\ast \mathfrak{s}}$ is a strong fishnet, as desired.

Since there is a composition of fishnets associated to multiplication in $S_m$ , we conclude from the surjectivity of $\phi$ that for every permutation in $S_m$ , there exists a fishnet diagram in $\mathcal L_\Delta$ whose underlying braid word is mapped to this permutation by $\phi$ . It is clear that the component number of a fishnet $L_\mathfrak{t}$ is determined by $\phi(\zeta_\mathfrak{t})$ and that, if every permutation of the strands can be achieved, then so can every component number. Since the image of $\zeta_\Delta$ under $\phi$ surjects onto $S_m$ , we conclude that fishnets in $\mathcal L_\Delta$ achieve every component number $k \in \{1,2,\dots,{m}/{2}\}$ .

Note that every fishnet in the set $\mathcal L_\Delta$ satisfies the gcd condition in Proposition 1·4, and so admits a maximal rank Coxeter quotient. A quotient admitted by all fishnets in $\mathcal L_\Delta$ is the Coxeter group given by

\[G_\Delta =\left\langle a_1, a_2, \dots, a_{\frac{m}{2}}\ \big\lvert \ a_1^2=a_2^2\dots = a_{\frac{m}{2}}^2=1 \text{ and } (a_ja_{j+1})^{\delta_{2j}}=1 \text{ for all } j \in\{1,...,{m}/{2}{-}1\} \right\rangle.\]

The group $G_\Delta$ is a quotient (possibly trivially so) of each of the Coxeter groups $G_\mathfrak{t}$ , for $\mathfrak{t}$ such that $L_\mathfrak{t}\in\mathcal{L}_\Delta.$ (Here, $G_\mathfrak{t}$ is as defined in the proof of Proposition 1·4 in terms of gcds of integers in $\mathfrak{t}$ .) By application of Lemma 3·1, the set of fishnets of width m which admit $G_\Delta$ as a quotient contains fishnets of every component number from 1 to ${m}/{2}$ .

3·2. Proof of Theorem 1·1

Let L be a link in bridge position with b bridges, presented in a diagram D with b local maxima with respect to the y-axis in the plane of projection. Note that we do not require $b=\beta(L)$ and indeed our argument applies to any link in any bridge position, except the case where L is an unknot and $b=\beta(L)=1$ .

Our proof can be split into two steps. First, we use D to construct a new diagram, D’, of L such that D’ is a loose fishnet of width 2b. We note that in most cases the process of obtaining D’ increases the crossing number. The second step is to interweave U, an unknot in bridge position with $b-1$ bridges, through D’ to obtain the desired link $L\cup U.$

Step one. A sequence of planar isotopies and Reidemeister II moves (potentially increasing the crossing number) transforms D into a diagram $D_1$ which is the plat closure of a braid. Moreover, we can assume that all local maxima of $D_1$ are at the same height, say $y=h_1$ , and similarly, that all local minima are at the same height, $y=h_2$ . We require that all crossings be at different heights between $h_1$ and $h_2$ , so that we can read off a braid word $\zeta^\ast=\sigma_{i_1}^{r_1}\sigma_{i_2}^{r_2}\cdots \sigma_{i_k}^{r_k}$ from $D_1$ . Here each $i_j\in\{1, 2, \dots, 2b{-}1\}$ and $\sigma_{i_j}$ denotes the corresponding generator of the braid group. Of course, $\zeta^\ast$ is not uniquely determined by D. We assume that $\zeta^\ast$ is reduced in the sense that $i_j\neq i_{j+1}$ for all j.

In our construction, for each $j=1, 2, \dots, k{-}1,$ the term $\sigma_{i_j}^{r_j}$ in $\zeta^\ast$ will take up two rows in a loose fishnet diagram of L. The last term, $\sigma_{i_k}^{r_k}$ , will take up a single row. As a result, there will be $2k-1$ rows in the diagram constructed, of which $k-1$ rows will consist entirely of zeroes. Clearly, when $k\gt1$ the fishnet produced in this manner will not have minimal height. This inefficiency does not increase the bridge number of $L\cup U$ , so we do not strive for minimality.

Consider $\sigma_{i_1}^{r_1}$ . It will contribute a box labelled $r_1$ in column $i_1$ . This box will be either in row 1 or in row 2, depending on the parity of $i_1$ . The remaining parameters in the first two rows will be zero. We proceed in the same manner for all $j\lt k$ . That is, $\sigma_{i_j}^{r_j}$ determines rows $2j-1$ and 2j. Again, all entries in these rows are zero, except for the entry in column $i_j,$ which is equal to $r_j$ .

Figure 4.

A link L in fishnet position with b bridges, colored black, and an unknot $U \subset S^3\backslash L$ , colored grey, embedded in its exterior. The resulting link $L\cup U$ surjects onto the symmetric group $S_{2b},$ mapping meridians to transpositions, as shown. Thus, $\beta(L\cup U) = \mu(L \cup U)= 2b-1$ .

Without loss of generality, we may assume that $i_k$ is even. (If it is odd, then the plat closures of $\sigma_{i_1}^{r_1}\sigma_{i_2}^{r_2}\dots \sigma_{i_k}^{r_k}$ and $\sigma_{i_1}^{r_1}\sigma_{i_2}^{r_2}\dots \sigma_{i_{k-1}}^{r_{k{-}1}}$ represent the same link.) The last, $(2k-1)$ th, row of D’ will then consist of $r_k$ in the $i_k$ -th column and zeroes in the other columns. We do not follow this by a row of zeroes. This completes the construction of a loose fishnet diagram D’ for L.

Step two. Having presented L as a loose fishnet, we now let the diagram D’ play the role of the black sublink in Figure 4. (This is the step where our construction uses the assumption $b\gt1.$ ) One possible way of choosing the desired unknot $U\subset S^3\backslash L$ is pictured in grey. In this construction, the link $L\cup U$ satisfies the hypotheses of Proposition 1·4 with $d_{2j}=3$ for all $j\in \{1, 2,\dots {m}/{2}{-}1\}$ .Footnote ⁴ Therefore, $L\cup U$ admits as a quotient the Coxeter group

\[G_{2b}\;:\!=\;\left\langle a_1, a_2, \dots, a_{\frac{m}{2}}\ \big\lvert \ a_1^2=a_2^2\dots = a_{\frac{m}{2}}^2=1 \text{ and } (a_ja_{j+1})^{3}=1 \text{ for all } j \in\{1,...,{m}/{2}{-}1\} \right\rangle.\]

This suffices to establish the equality

(3·1) \begin{equation} \beta(L\cup U)=\mu(L\cup U)= \frac{m}{2}=2b-1.\end{equation}

To determine the rank of $\pi_1(S^3\backslash(L\cup U)),$ we consider a further quotient of $G_{2b}$ (and hence of the group of $L\cup U$ ), defined as follows:

\[G_{2b}'\;:\!=\;\left\langle a_1, a_2, \dots, a_{\frac{m}{2}}\ \big\lvert \ a_1^2=a_2^2\dots = a_{\frac{m}{2}}^2=1 \text{ and } (a_ia_{j})^{3}=1 \text{ for all } i\neq j \right\rangle.\]

The proof of [ Reference Kaufmann and ZieschangKZ92 , lemma 3·2(c)] applies without modification to show that $\text{rank}(G_{2b}')={m}/{2}$ . This then implies that $\text{rank}(\pi_1(S^3\backslash(L\cup U)))=\mu(L\cup U) = \frac{m}{2},$ concluding the proof.

Remark 3·2. By letting all twist regions between L and U contain $\pm3$ crossings, we have arranged that $G_{2b}$ admits the symmetric group $S_{2b}$ as a further quotient. An explicit quotient

\[\pi_1(S^3\backslash(L\cup U))\twoheadrightarrow S_{2b},\]

mapping meridians to transpositions, is given in Figure 4. This quotient suffices to establish MRC for $L\cup U$ without appealing Proposition 1·4 or the more general theory of Coxeter groups.

Remark 3·3. Given a link $L\subset S^3$ , let E(L) denote its exterior. Recall that the tunnel number of L, denoted t(L), is the minimal number of arcs properly embedded in E(L) such that the exterior of the arcs in E(L) is a handlebody. By definition, $t(L)=g(E(L))-1$ , where g(E(L)) is the Heegaard genus of E(L). One can show $g(E(L))\geq {\rm rank}(\pi_1(S^3\backslash L))$ by an application of the Seifert–Van Kampen theorem to any minimal Heegaard splitting of E(L). Additionally, we have that $\mu(L)\geq {\rm rank}(\pi_1(S^3\backslash L))$ by the definition of meridional rank, and that $\beta(L)\geq g(E(L))$ since any b-bridge splitting of L gives a genus b Heegaard splitting for E(L). Putting these together, we have $\beta(L)\geq g(E(L))= t(L)+1\geq {\rm rank}(\pi_1(S^3\backslash L)).$ Therefore, the links $L\cup U$ obtained by our construction have the property that

\[\beta(L\cup U)=\mu(L\cup U)= t(L\cup U)+1 ={\rm rank}(\pi_1(S^3\backslash(L\cup U)))=2b-1.\]

4·1. Distance

Distance is a complexity measure for bridge spheres, obtained from the distance between disk sets in the curve complex. Let $L \subset S^3$ be a link and let $\Sigma$ be any bridge sphere for L separating $S^3$ into balls $B_1$ and $B_2$ . Define the curve complex of $\Sigma$ , denoted $\mathcal{C}(\Sigma)$ , to be the graph whose vertices are isotopy classes of essential simple closed curves in $\Sigma$ . Two vertices are connected by an edge if their corresponding curves can be isotoped in $\Sigma\backslash L$ to be disjoint. We endow the vertex set of the curve complex $\mathcal{C}(\Sigma)$ with a metric in the natural way: assign length one to each edge and let the distance between two vertices be the length of the shortest path between them. Let the disk set, $\mathcal{D}_i \subset \mathcal{C}(\Sigma)$ , be the set of vertices in $\mathcal{C}(\Sigma)$ which correspond to essential curves in $\Sigma$ that bound compressing disks in $B_i$ , $i = 1,2$ . We define the distance of $\Sigma$ , denoted $d(\Sigma)$ , to be

\[ d(\Sigma) = \min\{ d(c_1,c_2)\;:\; c_i \in \mathcal{D}_i\}.\]

If $\Sigma$ is punctured four or fewer times (i.e., when the curve complex is empty or disconnected), we define the distance of $\Sigma$ to be infinite. The distance of L, d(L), is the maximum possible distance $d(\Sigma)$ of any bridge sphere $\Sigma$ for a representative of L such that $\Sigma$ realises the bridge number of L. It follows from [ Reference TomovaTom07 ] that as long as $\beta(L) \geq 3$ , this maximum is a well-defined positive integer.

Given a bridge sphere $\Sigma$ for a link L, several authors have given upper bounds on the distance of $\Sigma$ in terms of the topology of certain essential [ Reference Bachman and SchleimerBS05, Reference Blair, Campisi, Johnson, Taylor and TomovaBCJ+17 ] or bi-compressible [ Reference TomovaTom07, Reference Blair, Campisi, Johnson, Taylor and TomovaBCJ+17 ] surfaces in the link exterior. For simplicity, we have restricted ourselves to considering only knots and using only three such results when proving Theorem 4·5. We remark that some of the bounds on distance in Theorem 4·5 can be improved by employing lengthier arguments or alternative upper bounds on distance.

First, we need an upper bound on the distance of a knot in terms of the genus and number of boundary components of an essential (possibly meridional) surface in the knot exterior. The following theorem is a restatement of Theorem 5·7 of [ Reference TomovaTom07 ] for knots in $S^3$ . This is very similar to the main theorem of [ Reference Bachman and SchleimerBS05 ], which uses a slightly different definition of distance. In the following theorem, E(L) denotes the exterior of a link L in $S^3$ .

Theorem 4·1. [ Reference TomovaTom07, Reference Bachman and SchleimerBS05 ]. Suppose that L is a non-trivial knot in $S^3$ . Let $\Sigma$ be a bridge sphere for L. Suppose that $\overline{S} \subset S^3$ is a surface transverse to L such that $S = \overline{S} \cap E(L)$ is not a sphere and is essential in E(L). Then

\[d(\Sigma) \leq \max\{3, 2g(S) + |\overline{S} \cap L|\}.\]

If, in addition, $\overline{S} = S$ and if S is a torus, then $d(\Sigma) \leq 2$ .

Next, we need an upper bound on the distance of a knot in terms of the genus and number of boundary components of an essential surface with non-meridional boundary components. The following theorem is a specialisation of Theorem 5·7 of [ Reference Blair, Campisi, Johnson, Taylor and TomovaBCJ+17 ] to knots in $S^3$ .

Theorem 4·2 [ Reference Blair, Campisi, Johnson, Taylor and TomovaBCJ+17 ]. Let K be a nontrivial knot in $S^3$ with $\beta(K)\geq 3$ . Suppose that F is a connected orientable essential surface properly embedded in the exterior of K with non-empty, non-meridional boundary. Then

$$\beta(K)(d(K)-4)\leq \frac{8g(F)-8}{|\partial F|}+4.$$

Finally, we need an upper bound on the distance of a knot in terms of the Heegaard genus of a manifold obtained by Dehn surgery. The following theorem is a specialisation of Theorem 9 in [ Reference Blair, Campisi, Johnson, Taylor and TomovaBCJ+17 ] to knots in $S^3$ .

Theorem 4·3. [ Reference Blair, Campisi, Johnson, Taylor and TomovaBCJ+17 ]. Let K be a non-trivial knot in $S^3$ . Let M’ be the result of non-trivial Dehn surgery on K and let g be the Heegaard genus of M’. Then

\[ d(K) \leq \max\Big\{\frac{2}{\beta(K)} + 4, \frac{4g}{\beta(K)} + 4, 2g+ 2\Big\}.\]

4·2. Distance and fishnet links

In [ Reference Jesse and YoavJM16 ], Johnson and Moriah study the distance of strong fishnet links, described in the language of 2k-plat projections. Given a diagram of a fishnet link as depicted in Figure 1, a horizontal line in the plane of projection that separates all maxima from all minima naturally corresponds to a bridge plane for the knot in $\mathbb{R}^3$ , or, equivalently, a bridge sphere for the knot in $S^3$ . We call any bridge sphere transversely isotopic to this the induced bridge sphere for a fishnet link. Denote by $\lceil x \rceil$ the ceiling function on x, equal to the smallest integer greater than or equal to x. Johnson and Moriah showed the following:

4·3. Distance and classes of links known to satisfy the MRC

In this section we show that knots for which the MRC has previously been established have either small bridge number or the property that the ratio of their bridge distance to their bridge number is less than 3. We have restricted ourselves to knots here. That said, the tools used would apply to links in all cases except generalised Montesinos links [ Reference Lustig and MoriahLM93 ], for which our proof would require a version of Theorem 4·3 for links. That said, we believe that Theorem 4·5 holds for links as well.

Proof. In [ Reference Boileau and ZieschangBZ85 ], [ Reference Baader, Blair and KjuchukovaBBK21 ] and [ Reference Baader, Blair, Kjuchukova and MisevBBKM23 ], the authors prove the MRC for classes of Montesinos and arborescent knots. Every knot in these classes has the property that it is either bridge number at most 3 or that its exterior contains an essential meridional 4-punctured sphere. In the latter case, the distance of the knot is less than or equal to 4 by Theorem 4·1.

In [ Reference Rost and ZieschangRZ87 ], [ Reference Boileau and ZimmermannBZ89 ], [ Reference CornwellCor14 ], [ Reference Cornwell and HemmingerCH16 ], [ Reference Boileau, Dutra, Jang and WeidmannBDJW17 ], [ Reference Baader and KjuchukovaBK20 ] and [ Reference DutraDut22 ], the authors prove the MRC for classes of 3-bridge knots, torus knots or satellite knots. Every knot in these classes has one or more of the following properties: its bridge number is at most 3, its exterior contains an essential non-meridional annulus, or its exterior contains an essential torus. In each of the latter two cases, the distance of the knot is less than or equal to three by Theorems 4·1 and 4·2.

In [ Reference Lustig and MoriahLM93 ], the authors prove the MRC for a class of knots known as “generalised Motesinos knots”. As a consequence of Theorem 0.1 in that same paper, every generalised Motesinos knot K which the authors show satisfies the MRC additionally admits a non-trivial Dehn surgery resulting in a 3-manifold M(K) such that the Heegaard genus of M(K) is equal to $\beta(K)$ . However, by Theorem 4·3, this non-trivial Dehn surgery implies that the distance of K is at most the maximum of 8 and $2\beta(K)+2$ .

A second class of knots which the authors of [ Reference Baader, Blair and KjuchukovaBBK21 ] prove satisfy the MRC are twisted links. Here we will give a brief overview of the construction of these links (restricting our attention to knots). For additional details, we refer the reader to [ Reference BrunnerBru92 ], where twisted links were introduced; and to [ Reference Baader, Blair and KjuchukovaBBK21 ], where they were shown to satisfy MRC.

Let D be a diagram of a knot L, and let F be one of the two surfaces with boundary L obtained from a checkerboard coloring of the regions in the plane determined by D. We regard the surface F as a union of disks and twisted bands. We assume D is reduced in the sense that it does not contain any nugatory crossings and that the sign of crossings in each band of F is constant. When every band of F is maximal (i.e. no disk is incident to exactly two bands) and contains at least one full twist, we call F a twisted surface. A diagram D is twisted if it determines such a surface via a checkerboard coloring, and a knot is twisted if it admits a twisted diagram. For instance, the standard diagram of the $P(a_1, a_2, \dots, a_n)$ pretzel knot is twisted so long as each $a_i$ satisfies $|a_i|\gt1$ .

Retract the spanning surface F to a planar graph $\Gamma$ , where vertices of $\Gamma$ represent disks of F and edges represent bands of F. The edges of $\Gamma$ are weighted with the number of half-twists in the corresponding band of F. Denote the dual weighted planar graph by $\Gamma^\ast$ , where each edge of $\Gamma^\ast$ inherits the weight of the corresponding edge of $\Gamma$ . The surface F is twisted if and only if all weights of $\Gamma$ are at least 2 in absolute value and $\Gamma$ has no vertices of valence one or two; L is then a twisted knot. In [ Reference Baader, Blair and KjuchukovaBBK21 ], the authors prove the MRC for twisted knots, and they show that the bridge number of a twisted knot L is equal to the number of vertices in $\Gamma^\ast$ .

Let K be a twisted knot, let $v^*$ be the number of vertices in $\Gamma^\ast$ , and let v, e, f be the number of vertices, edges, and faces, respectively, of the planar embedding of $\Gamma$ induced by F. Then $f=v^*$ and $2=v-e+v^*$ . Since F retracts to $\Gamma$ , then $\chi(F)=v-e$ and $2=\chi(F)+v^*$ . Since $v^*=\beta(K)$ by [ Reference Baader, Blair and KjuchukovaBBK21 ], we have that

(4·1) \begin{equation} \chi(F)= 2-\beta(K). \end{equation}

If F is orientable, then maximally compressing F results in an essential Seifert surface, $F^c$ , with genus at most the genus of F and with one, non-meridional, boundary component. By Theorem 4·2, $\beta(K)\leq 5$ , or $d(K)\leq ({8g(F^c)-8})/{5|\partial F^c|} + {4}/{5}+4\leq 2g(F)+{16}/{5}$ . Since d(K) is an integer, $d(K)\leq 2g(F)+3$ . Since $g(F)={1}/{2}(1-\chi(F))$ , using Equation (4·1) we conclude that $d(K)\leq \beta(K)+2$ .

If F is non-orientable, we can use a similar construction. Let $H_F$ be a closed regular neighbourhood of F in $S^3$ . So, $H_F$ is a handlebody of genus $1-\chi(\Gamma)=1-\chi(F)$ . Since K can be assumed to lie in the surface $\partial H_F$ , we have that $\partial H_F$ meets a neighbourhood of K in exactly one annulus, denoted A. Hence $F'\;:\!=\;\partial H_F \setminus int(A)$ is a compact, connected, orientable surface properly embedded in the exterior of K. Since $F'$ is obtained from $\partial H_F$ by deleting an annulus whose core is non-separating in $\partial H_F$ , we obtain $g(F')=g(\partial H_F)-1 =-\chi(F)$ . Moreover, $F'$ has two boundary components. Note that $F'$ may be compressible. Maximally compressing $F'$ in the exterior of L produces an essential surface $F''$ with genus at most $-\chi(F)$ and with at most two boundary components. If $\beta(K)\leq 5$ , then the theorem follows. So, we will assume $\beta(K)\geq 6$ . By Theorem 4·2, $\beta(K)(d(K)-4)\leq ({8g(F'')-8})/{|\partial F''|}+4.$ Since $\beta(K)\geq 6$ , $d(K)\leq ({8g(F'')-8})/{6|\partial F''|}+ {14}/{3}$ . Since $1\leq |\partial F''|\leq 2$ , then $d(K)\leq ({8g(F'')-8})/{6}+ {14}/{3}=({4}/{3})g(F'')+{10}/{3}.$ Since $g(F'')\leq -\chi(F)=\beta(K)-2$ , then $d(K)\leq ({4}/{3})\beta(K)+{2}/{3}$ .