2. Symmetric unions and their braidings

Let $D_J \subset \mathbb{R}^2$ be an unoriented diagram of a knot J. After choosing a straight line axis $A \subset \mathbb{R}^2$ with $D_J \cap A = \varnothing$ , reflect $D_J$ about A to get a diagram $-D_J$ of the mirror image of J. Then, for some $k \geq 0$ , choose $k+1$ discs $T_0, T_1, \dots, T_k$ embedded disjointly in $\mathbb{R}^2$ such that each $T_i$ is invariant under reflection about A and intersects the diagram $D_J \sqcup -D_J$ in two unknotted arcs. Fix an integer $\mu$ such that $1 \leq \mu \leq k+1$ . Viewing each $T_i$ as a rational 0-tangle, replace it by an $m_i$ -tangle $T_i'$ , where $m_i = \infty$ for $0 \leq i \leq \mu-1$ and $m_i \in \mathbb{Z} \setminus \{0\}$ for $\mu \leq i \leq k$ . Choose an orientation on the resulting diagram and denote it by $D_L = (D_J \sqcup -D_J)(\infty_\mu, n_1, \dots, n_l)$ , where $l = k - \mu + 1$ and $n_i = m_{\mu -1 + i}$ , and call the tangles $T_{\mu}', \dots, T_k'$ twist regions. The construction is illustrated in Figure  3 .

Fig. 3.

Construction of the symmetric union diagram $(D_J \sqcup -D_J)(\infty_\mu, n_1, \dots, n_l)$ ; rectangles labelled $n_1, \dots, n_l$ contain the respective number of ‘vertical’ half-twists.

Observe (cf. [Reference Lamm11, remark 2.2]) that an SU link L has $\mu$ components, hence L is a knot if and only if $\mu = 1$ ; in the following, we shall primarily be concerned with this case.

In general, an SU knot K can admit many different partial knots; however, the determinant of any partial knot is prescribed by that of K.

By [Reference Lamm11, Theorem 4.3], a link L is an SU link if and only if it can be represented as the closure of an SU braid $\beta$ , i.e., a braid given by

(†) \begin{equation} \beta = \gamma C_1 \gamma^{-1} C_2 \in B_n \end{equation}

for some $n \geq 1 $ , where $\gamma \in B_n$ is arbitrary and $C_i$ are given by the following:

1. (1) if $n = 1$ , then $C_1$ and $C_2$ are trivial;

2. (2) if $n = 2$ , then $C_1 = \textrm{id} \in B_2$ and $C_2 \in \{ \sigma_1^{\pm 1} \}$ ;

3. (3) if $n \geq 3$ , then

   \begin{align*} C_1 &= \begin{cases} \sigma_2^{\varepsilon_{12}} \sigma_4^{\varepsilon_{14}} \dots \sigma_{n-1}^{\varepsilon_{1(n-1)}} & \textrm{if $n$ is odd} \\[5pt] \sigma_3^{\varepsilon_{13}} \sigma_5^{\varepsilon_{15}} \dots \sigma_{n-1}^{\varepsilon_{1(n-1)}} & \textrm{if $n$ is even, and} \end{cases}\\[5pt]C_2 &= \begin{cases} \sigma_2^{\varepsilon_{22}} \sigma_4^{\varepsilon_{24}} \dots \sigma_{n-1}^{\varepsilon_{2(n-1)}} & \textrm{if $n$ is odd} \\[5pt]\sigma_1^{\varepsilon_{21}} \sigma_3^{\varepsilon_{23}} \dots \sigma_{n-1}^{\varepsilon_{2(n-1)}} & \textrm{if $n$ is even} \end{cases} \end{align*}
   such that $\varepsilon_{ij} \in \{\pm 1\}$ for all i and j.

Fix $m \geq 1$ and suppose that K is an SU knot with $b_s(K) \in \{ 2m-1, 2m \}$ . Then K can be constructed by taking $D_J$ to be the plat closure of a braid $\gamma \in B_{2m}$ . We illustrate the case $m = 2$ in Figure  4 , where the diagram $D_J$ is a 4-plat closure, and hence, provided $\det(K) \neq 1$ , the partial knot J is a two-bridge knot $K_{p,q}$ for some $0 \lt q \lt p$ (see, e.g., [Reference Burde and Zieschang3, Section 12.B]). Since $\det(K_{p,q}) = p$ for any p, by Theorem 2·2 the following result holds.

Fig. 4.

A knot K with $b_s(K) \in \{ 3, 4 \}$ admits an SU diagram with the partial knot J given by the plat closure of $\gamma \in B_4$ , where each twist region $\tau_i$ contains one crossing. If $\gamma$ does not contain any $\sigma_1$ letters, then the topmost free strand in the right diagram can be moved all the way down so that the $\tau_3$ region can be removed via the Reidemeister I move; this yields the closure of an SU 3-braid. (Cf. [Reference Lamm11, Figure 13].)

Proposition 2·4. If K has $b_s(K) \in \{3, 4\}$ and $\det(K) \neq 1$ , then K must admit a partial knot J belonging to the finite set

\begin{equation*} \mathcal{J}_p = \{ K_{p,q} : 0 \lt q \lt p \text{ and } p^2 = \det(K) \}. \end{equation*}

On the other hand, if $\det(K)=1$ , then $K_{p,q}$ is necessarily the unknot $ K_{1,q}$ .

3. Classification of knots with symmetric braid index 3

First let us discuss the classification of finite concordance order 3-braid knots proved by Lisca in [Reference Lisca 13 ] and briefly introduced in Section 1. The classification consists of three families (1), (2) and (3); the following is an outline of their main properties:

1. (1) Family (1) consists of knots K such that K or $-K$ is the closure of a braid of the form

   (*) \begin{equation} \beta = (\sigma_1 \sigma_2)^3 \sigma_1^{x_1} \sigma_2^{-y_1} \dots \sigma_1^{x_n} \sigma_2^{-y_n}, \end{equation}
   where $n, x_i, y_i \geq 1$ , $\sum_i y_i = \sum_i x_i + 4$ , and all $x_i$ and $y_i$ are subject to a technical combinatorial condition; moreover, $\beta$ is quasipositive and K is ribbon;

2. (2) Family (2) consists of braided SU knots obtained as closures of braids of the form

   \begin{equation*}\beta = \gamma \sigma_2^{\pm 1} \gamma^{-1} \sigma_2^{\mp 1},\end{equation*}
   where $\gamma \in B_3$ ; in particular, (2) contains the unknot;

3. (3) Family (3) consists of certain alternating amphichiral knots with dihedral symmetry obtained as closures of braids of the form

   \begin{equation*} \beta = \sigma_1^{x_1} \sigma_2^{-y_1} \dots \sigma_1^{x_n} \sigma_2^{-y_n}, \end{equation*}
   where $n \geq 2$ and $x_i, y_i \geq 1$ .

Family (1) is disjoint from $(2) \cup (3)$ , but families (2) and (3) have non-trivial intersection. According to the KnotInfo database [Reference Livingston and Moore 14 ], there are knots in (2), such as $10_{48}$ and $12_{a1011}$ , that are chiral, hence $ (2) \setminus (3)$ is non-empty. On the other hand, $(3) \setminus (2)$ contains all non-slice finite concordance order 3-braid knots, such as the figure-eight knot. In Theorem 2 we will show that, in fact, families (1) and (2) consist precisely of knots with $b_s(K) \leq 3$ , whilst Theorem 1·4 exhibits ribbon knots in $(3) \setminus (2)$ .

In order to state and prove Theorem 2, we now recall some relevant terminology regarding 3-braids, established in [Reference Lisca 13 ]. For any $\beta \in B_3$ of the form

\begin{equation*} \beta = (\sigma_1 \sigma_2)^{3t} \sigma_1^{x_1} \sigma_2^{-y_1} \dots \sigma_1^{x_n} \sigma_2^{-y_n} \end{equation*}

with $n, x_i, y_i \geq 1$ , $t \in \mathbb{Z}$ , we define the associated string $\mathbf{a}(\beta)$ of $\beta$ to be the string

\begin{equation*} (2^{[x_1-1]}, y_1 + 2, 2^{[x_2-1]}, y_2 + 2, \dots, 2^{[x_n - 1]}, y_n + 2), \end{equation*}

where $2^{[m]}$ denotes the integer 2 repeated m times. The pair $(t, \mathbf{a}(\beta))$ uniquely recovers $\beta$ and determines the isotopy type of $\mathrm{cl}(\beta)$ up to cyclic rotation and reflection of $\mathbf{a}(\beta)$ .

The linear dual of a string $\mathbf{b} = (b_1, \dots, b_k)$ with all $b_i \geq 2$ is defined as follows:

1. (i) if $\mathbf{b} = (2^{[m]})$ , then its linear dual is the string $\mathbf{c} = (m+1)$ ;

2. (ii) if there exists j such that $b_j \geq 3$ , write $\mathbf{b}$ as

   \begin{equation*}\mathbf{b} = (2^{[m_1]}, 3 + n_1, 2^{[m_2]}, 3 + n_2, \dots, 2^{[m_l]}, 2 + n_l)\end{equation*}
   with all $m_i, n_i \geq 0$ ; then its linear dual is the string
   \begin{equation*} \mathbf{c} = (2 + m_1, 2^{[n_1]}, 3 + m_2, 2^{[n_2]}, 3 + m_3, \dots, 3 + m_l, 2^{[n_l]}).\end{equation*}

One can verify that the notion of a linear dual is symmetric. Now we prove the following.

Moreover, in case $(\mathrm{A})$ , the knot K is always ribbon and it is quasipositive when $\varepsilon_1 = \varepsilon_2 = 1$ , and in case $(\mathrm{B})$ , it is amphichiral.

Proof of Theorem 1·3. First, we show that Lisca’s families (1) and (2) are included in case $(\mathrm{A})$ . In view of the discussion at the start of the section, it is, in fact, sufficient to show that knots in family (1) are closures of 3-braids of the form $\gamma \sigma_2^{\pm 1} \gamma^{-1} \sigma_2^{\pm 1}$ . To this end, we leverage previous work [Reference Brejevs and Simone2, Reference Simone 17 ] by Simone and the first author and Simone; it contains a classification of all links arising as closures of 3-braids of the form

\begin{equation*}\beta = (\sigma_1 \sigma_2)^{3t} \sigma_1^{x_1} \sigma_2^{-y_1} \dots \sigma_1^{x_n} \sigma_2^{-y_n}\end{equation*}

with $n, x_i, y_i \geq 1$ and $t \in \{ \pm 1 \}$ , that are $\chi$ -slice, which is a generalisation of the usual notion of knot sliceness to links. In particular, this classification includes all knots in family (1). It follows from [Reference Brejevs and Simone 2 , theorem 1·11] [Reference Brejevs and Simone 2 ] that, up to mirroring, knots in family (1) are closures of 3-braids with $t = -1$ whose associated strings lie in the family

\begin{equation*}\mathcal{S}_{1a} = \{ (b_1, \dots, b_k, 2, c_l, \dots, c_1, 2 \mid k+l \geq 3 \},\end{equation*}

where $(b_1, \dots, b_k)$ and $(c_1, \dots, c_l)$ are linear duals. Furthermore, [Reference Brejevs 1 , section 2] and [Reference Brejevs and Simone 2 , figure 14] imply that such knots emerge as closures of 3-braids of the form

\begin{equation*}\beta' = \Delta^{-2} \sigma_1 \overline{\zeta} \sigma_1^2 \zeta^{-1} \sigma_1,\end{equation*}

where $\zeta \in B_3$ is arbitrary, $\Delta = \sigma_1 \sigma_2 \sigma_1 = \sigma_2 \sigma_1 \sigma_2$ is the Garside element in $B_3$ , and $\overline{\,\cdot\,} : B_3 \rightarrow B_3$ is the map defined by $\sigma_1 \mapsto \sigma_2$ and $\sigma_2 \mapsto \sigma_1$ . Recalling that $\Delta^{-1} \sigma_1 = \sigma_2 \Delta^{-1}$ and $\sigma_2 \Delta^{-1} = \Delta^{-1} \sigma_1$ , we can write $\Delta^{-1} \sigma_1 \overline{\zeta} = \sigma_2 \zeta \Delta^{-1}$ , and hence

\begin{align*}\beta' &= \Delta^{-1} \sigma_2 \zeta \Delta^{-1} \sigma_1^2 \zeta^{-1} \sigma_1 \\&= (\sigma_2^{-1} \sigma_1^{-1} \sigma_2^{-1}) \sigma_2 \zeta (\sigma_1^{-1} \sigma_2^{-1} \sigma_1^{-1}) \sigma_1^2 \zeta^{-1} \sigma_1 \\&= \sigma_2^{-1} \sigma_1^{-1} \zeta \sigma_1^{-1} \sigma_2^{-1} \sigma_1 \zeta^{-1} \sigma_1 \\ & = \sigma_2^{-1} \gamma \sigma_2^{-1} \gamma^{-1} \\ &\sim \gamma \sigma_2^{-1} \gamma^{-1} \sigma_2^{-1},\end{align*}

where $\gamma = \sigma_1^{-1} \zeta \sigma_1^{-1}$ and $\sim$ denotes conjugacy in $B_3$ . Since closures of conjugate braids are isotopic, it follows that knots in family (1), up to mirroring, are SU knots with symmetric braid index 3 and corresponding SU braids given by

\begin{equation*} \beta'' = \gamma \sigma_2^{\pm 1} \gamma^{-1} \sigma_2^{\pm 1}. \end{equation*}

Now we show that Lisca’s family (3) is included in case $(\mathrm{B})$ . Comparing the definition of family (3) in [Reference Lisca 13 ] with [Reference Simone 17 , section 1] (particularly with example 1.3 and the discussion after corollary 1.11), one can see that knots in family (3) correspond to closures of alternating 3-braids whose associated strings belong to the family

\begin{equation*} \mathcal{S}_{2c} = \{ b_1 + 1, b_2, \dots, b_{k-1}, b_k + 1, c_1, \dots, c_l \mid k + l \geq 2 \}. \end{equation*}

This yields case $(\mathrm{B})$ .

Clearly, in case $(\mathrm{A})$ the knot K is ribbon as it is an SU knot. The statements about quasipositivity and amphichirality also follow immediately from Theorem 1.1 in [Reference Lisca 13 ], which concludes the proof.

4. Khovanov homology of knots with symmetric braid index 3

Having understood the placement of SU knots with $b_s(K) \leq 3$ among all finite concordance order 3-braid closures, we now derive their Khovanov-homological characterisation. Let us first define an auxiliary class of SU knots to be used in the following, studied previously in [Reference Moore 15 ].

Such knots $K_n$ are shown to be exactly the Montesinos knots $ K[{q}/{p}, {1}/{n},- {q}/{p}]$ and their Khovanov homology is computed in [Reference Köse 7 ]. This generalises [Reference Moore15, Lemma 10].

Recall that the Khovanov homology of a link L can be compactly described by its Khovanov polynomial $\mathrm{Kh}(L)(t,q)$ , given by

\begin{align*} \mathrm{Kh}(L)(t,q) = \sum_{i,j} \mathrm{rank}\,(\mathrm{Kh}^{i,j}(L))\,t^iq^j.\end{align*}

Lemma 4·3 ([Reference Köse 7 ]). Let $K_n$ denote the Montesinos knot $ K[{r}/{p}, {1}/{n},- {r}/{p}]$ and let

\begin{equation*} V(K_{p,r} \# -K_{p,r}) = \sum_{i=-m}^{m} a_i q^{2i}\end{equation*}

such that $a_m \neq 0$ be the Jones polynomial of $K_{p,r} \# -K_{p,r}$ . Then the Khovanov homology of $K_n$ is determined by n and the coefficients of $V(K_{p,r} \# -K_{p,r})$ as follows:

\begin{align*} \mathrm{Kh}(K_n)(t,q) =\begin{cases}q^{-1} + q & \text{ if $p=1$,} \\[5pt] q^{-1} \left( 1 +q^2 + (1+tq^4) t^n q^{2n} \sum_{k=-m}^{m-1} b_k t^k q^{2k} \right) & \text{otherwise},\end{cases}\end{align*}

where $b_k = b_{-k-1}$ and $b_k = (-1)^{k+1}\sum_{i=k+1}^{m} a_i$ for $k \in \{0,1, \dots , m-1\}$ .

The SU knots with symmetric braid index 3 appear in [Reference Watson 19 ] as a subset of a family of knots $K_\gamma(T,U)$ , where $\gamma \in B_3$ , and T and U are tangles. The tangle $T^\sigma$ is obtained by adding a half-twist to T, as shown in Figure  5 , and the tangle $T^{\overline{\sigma}}$ is obtained by adding the opposite half-twist to T in the same way.

Fig. 5.

The knots $K_\gamma(T,U)$ on the left and $K_\gamma(T^\sigma, U^{\overline{\sigma}})$ on the right defined in [Reference Watson 19 ]. Here when T and U are the rational tangles ${p}/{q}$ for $p=1$ , $K_\gamma(T,U)$ is an SU knot. In particular, if T and U are $(\pm 1)$ -tangles, it is an SU knot with $b_s=3$ .

We say that the tangle pair (T,U) is compatible if the writhe $w(K_\gamma(T,U)) = w(K_\gamma(T^\sigma, U^{\overline{\sigma}}))$ ; the notion of compatibility is independent of $\gamma$ and is purely diagrammatic since writhe is not a knot invariant. The sum $T + U$ of two tangles is defined by stacking T on top of U. This generalises the half-twist action: $T^\sigma$ (resp., $T^{\overline{\sigma}}$ ) may be denoted $T + $ (resp., $T + $ ). A tangle T is called simple if $T + $ is isotopic (fixing the endpoints) to the tangle . Note that $T^\sigma$ is simple if and only if T is simple.

Theorem 4·5 ([Reference Watson 19 ]). Suppose that (T,U) is a compatible pair of simple tangles. Then

\begin{equation*} \mathrm{Kh}(K_\gamma(T,U)) \cong \mathrm{Kh}(K_\gamma(T^\sigma, U^{\overline{\sigma}})).\end{equation*}

We are now equipped for the proof of our main result.

Theorem 1·4 now follows by combining Theorem 1·2 and Proposition 2·4.

Proof of Theorem 1·4. Let $K = 10_{99}$ and suppose $b_s(K) = 3$ . Since $\det(K) = 81$ , by Proposition 2·4, K admits an element of $\mathcal{J}_9 = \{ K_{9,1}, K_{9,4} \} = \{ 9_1, 6_1 \}$ as a partial knot. The Jones polynomial of $K_{9,1} \# -K_{9,1}$ is given by

\begin{align*} V(K_{9,1} \# -K_{9,1}) &= q^{-18} + {q^{-16}} -2{q^{-14}} + 3{q^{-12}} - 4{q^{-10}}\\ &+ 5{q^{-8}} - 6{q^{-6}} + 7{q^{-4}} - 7{q^{-2}} + 9\\ &- 7q^{2} + 7q^{4} - 6q^{6} + 5q^{8} - 4q^{10}\\ &+ 3q^{12} - 2q^{14} + q^{16} - q^{18}, \end{align*}

hence by Lemma 4·3 we have $\mathrm{q_{max}}(K_{9,1} \# -K_{9,1})=19$ and

\begin{equation*}\mathrm{q_{max}}(K[\tfrac{1}{9},\tfrac{1}{2},-\tfrac{1}{9}]) = -\mathrm{q_{min}}(K[\tfrac{1}{9},-\tfrac{1}{2},-\tfrac{1}{9}]) = 23.\end{equation*}

Since $\mathrm{q_{max}}(K) = -\mathrm{q_{min}}(K) = 11$ , we have that $K_{9,1}$ is not a partial knot for any braided SU diagram of K on three strands. Similarly, we compute that $\mathrm{q_{max}}(K_{9,4} \# -K_{9,4}) = 13$ and

\begin{equation*}\mathrm{q_{max}}(K[\tfrac{4}{9},\tfrac{1}{2},-\tfrac{4}{9}]) = -\mathrm{q_{min}}(K[\tfrac{4}{9},-\tfrac{1}{2},-\tfrac{4}{9}]) = 17,\end{equation*}

which yields a contradiction.

The proof for $K = 10_{123}$ is analogous, with $\det(K) = 121$ and $\mathcal{J}_{11} = \{ K_{11,1}, K_{11,3}, K_{11,5} \} = \{ 11_{a367}, 6_2, 7_2 \}$ . The statement that $b_s(K) \in \{ 4,5 \}$ for both knots follows from the existence of index five SU braids, presented in Table 1, whose closures yield SU diagrams for $10_{99}$ and $10_{123}$ .

Proof of Corollary 1·5 The result is immediate if K is in family (3) as the knots in family (3) are amphichiral. If K is in family (1) or family (2), by Theorem 1·2 and Lemma 4·3, it follows that

\begin{align*} \mathrm{q_{max}}(K) &= \max \{1, 2n+2m-1\} \text{ and} \\ \mathrm{q_{min}}(K) &= \min \{-1, 2n-2m+1\}\end{align*}

since, in the notation of Lemma 4·3, we have $b_{m-1}=b_{-m}=(-1)^ma_m \neq 0$ , where $a_m$ is the leading coefficient of $V(K_{p,r} \# -K_{p,r})$ and $n =\pm 2$ or 0. Let $\mathrm{br}(K)$ denote the breadth of the Jones polynomial of K defined by

\begin{equation*}\mathrm{br}(K) = M(K) - m(K),\end{equation*}

where M(K) and m(K) are the largest and the smallest powers of q in V(K), respectively. In general it holds that $\mathrm{br}(K) \leq 2c(K)$ for any K, where c(K) is the crossing number of K, but, in particular, $\mathrm{br}(K) = 2c(K)$ if K is alternating [Reference Kauffman5, Reference Murasugi16, Reference Thistlethwaite 18 ].

Two-bridge knots are alternating, and so are their connected sums. Thus, $m \geq 3$ since

\begin{align*} 4m = \mathrm{br}(K_{p,r} \# -K_{p,r}) = 2 c(K_{p,r} \# -K_{p,r}) \geq 2 c( K_{3,1} \# -K_{3,1} )= 12. \end{align*}

Since $n =\pm 2$ or 0 and $m \geq 3$ , it follows that $\mathrm{q_{max}}(K) = 2n+2m-1$ and $\mathrm{q_{min}}(K)=2n-2m+1$ , hence $\mathrm{q_{max}}(K)+\mathrm{q_{min}}(K) = 4n$ . This completes the proof as $n= \pm 2$ if K is in family (1) and $n=0$ if K is in family (2).

Table 1

Bounds for $b_s(K)$ for ribbon knots with at most 11 crossings, up to taking the mirror image. Corresponding SU braids are presented in the form (†), with i and $\underline{i}$ denoting $\sigma_i$ and $\sigma_i^{-1}$ , respectively. Knots that do not possess known SU diagrams are highlighted in red, and knots with distinct symmetric and regular braid indices in grey.

5. Symmetric braid indices of small ribbon knots

In Table 1 we present our computations of symmetric braid indices for the 50 ribbon knots with at most 11 crossings. Among these, there are six which do not have known SU diagrams; among the rest, we could not exactly determine the symmetric braid index for 10 knots, whilst for the remaining 34 examples, symmetric and regular braid indices coincide. For most of the knots in the table, we found the corresponding SU braids $\gamma C_1 \gamma^{-1} C_2$ by iterating over candidate $\gamma$ up to length 12 and index six, and all possible combinations of $C_1$ and $C_2$ , using SnapPy [Reference Culler, Dunfield, Goerner and Weeks 4 ] to identify the isotopy type of the closure, up to mirroring; the knot $11_{a326}$ , however, we braided by hand using the diagram from [Reference Lamm 12 ].

There are eight known SU knots in Table 1 that are not obstructed from having $b_s(K) = 4$ , yet for which we were unable to find the corresponding SU 4-braids via the initial approach. For these knots, we deepened our search by applying the fact that each 4-plat diagram of a rational knot $K_{p,q}$ is determined by a continued fraction expansion of $p/q$ , where we allow positive and negative coefficients within the same expansion, which correspond to positive and negative crossings, respectively. This enabled us to efficiently generate more candidate $\gamma$ of index 4 for these eight knots, arising from values of $p/q$ of all knots $K_{p,q} \in \mathcal{J}_p$ for appropriate p. However, we were unable to find SU 4-braids for any of the knots in question using this procedure, which serves as evidence that their symmetric braid indices may be strictly greater than four.

6. Some future directions

In order to resolve Conjecture 5·1, one could seek a Khovanov-theoretic description of knots with $b_s(K) = 4$ , particularly since they enjoy many of the properties of knots with $b_s(K) = 3$ : each knot with symmetric braid index 4 must admit a partial knot from a finite set and has symmetric fusion number one, as replacing the $\tau_1$ twist region in Figure  4 by the $\infty$ -tangle produces the two-component unlink. However, we do not have a classification of symmetric fusion number one knots with two twist regions, and there does not exist an analogue of Theorem 4·2 applicable to SU 4-braid closures. Hence, we leave the following question open.

In addition, observe that symmetric and regular braid indices agree for the two-bridge knots in Table 1, namely $6_1$ , $8_8$ , $8_9$ , $9_{27}$ , $10_3$ , $10_{22}$ , $10_{35}$ , $10_{42}$ and $11_{a96}$ . Since ribbon two-bridge knots are SU knots by [Reference Lamm 10 ], one may wonder if this apparent coincidence reflects a general fact.