2.1 Geometrical set-up

We now describe the general geometrical set-up and introduce ideas from homology which are necessary for treating multiply connected domains. A comprehensive review of the application of homology to magnetic fields can be found in Blank, Friedrichs & Grad (Reference Blank, Friedrichs and Grad1957). A more recent and accessible account can be found in Cantarella et al. (Reference Cantarella, DeTurck and Gluck2002).

We consider a domain $\unicode[STIX]{x1D6FA}\in \mathbb{R}^{3}$ that is a bounded open connected set with Lipschitz continuous boundary $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ and outer unit normal vector $\boldsymbol{n}$ . If the first Betti number of $\unicode[STIX]{x1D6FA}$ is (the genus) $g>0$ , then the first Betti number of $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ is $2g$ (e.g. Cantarella et al. Reference Cantarella, DeTurck and Gluck2002). We can consider $2g$ non-bounding cycles on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ , $\{\unicode[STIX]{x1D6FE}_{j}\}_{j=1}^{g}\cup \{\unicode[STIX]{x1D6FE}_{j}^{\prime }\}_{j=1}^{g}$ , that represent the generators of the first homology group of $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ . Each set of cycles is associated with the closed domain and its complement in a ball containing the domain. The $\{\unicode[STIX]{x1D6FE}_{j}\}_{j=1}^{g}$ represent the generators of the first homology group of $\overline{\unicode[STIX]{x1D6FA}}$ and have unit tangent vectors denoted by $\boldsymbol{t}_{j}$ . The $\{\unicode[STIX]{x1D6FE}_{j}^{\prime }\}_{j=1}^{g}$ represent the generators of the first homology group of $\overline{\unicode[STIX]{x1D6FA}^{\prime }}$ , where $\unicode[STIX]{x1D6FA}^{\prime }=B\setminus \overline{\unicode[STIX]{x1D6FA}}$ and $B$ is an open ball containing $\overline{\unicode[STIX]{x1D6FA}}$ . The unit tangent vector of a cycle $\unicode[STIX]{x1D6FE}_{j}^{\prime }$ is denoted $\boldsymbol{t}_{j}^{\prime }$ .

In $\unicode[STIX]{x1D6FA}$ there are $g$ cutting surfaces, $\{\unicode[STIX]{x1D6F4}\}_{j=1}^{g}$ , that are connected orientable Lipschitz surfaces satisfying $\unicode[STIX]{x1D6F4}_{j}\subset \unicode[STIX]{x1D6FA}$ and $\unicode[STIX]{x2202}\unicode[STIX]{x1D6F4}_{j}\subset \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ . Each surface $\unicode[STIX]{x1D6F4}_{j}$ satisfies $\unicode[STIX]{x2202}\unicode[STIX]{x1D6F4}_{j}=\unicode[STIX]{x1D6FE}_{j}^{\prime }$ and cuts the cycle $\unicode[STIX]{x1D6FE}_{j}$ . A similar set of cutting surfaces, $\{\unicode[STIX]{x1D6F4}^{\prime }\}_{j=1}^{g}$ , exists in $\unicode[STIX]{x1D6FA}^{\prime }$ . An illustration of the geometrical set-up for a domain with $g=2$ is shown in figure 1. For the sake of definiteness, we choose the unit normal vector $\boldsymbol{n}_{\unicode[STIX]{x1D6F4}_{j}}$ on $\unicode[STIX]{x1D6F4}_{j}$ oriented in such a way that (i) the unit tangent vector $\boldsymbol{t}_{j}^{\prime }$ on $\unicode[STIX]{x1D6FE}_{j}^{\prime }=\unicode[STIX]{x2202}\unicode[STIX]{x1D6F4}_{j}$ is oriented counterclockwise with respect to $\boldsymbol{n}_{\unicode[STIX]{x1D6F4}_{j}}$ (the ‘right-hand rule’) and (ii) the unit tangent vector $\boldsymbol{t}_{j}$ crosses $\unicode[STIX]{x1D6F4}_{j}$ consistently with the direction of $\boldsymbol{n}_{\unicode[STIX]{x1D6F4}_{j}}$ .

Figure 1. A domain with $g=2$ . The cycles are shown in brown and cyan, following the notation in the main text. The oriented surfaces bounded by the cycles $\unicode[STIX]{x1D6FE}_{j}^{\prime }$ and $\unicode[STIX]{x1D6FE}_{j}$ define the cutting surfaces $\unicode[STIX]{x1D6F4}_{j}$ and $\unicode[STIX]{x1D6F4}_{j}^{\prime }$ , respectively.

In the above description, if the boundary of $\unicode[STIX]{x1D6FA}$ is not connected then some care needs to be taken to make sure that a particular cutting surface remains in $\unicode[STIX]{x1D6FA}$ or $\unicode[STIX]{x1D6FA}^{\prime }$ but not in both. For the $n$ -holed torus, such as the two-holed torus in figure 1, this is not an issue. For a toroidal shell (e.g. O’Neil & Cerfon Reference O’Neil and Cerfon2018), however, the boundary is not connected, that is, the outer boundary of the solid torus is separated from the inner boundary of the toroidal hole. A toroidal shell has $g=2$ and so there are two cycles for $\unicode[STIX]{x1D6FA}$ and two for $\unicode[STIX]{x1D6FA}^{\prime }$ . It is easy to see that the cutting surfaces $\unicode[STIX]{x1D6F4}_{j}^{\prime }$ , corresponding to the $\unicode[STIX]{x1D6FE}_{j}$ , lie entirely in $\unicode[STIX]{x1D6FA}^{\prime }$ . Naively performing the same procedure for the cutting surfaces $\unicode[STIX]{x1D6F4}_{j}$ of the $\unicode[STIX]{x1D6FE}_{j}^{\prime }$ , however, results in their overlapping with parts of the complementary domain $\unicode[STIX]{x1D6FA}^{\prime }$ . Restricting the $\unicode[STIX]{x1D6F4}_{j}$ to lie entirely in $\unicode[STIX]{x1D6FA}$ can be achieved using homological properties. We explain the procedure by describing the cutting surfaces for the cycles, $\unicode[STIX]{x1D6FE}_{1}^{\prime }$ and $\unicode[STIX]{x1D6FE}_{2}^{\prime }$ , which orbit the central hole of the torus and the toroidal hole, respectively. This situation is illustrated in figure 2.

Figure 2. Toroidal shell with cutting surfaces. (a) A three-dimensional illustration of a toroidal shell cut in half. The cutting surfaces $\unicode[STIX]{x1D6F4}_{1}$ and $\unicode[STIX]{x1D6F4}_{2}$ are indicated. (b) The major cross-section (toroidal hole shown as dashed lines) where $\unicode[STIX]{x1D6FE}_{1}^{\prime }$ is the boundary of the annulus $\unicode[STIX]{x1D6F4}_{1}$ represented by the blue and red cycles. (c) The minor cross-section where $\unicode[STIX]{x1D6FE}_{2}^{\prime }$ is the boundary of the annulus $\unicode[STIX]{x1D6F4}_{2}$ represented by the orange and green cycles. Note that the cross-sections are not to scale.

The boundary of the first cutting surface, $\unicode[STIX]{x2202}\unicode[STIX]{x1D6F4}_{1}=\unicode[STIX]{x1D6FE}_{1}^{\prime }$ , is indicated in figure 2(b) by the red and blue cycles. Similarly, the boundary of the other cutting surface, $\unicode[STIX]{x2202}\unicode[STIX]{x1D6F4}_{2}=\unicode[STIX]{x1D6FE}_{2}^{\prime }$ , is indicated in figure 2(c) by orange and green cycles. The surfaces $\unicode[STIX]{x1D6F4}_{1}$ and $\unicode[STIX]{x1D6F4}_{2}$ lie entirely in $\unicode[STIX]{x1D6FA}$ and are annuli.

For a given orientation of the normal $\boldsymbol{n}_{1}$ of $\unicode[STIX]{x1D6F4}_{1}$ ,

(2.1) $$\begin{eqnarray}\displaystyle \int _{\unicode[STIX]{x1D6F4}_{1}}\text{curl}\boldsymbol{w}\boldsymbol{\cdot }\boldsymbol{n}_{1}=\int _{\text{blue}}\boldsymbol{w}\boldsymbol{\cdot }\boldsymbol{t}_{1}^{\prime }-\int _{\text{red}}\boldsymbol{w}\boldsymbol{\cdot }\boldsymbol{t}_{1}^{\prime }, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{w}$ is a vector field and ‘blue’ and ‘red’ represent the cycles displayed in figure 2(b). A similar result holds for $\unicode[STIX]{x1D6F4}_{2}$ . Note, however, that the red cycle is bounding in $\unicode[STIX]{x1D6FA}^{\prime }$ , thus $\unicode[STIX]{x1D6FE}_{1}^{\prime }$ and the blue cycle are equivalent homologically. The same is true for $\unicode[STIX]{x1D6FE}_{2}^{\prime }$ and the green cycle in figure 2(c).

In this paper we will focus on ( $n$ -holed) tori and toroidal shells since these domains have the most immediate applications, both as domains in their own right (e.g. Dewar et al. Reference Dewar, Yoshida, Bhattacharjee and Hudson2015; O’Neil & Cerfon Reference O’Neil and Cerfon2018) and in their connection to the common simulation domains of periodic and doubly periodic cubes (we will return to discuss helicity in periodic domains later). The ‘cut’ domains (those formed by removing the cutting surfaces) for these cases are simply connected. This fact, however, is not true of all domains in $\mathbb{R}^{3}$ . For example, consider a domain $\unicode[STIX]{x1D6FA}=B\setminus K$ where $B$ is a ball, as before, and $K$ is a trefoil knot (Benedetti, Frigerio & Ghiloni Reference Benedetti, Frigerio and Ghiloni2012; Alonso Rodríguez et al. Reference Alonso Rodríguez, Camaño, Rodríguez, Valli and Venegas2018). An illustration of such a domain is displayed in figure 3. The procedure that we will now describe can cope with all the cases described above, including this last one.

Figure 3. An example of a ‘cut’ domain that is not simply connected. Following the description in the main text, the orange domain is the trefoil knot $K$ . The green surface is one of the cutting surfaces, shown here as two ‘discs’ with three ‘twisting bands’. This image was produced with SeifertView (Jarke J. van Wijk, Technische Universiteit Eindhoven).

2.2 Helmholtz decomposition and Neumann harmonic vector fields

The gauge transformation normally used in helicity studies, $\boldsymbol{A}\rightarrow \boldsymbol{A}+\text{grad}\,\unicode[STIX]{x1D712}$ , is only applicable in simply connected domains if $\unicode[STIX]{x1D712}$ is to remain single-valued. For the regions under consideration with more complex topologies, we require the full Helmholtz decomposition (e.g. Blank et al. Reference Blank, Friedrichs and Grad1957; Cantarella et al. Reference Cantarella, DeTurck and Gluck2002).

Theorem 1 (Helmholtz).

Any $\boldsymbol{u}\in (L^{2}(\unicode[STIX]{x1D6FA}))^{3}$ can be decomposed as

(2.2) $$\begin{eqnarray}\displaystyle \boldsymbol{u}=\text{curl}\,\boldsymbol{P}+\text{grad}\,\unicode[STIX]{x1D719}+\unicode[STIX]{x1D746}, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{P}$ is a vector field, $\unicode[STIX]{x1D719}$ a single-valued scalar function and $\unicode[STIX]{x1D746}$ is in the space of Neumann harmonic fields ${\mathcal{H}}$ , defined as

(2.3) $$\begin{eqnarray}\displaystyle {\mathcal{H}}=\{\unicode[STIX]{x1D746}\in (L^{2}(\unicode[STIX]{x1D6FA}))^{3}:\text{curl}\unicode[STIX]{x1D746}=\mathbf{0},\text{div}\unicode[STIX]{x1D746}=0,\unicode[STIX]{x1D746}\boldsymbol{\cdot }\boldsymbol{n}=0\}. & & \displaystyle\end{eqnarray}$$

In order to make use of ${\mathcal{H}}$ , we need its basis $\{\unicode[STIX]{x1D746}_{j}\}_{j=1}^{g}$ . This can be found, for both $\unicode[STIX]{x1D6FA}$ and $\unicode[STIX]{x1D6FA}^{\prime }$ separately, by finding the solutions $\unicode[STIX]{x1D719}_{j}$ of suitable elliptic problems where the appropriate cutting surfaces are removed from the domain under study. The basis functions for the space of harmonic fields in $\unicode[STIX]{x1D6FA}$ take the form $\unicode[STIX]{x1D746}_{j}=\widetilde{\text{grad}\,}\unicode[STIX]{x1D719}_{j}$ , where $\widetilde{\text{grad}\,}\unicode[STIX]{x1D719}_{j}$ is the extension of $\text{grad}\,\unicode[STIX]{x1D719}_{j}$ to $(L^{2}(\unicode[STIX]{x1D6FA}))^{3}$ and each $\unicode[STIX]{x1D719}_{j}$ has a jump equal to 1 on the corresponding cutting surface. The basis functions of the space of harmonic fields in $\unicode[STIX]{x1D6FA}^{\prime }$ are constructed in a similar way and are denoted by $\{\unicode[STIX]{x1D746}_{j}^{\prime }\}_{j=1}^{g}$ . For more details on the construction of the basis functions, we direct the reader to Alonso Rodríguez et al. (Reference Alonso Rodríguez, Camaño, Rodríguez, Valli and Venegas2018).

2.3 Zero flux magnetic fields

Consider two different vector potentials $\boldsymbol{A}_{1}$ and $\boldsymbol{A}_{2}$ for $\boldsymbol{B}\in {\mathcal{V}}$ . Since $\text{curl}\,(\boldsymbol{A}_{1}-\boldsymbol{A}_{2})=\mathbf{0}$ in $\unicode[STIX]{x1D6FA}$ , it follows from Theorem 1 that we can write

(2.4) $$\begin{eqnarray}\displaystyle \boldsymbol{A}_{1}-\boldsymbol{A}_{2}=\text{grad}\,\unicode[STIX]{x1D712}+\unicode[STIX]{x1D746}\text{ in }\unicode[STIX]{x1D6FA}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D712}$ is a scalar function and $\unicode[STIX]{x1D746}\in {\mathcal{H}}$ . Some simple manipulation reveals

(2.5) $$\begin{eqnarray}\displaystyle \int _{\unicode[STIX]{x1D6FA}}\boldsymbol{A}_{1}\boldsymbol{\cdot }\boldsymbol{B}-\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{A}_{2}\boldsymbol{\cdot }\boldsymbol{B}=\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D746}. & & \displaystyle\end{eqnarray}$$

Therefore, if $\boldsymbol{B}\bot {\mathcal{H}}$ then helicity is independent of the vector potential. Writing $\unicode[STIX]{x1D746}$ in terms of its basis functions, we require that

(2.6) $$\begin{eqnarray}\displaystyle \int _{\unicode[STIX]{x1D6FA}}\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D746}_{j}=0,\quad \text{for}~j=1,\ldots ,g. & & \displaystyle\end{eqnarray}$$

This statement is nothing more than enforcing zero magnetic flux through every cutting surface,

(2.7) $$\begin{eqnarray}\displaystyle \int _{\unicode[STIX]{x1D6FA}}\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D746}_{j} & = & \displaystyle \int _{\unicode[STIX]{x1D6FA}\setminus \unicode[STIX]{x1D6F4}_{j}}\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D746}_{j}=\int _{\unicode[STIX]{x1D6FA}\setminus \unicode[STIX]{x1D6F4}_{j}}\boldsymbol{B}\boldsymbol{\cdot }\text{grad}\,\unicode[STIX]{x1D719}_{j}\nonumber\\ \displaystyle & = & \displaystyle \int _{\unicode[STIX]{x2202}(\unicode[STIX]{x1D6FA}\setminus \unicode[STIX]{x1D6F4}_{j})}\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{n}_{j}\unicode[STIX]{x1D719}_{j}-\int _{\unicode[STIX]{x1D6FA}\setminus \unicode[STIX]{x1D6F4}_{j}}(\text{div}\,\boldsymbol{B})\unicode[STIX]{x1D719}_{j}\nonumber\\ \displaystyle & = & \displaystyle \int _{\unicode[STIX]{x1D6F4}_{j}}\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{n}_{j}[[\unicode[STIX]{x1D719}_{j}]]=\int _{\unicode[STIX]{x1D6F4}_{j}}\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{n}_{j},\end{eqnarray}$$

where $[[\unicode[STIX]{x1D719}_{j}]]=\unicode[STIX]{x1D719}_{j}|_{\unicode[STIX]{x1D6F4}_{j}^{+}}-\unicode[STIX]{x1D719}_{j}|_{\unicode[STIX]{x1D6F4}_{j}^{-}}=1$ , from the construction of the basis functions (Alonso Rodríguez et al. Reference Alonso Rodríguez, Camaño, Rodríguez, Valli and Venegas2018). Thus, by enforcing zero magnetic flux through all the cutting surfaces of $\unicode[STIX]{x1D6FA}$ , the helicity in the domain can be written as in (1.2).

This approach to defining helicity in multiply connected domains has been used in several theoretical works (e.g. Jordan, Yoshida & Ito Reference Jordan, Yoshida and Ito1998; Faraco & Lindberg Reference Faraco and Lindberg2018). For cases where there is non-zero magnetic flux, more work is needed to produce a gauge-invariant helicity. We will now derive a more general formula which includes the zero flux condition as a special case.

2.4 Generalized Bevir–Gray formula

Let us consider the following quantities,

(2.8a,b ) $$\begin{eqnarray}\displaystyle H_{1}=\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{A}_{1}\boldsymbol{\cdot }\boldsymbol{B},\quad H_{2}=\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{A}_{2}\boldsymbol{\cdot }\boldsymbol{B}, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{B}=\text{curl}\,\boldsymbol{A}_{1}$ and $\boldsymbol{B}=\text{curl}\,\boldsymbol{A}_{2}$ , respectively. Without the zero flux condition, $H_{1}$ and $H_{2}$ are not, in general, gauge invariant in multiply connected domains. Considering the difference of these quantities,

(2.9) $$\begin{eqnarray}\displaystyle H_{1}-H_{2} & = & \displaystyle \int _{\unicode[STIX]{x1D6FA}}\boldsymbol{B}\boldsymbol{\cdot }(\boldsymbol{A}_{1}-\boldsymbol{A}_{2})=\int _{\unicode[STIX]{x1D6FA}}\text{curl}\,\boldsymbol{A}_{1}\boldsymbol{\cdot }(\boldsymbol{A}_{1}-\boldsymbol{A}_{2})\nonumber\\ \displaystyle & = & \displaystyle \int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}\boldsymbol{n}\times \boldsymbol{A}_{1}\boldsymbol{\cdot }(\boldsymbol{A}_{1}-\boldsymbol{A}_{2})=\int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}\boldsymbol{A}_{1}\times \boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{A}_{2}.\end{eqnarray}$$

The second line follows from integration by parts and using $\text{curl}\,(\boldsymbol{A}_{1}-\boldsymbol{A}_{2})=\mathbf{0}$ . Since $\text{curl}\,\boldsymbol{A}_{1}\boldsymbol{\cdot }\boldsymbol{n}=0=\text{curl}\,\boldsymbol{A}_{2}\boldsymbol{\cdot }\boldsymbol{n}$ , the tangential traces $\boldsymbol{A}_{1}\times \boldsymbol{n}$ and $\boldsymbol{A}_{2}\times \boldsymbol{n}$ can be expressed, via a Helmholtz decomposition on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ (e.g. Hiptmair, Kotiuga & Tordeux Reference Hiptmair, Kotiuga and Tordeux2012; Alonso Rodríguez et al. Reference Alonso Rodríguez, Camaño, Rodríguez, Valli and Venegas2018), as

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{A}_{1}\times \boldsymbol{n}=\text{grad}\,\unicode[STIX]{x1D702}\times \boldsymbol{n}+\mathop{\sum }_{j=1}^{g}\unicode[STIX]{x1D6FC}_{j}\unicode[STIX]{x1D746}_{j}\times \boldsymbol{n}+\mathop{\sum }_{j=1}^{g}\unicode[STIX]{x1D6FD}_{j}\unicode[STIX]{x1D746}_{j}^{\prime }\times \boldsymbol{n}, & \displaystyle\end{eqnarray}$$

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{A}_{2}\times \boldsymbol{n}=\text{grad}\,\unicode[STIX]{x1D709}\times \boldsymbol{n}+\mathop{\sum }_{j=1}^{g}\unicode[STIX]{x1D6FF}_{j}\unicode[STIX]{x1D746}_{j}\times \boldsymbol{n}+\mathop{\sum }_{j=1}^{g}\unicode[STIX]{x1D707}_{j}\unicode[STIX]{x1D746}_{j}^{\prime }\times \boldsymbol{n}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D702}$ , $\unicode[STIX]{x1D709}$ are scalar functions and $\unicode[STIX]{x1D6FC}_{j},\unicode[STIX]{x1D6FD}_{j},\unicode[STIX]{x1D6FF}_{j},\unicode[STIX]{x1D707}_{j}\in \mathbb{R}~(j=1,\ldots ,g)$ . Alonso Rodríguez et al. (Reference Alonso Rodríguez, Camaño, Rodríguez, Valli and Venegas2018) derived the following useful identities:

(2.12) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}\text{grad}\,\unicode[STIX]{x1D702}\times \boldsymbol{n}\boldsymbol{\cdot }\text{grad}\,\unicode[STIX]{x1D709}=0,\\[5.0pt] \displaystyle \int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}\text{grad}\,\unicode[STIX]{x1D702}\times \boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D746}_{j}=0,\quad \displaystyle \int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}\text{grad}\,\unicode[STIX]{x1D702}\times \boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D746}_{i}^{\prime }=0,\\[10.0pt] \displaystyle \int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D746}_{i}\times \boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D746}_{j}=0,\quad \displaystyle \int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D746}_{i}^{\prime }\times \boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D746}_{j}^{\prime }=0,\\[10.0pt] \displaystyle \int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D746}_{j}\times \boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D746}_{i}^{\prime }=\unicode[STIX]{x1D6FF}_{ij}=\displaystyle -\int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D746}_{i}^{\prime }\times \boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D746}_{j},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

for $1\leqslant i,j\leqslant g$ . Note that $\unicode[STIX]{x1D6FF}_{ij}$ is the Kronecker delta. By making use of (2.10) and (2.11) and the above identities, it can be shown that

(2.13) $$\begin{eqnarray}\displaystyle \int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}\boldsymbol{A}_{1}\times \boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{A}_{2}=\mathop{\sum }_{j=1}^{g}\unicode[STIX]{x1D6FC}_{j}\unicode[STIX]{x1D707}_{j}-\mathop{\sum }_{j=1}^{g}\unicode[STIX]{x1D6FD}_{j}\unicode[STIX]{x1D6FF}_{j}, & & \displaystyle\end{eqnarray}$$

where

(2.14a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}_{j}=\oint _{\unicode[STIX]{x1D6FE}_{j}}\boldsymbol{A}_{1}\boldsymbol{\cdot }\boldsymbol{t}_{j},\quad \unicode[STIX]{x1D6FD}_{j}=\oint _{\unicode[STIX]{x1D6FE}_{j}^{\prime }}\boldsymbol{A}_{1}\boldsymbol{\cdot }\boldsymbol{t}_{j}^{\prime }, & & \displaystyle\end{eqnarray}$$

(2.15a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{j}=\oint _{\unicode[STIX]{x1D6FE}_{j}}\boldsymbol{A}_{2}\boldsymbol{\cdot }\boldsymbol{t}_{j},\quad \unicode[STIX]{x1D707}_{j}=\oint _{\unicode[STIX]{x1D6FE}_{j}^{\prime }}\boldsymbol{A}_{2}\boldsymbol{\cdot }\boldsymbol{t}_{j}^{\prime }. & & \displaystyle\end{eqnarray}$$

Since $\unicode[STIX]{x1D6FE}_{j}^{\prime }=\unicode[STIX]{x2202}\unicode[STIX]{x1D6F4}_{j}$ , we have, by Stokes’s theorem,

(2.16) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}_{j}=\oint _{\unicode[STIX]{x1D6FE}_{j}^{\prime }}\boldsymbol{A}_{1}\boldsymbol{\cdot }\boldsymbol{t}_{j}^{\prime }=\int _{\unicode[STIX]{x1D6F4}_{j}}\text{curl}\,\boldsymbol{A}_{1}\boldsymbol{\cdot }\boldsymbol{n}_{j}=\int _{\unicode[STIX]{x1D6F4}_{j}}\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{n}_{j}. & & \displaystyle\end{eqnarray}$$

By exactly the same reasoning,

(2.17) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{j}=\oint _{\unicode[STIX]{x1D6FE}_{j}^{\prime }}\boldsymbol{A}_{2}\boldsymbol{\cdot }\boldsymbol{t}_{j}^{\prime }=\int _{\unicode[STIX]{x1D6F4}_{j}}\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{n}_{j}. & & \displaystyle\end{eqnarray}$$

Putting all these results together, we can now write

(2.18) $$\begin{eqnarray}\displaystyle H_{1}-H_{2}=\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{A}_{1}\boldsymbol{\cdot }\boldsymbol{B}-\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{A}_{2}\boldsymbol{\cdot }\boldsymbol{B}=\mathop{\sum }_{j=1}^{g}\left(\oint _{\unicode[STIX]{x1D6FE}_{j}}\boldsymbol{A}_{1}\boldsymbol{\cdot }\boldsymbol{t}_{j}-\oint _{\unicode[STIX]{x1D6FE}_{j}}\boldsymbol{A}_{2}\boldsymbol{\cdot }\boldsymbol{t}_{j}\right)\left(\int _{\unicode[STIX]{x1D6F4}_{j}}\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{n}_{j}\right).\;\quad \; & & \displaystyle\end{eqnarray}$$

An alternative route to (2.18) is to consider a result from Blank et al. (Reference Blank, Friedrichs and Grad1957). Based on the consideration of vector identities, they derive a particular formula (see their formula (6.5) on page 65) for the inner product of an irrotational vector $\boldsymbol{x}$ and a solenoidal vector $\boldsymbol{y}$ ,

(2.19) $$\begin{eqnarray}\displaystyle \int _{\unicode[STIX]{x1D6FA}}\boldsymbol{x}\boldsymbol{\cdot }\boldsymbol{y}=\mathop{\sum }_{j=1}^{g}\oint _{\unicode[STIX]{x1D6FE}_{j}}\boldsymbol{x}\boldsymbol{\cdot }\boldsymbol{t}_{j}\int _{\unicode[STIX]{x1D6F4}_{j}}\boldsymbol{y}\boldsymbol{\cdot }\boldsymbol{n}_{j}, & & \displaystyle\end{eqnarray}$$

where the notation for cycles and surfaces is as before and the vector fields are everywhere tangent to the boundary. In our application, $\boldsymbol{x}=\boldsymbol{A}_{1}-\boldsymbol{A}_{2}$ and $\boldsymbol{y}=\boldsymbol{B}$ .

Returning to (2.18), it is clear that the quantity

(2.20) $$\begin{eqnarray}\displaystyle \int _{\unicode[STIX]{x1D6FA}}\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{B}-\mathop{\sum }_{j=1}^{g}\left(\oint _{\unicode[STIX]{x1D6FE}_{j}}\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{t}_{j}\right)\left(\int _{\unicode[STIX]{x1D6F4}_{j}}\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{n}_{j}\right) & & \displaystyle\end{eqnarray}$$

is independent of the choice of vector potential. We are, therefore, led to define the gauge-invariant magnetic helicity as

(2.21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F6}(\boldsymbol{B})=\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{B}-\mathop{\sum }_{j=1}^{g}\left(\oint _{\unicode[STIX]{x1D6FE}_{j}}\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{t}_{j}\right)\left(\int _{\unicode[STIX]{x1D6F4}_{j}}\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{n}_{j}\right). & & \displaystyle\end{eqnarray}$$

It is clear that for $g=1$ , equation (2.21) reduces to the Bevir–Gray formula. Also, if $\boldsymbol{B}\bot {\mathcal{H}}$ (zero flux), the helicity reduces to

(2.22) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F6}(\boldsymbol{B})=\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{B}. & & \displaystyle\end{eqnarray}$$

This is also the case for simply connected domains, for which ${\mathcal{H}}=\{\mathbf{0}\}$ .

2.5 Biot–Savart helicity

As mentioned before, apart from the application of the Bevir–Gray formula in toroidal domains, helicity in multiply connected domains has, generally, taken the form of Biot–Savart helicity,

(2.23) $$\begin{eqnarray}\displaystyle H(\boldsymbol{B})=\int _{\unicode[STIX]{x1D6FA}}\text{BS}(\boldsymbol{B})\boldsymbol{\cdot }\boldsymbol{B}. & & \displaystyle\end{eqnarray}$$

The Biot–Savart helicity formula resembles that of helicity in simply connected domains, that is to say, the second term on the right-hand side of (2.21) is missing. We will now show that (2.21) and (2.23) are coincident. Following Cantarella et al. (Reference Cantarella, DeTurck and Gluck2001) and Valli (Reference Valli2019), it can be shown that the Biot–Savart operator, defined on $\unicode[STIX]{x1D6FA}$ , can be extended to $\mathbb{R}^{3}$ .

For $\boldsymbol{B}\in {\mathcal{V}}$ , $\text{BS}(\boldsymbol{B})\in {\mathcal{V}}$ . Since $\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{n}=0$ on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ and $\text{div}\,\boldsymbol{B}=0$ in $\unicode[STIX]{x1D6FA}$ , the extended magnetic field

(2.24) $$\begin{eqnarray}\displaystyle \widetilde{\boldsymbol{B}}=\left\{\begin{array}{@{}ll@{}}\boldsymbol{B} & \text{in}~\unicode[STIX]{x1D6FA},\\ \mathbf{0} & \text{in}~\mathbb{R}^{3}\setminus \overline{\unicode[STIX]{x1D6FA}},\end{array}\right. & & \displaystyle\end{eqnarray}$$

satisfies $\text{div}\,\widetilde{\boldsymbol{B}}=0$ in $\mathbb{R}^{3}$ .

Equation (1.4) can be modified to

(2.25) $$\begin{eqnarray}\displaystyle \text{BS}(\boldsymbol{B})(\boldsymbol{x})=\frac{1}{4\unicode[STIX]{x03C0}}\int _{\mathbb{R}^{3}}\widetilde{\boldsymbol{B}}(\boldsymbol{y})\times \frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}\,\text{d}\boldsymbol{y}. & & \displaystyle\end{eqnarray}$$

We know from Cantarella et al. (Reference Cantarella, DeTurck and Gluck2001) that $\text{curl}\,\text{BS}(\boldsymbol{B})=\widetilde{\boldsymbol{B}}$ and $\text{div}\,\boldsymbol{B}=0$ in $\mathbb{R}^{3}$ . Therefore, we can consider line integrals along closed paths and apply Stokes’s theorem.

2.5.1 $n$ -holed tori

We know that each $\unicode[STIX]{x1D6F4}_{j}^{\prime }\subset \unicode[STIX]{x1D6FA}^{\prime }~(j=1,\ldots ,g)$ is a surface bounded by a simple cycle $\unicode[STIX]{x1D6FE}_{j}$ (one with a connected boundary). We then have

(2.26) $$\begin{eqnarray}\displaystyle \oint _{\unicode[STIX]{x1D6FE}_{j}}\text{BS}(\boldsymbol{B})\boldsymbol{\cdot }\boldsymbol{t}_{j}=\int _{\unicode[STIX]{x1D6F4}_{j}^{\prime }}\text{curl}\,\text{BS}(\boldsymbol{B})\boldsymbol{\cdot }\boldsymbol{n}_{j}^{\prime }=0, & & \displaystyle\end{eqnarray}$$

since $\text{curl}\,\text{BS}(\boldsymbol{B})=\mathbf{0}$ in $\unicode[STIX]{x1D6FA}^{\prime }$ . Thus, choosing $\boldsymbol{A}=\text{BS}(\boldsymbol{B})$ in (2.21), the helicity reduces to

(2.27) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F6}(\boldsymbol{B})=\int _{\unicode[STIX]{x1D6FA}}\text{BS}(\boldsymbol{B})\boldsymbol{\cdot }\boldsymbol{B}. & & \displaystyle\end{eqnarray}$$

Note that (2.27) can also be used for magnetic fields with zero flux, without having to impose this as a condition to ensure gauge invariance.

In above approach to deriving (2.27) we have made the standard construction of extending the Biot–Savart operator to $\mathbb{R}^{3}$ by assuming that the magnetic field is zero outside the domain. By virtue of (2.27), the magnetic field in the domain can inherit the field line topology interpretation of Moffatt (Reference Moffatt1969) and Arnold & Khesin (Reference Arnold and Khesin1992). For applications where linkage with magnetic field outside the domain is important, see § 2.6 below. First, however, we will investigate how the Biot–Savart operator can be used to understand the linkage of field lines on the surface of the domain with the domain itself.

2.5.2 Field line helicity on a toroidal boundary

Although the second term on the right-hand side of (2.21), related to the domain topology, is zero for Biot–Savart helicity, this does not mean that integrals on the domain boundary are unimportant. For the case of a standard torus $(g=1)$ , the property in (2.26) can be extended to define the field line helicity (e.g. Berger Reference Berger1988; Yeates & Hornig Reference Yeates and Hornig2013) on the boundary when the path of integration follows a closed field line. Such field lines are possible everywhere on the boundary of a torus as the Euler characteristic of the boundary is zero. A prominent example of such a field line is a torus knot (e.g. Oberti & Ricca Reference Oberti and Ricca2018). For domains with $g>1$ , closed field lines on the boundary are possible, but not everywhere on the boundary. This is due to the Euler characteristic being non-zero and, as a consequence of the ‘hairy ball’ theorem (e.g. Frankel Reference Frankel2004), smooth vector fields on the surfaces of these domains must have at least one point where they vanish.

A closed path $\unicode[STIX]{x1D6FE}$ on the surface of a torus $\unicode[STIX]{x1D6FA}$ can be expressed in terms of the basis cycles $\unicode[STIX]{x1D6FE}_{1}$ and $\unicode[STIX]{x1D6FE}_{1}^{\prime }$ as

(2.28) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}=L^{\prime }\unicode[STIX]{x1D6FE}_{1}+L\unicode[STIX]{x1D6FE}_{1}^{\prime }, & & \displaystyle\end{eqnarray}$$

up to bounding cycles on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ , that are homologically trivial. Here $L^{\prime }\in \mathbb{Z}$ is the linking number between $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FE}_{1}^{\prime }$ (slightly deformed outside $\unicode[STIX]{x1D6FA}$ , in order that there is no intersection with $\unicode[STIX]{x1D6FE}$ ); similarly, $L\in \mathbb{Z}$ is the linking number between $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FE}_{1}$ (slightly deformed inside $\unicode[STIX]{x1D6FA}$ , in order that there is no intersection with $\unicode[STIX]{x1D6FE}$ ). Therefore, if $\boldsymbol{w}$ is a vector field such that $\text{curl}\,\boldsymbol{w}\boldsymbol{\cdot }\boldsymbol{n}=0$ on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ and $\boldsymbol{t}$ is the unit tangent vector on $\unicode[STIX]{x1D6FE}$ ,

(2.29) $$\begin{eqnarray}\displaystyle \oint _{\unicode[STIX]{x1D6FE}}\boldsymbol{w}\boldsymbol{\cdot }\boldsymbol{t}=L^{\prime }\oint _{\unicode[STIX]{x1D6FE}_{1}}\boldsymbol{w}\boldsymbol{\cdot }\boldsymbol{t}_{1}+L\oint _{\unicode[STIX]{x1D6FE}_{1}^{\prime }}\boldsymbol{w}\boldsymbol{\cdot }\boldsymbol{t}_{1}^{\prime }. & & \displaystyle\end{eqnarray}$$

Applying this result to the Biot–Savart vector field $\text{BS}(\boldsymbol{B})$ , for which (from (2.26)),

(2.30) $$\begin{eqnarray}\displaystyle \oint _{\unicode[STIX]{x1D6FE}_{1}}\text{BS}(\boldsymbol{B})\boldsymbol{\cdot }\boldsymbol{t}_{1}=0, & & \displaystyle\end{eqnarray}$$

it follows that

(2.31) $$\begin{eqnarray}\displaystyle \oint _{\unicode[STIX]{x1D6FE}}\text{BS}(\boldsymbol{B})\boldsymbol{\cdot }\boldsymbol{t}=L\oint _{\unicode[STIX]{x1D6FE}_{1}^{\prime }}\text{BS}(\boldsymbol{B})\boldsymbol{\cdot }\boldsymbol{t}_{1}^{\prime }. & & \displaystyle\end{eqnarray}$$

Thus, by Stokes’s theorem,

(2.32) $$\begin{eqnarray}\displaystyle \oint _{\unicode[STIX]{x1D6FE}}\text{BS}(\boldsymbol{B})\boldsymbol{\cdot }\boldsymbol{t}=L\int _{\unicode[STIX]{x1D6F4}_{1}}\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{n}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6F4}_{1}$ is a cutting surface of $\unicode[STIX]{x1D6FA}$ with $\unicode[STIX]{x2202}\unicode[STIX]{x1D6F4}_{1}=\unicode[STIX]{x1D6FE}_{1}^{\prime }$ . The value $L$ can be interpreted as the number of loops of $\unicode[STIX]{x1D6FE}$ around the minor cross-section of the torus (say, the ‘linking number’ of the field line with the torus).

It is interesting to note that the field line helicity (the integral on the left-hand side of (2.32)) ‘knows’ about the linkage of the boundary curve with field lines inside the torus, by virtue of (2.25). This result is true for any magnetic field inside the torus, no matter how complex the field line topology. The value of the field line helicity, however, depends simply on the magnetic flux and a purely topological quantity depending only on the curve and the domain. Equation (2.32) is analogous to the voltage formula of a transformer (for a topological perspective on this formula, see Gross & Kotiuga Reference Gross and Kotiuga2004).

2.5.3 Field line helicity on the boundary of a toroidal shell

The Biot–Savart helicity is coincident with the general helicity, equation (2.21), in a toroidal shell by the same procedure as described for $n$ -tori. The toroidal shell does, however, have some extra physical interpretations related to the cutting surfaces and field line helicities on the boundaries.

As described in § 2.1, some care is required in order to identify the $\unicode[STIX]{x1D6F4}_{j}$ cutting surfaces. Given suitable cutting surfaces, the helicity for a toroidal shell is

(2.33) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F6}(\boldsymbol{B})=\int _{\unicode[STIX]{x1D6FA}}\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{B}-\oint _{\unicode[STIX]{x1D6FE}_{1}}\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{t}_{1}\int _{\unicode[STIX]{x1D6F4}_{1}}\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{n}_{1}-\oint _{\unicode[STIX]{x1D6FE}_{2}}\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{t}_{2}\int _{\unicode[STIX]{x1D6F4}_{2}}\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{n}_{2}, & & \displaystyle\end{eqnarray}$$

where the notation is standard. The cutting surface $\unicode[STIX]{x1D6F4}_{1}$ is that shown in figure 2(b) and $\unicode[STIX]{x1D6F4}_{2}$ is that shown in figure 2(c). Therefore, we can label the fluxes as

(2.34a,b ) $$\begin{eqnarray}\displaystyle \int _{\unicode[STIX]{x1D6F4}_{1}}\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{n}_{1}=\unicode[STIX]{x1D6F9}_{P},\quad \int _{\unicode[STIX]{x1D6F4}_{2}}\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{n}_{2}=\unicode[STIX]{x1D6F9}_{T}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6F9}_{P}$ and $\unicode[STIX]{x1D6F9}_{T}$ are the poloidal and toroidal fluxes respectively. If we select $\boldsymbol{A}=\text{BS}(\boldsymbol{B})$ , then the general helicity formula naturally reduces to the form of (2.27) due to the property in (2.26).

Considering the field line helicity in (2.32), there are now two separate boundaries for a (closed) field line to lie on. For a field line lying on the outer boundary of a toroidal shell, the interpretation is the same as that of a standard torus. That is, a field line twisting around the minor circumference of the torus $L$ times will have a field line helicity of $L\unicode[STIX]{x1D6F9}_{T}$ . For a field line on the inner boundary, the path of the line integral can also be deformed into a union of circles around the major and minor circumferences of the toroidal hole. This time, however, due to the property in (2.26), the contribution to the integral from circles around the minor cross-section of the hole is zero. The non-zero contribution comes from the major cross-section, and if a field line on the inner boundary wraps around the major cross-section $L$ times, its field line helicity is $L\unicode[STIX]{x1D6F9}_{P}$ .

2.6 Mutual helicity

Through the summation term on the right-hand side of (2.21), the domain topology enters explicitly into the helicity formula for multiply connected domains. This term is related to the mutual helicity (Laurence & Avellaneda Reference Laurence and Avellaneda1993; Cantarella Reference Cantarella2000), a quantity that measures the linkage of the magnetic field in two domains, $\unicode[STIX]{x1D6FA}$ and $\unicode[STIX]{x1D6FA}^{\prime }$ , say. Indeed, through (2.21) we can prove a representation formula for the mutual helicity which is equivalent to that obtained by Cantarella (Reference Cantarella2000) for domains $\unicode[STIX]{x1D6FA}$ , $\unicode[STIX]{x1D6FA}^{\prime }\subset \mathbb{R}^{3}$ of arbitrary topology.

With its topological connection to the Gauss linking number (e.g. Moffatt Reference Moffatt1969; Laurence & Avellaneda Reference Laurence and Avellaneda1993), helicity is often written as

(2.35) $$\begin{eqnarray}\displaystyle H_{U}(\boldsymbol{B})=\frac{1}{4\unicode[STIX]{x03C0}}\int _{U\times U}\left(\boldsymbol{B}(\boldsymbol{x})\times \boldsymbol{B}(\boldsymbol{y})\boldsymbol{\cdot }\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}\right)\text{d}\boldsymbol{x}\,\text{d}\boldsymbol{y}, & & \displaystyle\end{eqnarray}$$

where $U$ is a bounded open set (but not necessarily connected) and $\boldsymbol{B}$ is a magnetic field tangent to $\unicode[STIX]{x2202}U$ . If we consider two disjoint bounded domains $M$ and $N$ and we set $U=M\cup N$ and $\boldsymbol{B}_{M}=\boldsymbol{B}_{|M}$ , $\boldsymbol{B}_{N}=\boldsymbol{B}_{|N}$ , then it is easily checked that

(2.36) $$\begin{eqnarray}\displaystyle H_{M\cup N}(\boldsymbol{B})=H_{M}(\boldsymbol{B}_{M})+H_{N}(\boldsymbol{B}_{N})+2H(\boldsymbol{B}_{M},\boldsymbol{B}_{N}), & & \displaystyle\end{eqnarray}$$

where

(2.37) $$\begin{eqnarray}\displaystyle H(\boldsymbol{B}_{M},\boldsymbol{B}_{N})=\frac{1}{4\unicode[STIX]{x03C0}}\int _{M\times N}\left(\boldsymbol{B}_{M}(\boldsymbol{x})\times \boldsymbol{B}_{N}(\boldsymbol{y})\boldsymbol{\cdot }\frac{\boldsymbol{x}-\boldsymbol{y}}{|\boldsymbol{x}-\boldsymbol{y}|^{3}}\right)\text{d}\boldsymbol{x}\,\text{d}\boldsymbol{y} & & \displaystyle\end{eqnarray}$$

is the mutual helicity. The self-helicities, $H_{M}(\boldsymbol{B}_{M})$ and $H_{N}(\boldsymbol{B}_{N})$ , can be readily expressed by means of the Biot–Savart operator,

(2.38a,b ) $$\begin{eqnarray}\displaystyle H_{M}(\boldsymbol{B}_{M})=\int _{M}\text{BS}(\boldsymbol{B}_{M})\boldsymbol{\cdot }\boldsymbol{B}_{M},\quad H_{N}(\boldsymbol{B}_{N})=\int _{N}\text{BS}(\boldsymbol{B}_{N})\boldsymbol{\cdot }\boldsymbol{B}_{N}. & & \displaystyle\end{eqnarray}$$

Consider an open cube $Q$ such that $\overline{M\cup N}\subset Q$ . In this domain, we extend $\boldsymbol{B}$ by $\mathbf{0}$ in $Q\setminus \overline{M\cup N}$ . We label this extension $\widehat{\boldsymbol{B}}$ and note that it is divergence-free in $Q$ and tangent to $\unicode[STIX]{x2202}Q$ . Further, its helicity in $Q$ coincides with that of $\boldsymbol{B}$ in $M\cup N$ ,

(2.39) $$\begin{eqnarray}\displaystyle H_{Q}(\widehat{\boldsymbol{B}})=H_{M\cup N}(\boldsymbol{B}). & & \displaystyle\end{eqnarray}$$

Since $Q$ is simply connected, we can use (2.21) with $g=0$ to write

(2.40) $$\begin{eqnarray}\displaystyle H_{Q}(\widehat{\boldsymbol{B}})=\int _{Q}\widehat{\boldsymbol{A}}\boldsymbol{\cdot }\widehat{\boldsymbol{B}}, & & \displaystyle\end{eqnarray}$$

where $\widehat{\boldsymbol{A}}$ is any vector potential of $\widehat{\boldsymbol{B}}$ in $Q$ . Since $\widehat{\boldsymbol{B}}$ is vanishing outside $M\cup N$ , it follows that

(2.41) $$\begin{eqnarray}\displaystyle H_{Q}(\widehat{\boldsymbol{B}})=\int _{M}\widehat{\boldsymbol{A}}_{|M}\boldsymbol{\cdot }\widehat{\boldsymbol{B}}_{M}+\int _{N}\widehat{\boldsymbol{A}}_{|N}\boldsymbol{\cdot }\widehat{\boldsymbol{B}}_{N}. & & \displaystyle\end{eqnarray}$$

Using (2.21) in $M$ (with genus $g_{M}$ ) and in $N$ (with genus $g_{N}$ ), we have

(2.42) $$\begin{eqnarray}\displaystyle \int _{M}\widehat{\boldsymbol{A}}_{|M}\boldsymbol{\cdot }\widehat{\boldsymbol{B}}_{M}=\unicode[STIX]{x1D6F6}_{M}(\boldsymbol{B}_{M})+\mathop{\sum }_{j=1}^{g_{M}}\left(\oint _{\unicode[STIX]{x1D6FE}_{j}^{M}}\widehat{\boldsymbol{A}}_{|M}\boldsymbol{\cdot }\boldsymbol{t}_{j}^{M}\right)\left(\int _{\unicode[STIX]{x1D6F4}_{j}^{M}}\boldsymbol{B}_{M}\boldsymbol{\cdot }\boldsymbol{n}_{j}^{M}\right) & & \displaystyle\end{eqnarray}$$

and

(2.43) $$\begin{eqnarray}\displaystyle \int _{N}\widehat{\boldsymbol{A}}_{|N}\boldsymbol{\cdot }\widehat{\boldsymbol{B}}_{N}=\unicode[STIX]{x1D6F6}_{N}(\boldsymbol{B}_{N})+\mathop{\sum }_{l=1}^{g_{N}}\left(\oint _{\unicode[STIX]{x1D6FE}_{l}^{N}}\widehat{\boldsymbol{A}}_{|N}\boldsymbol{\cdot }\boldsymbol{t}_{l}^{N}\right)\left(\int _{\unicode[STIX]{x1D6F4}_{l}^{N}}\boldsymbol{B}_{N}\boldsymbol{\cdot }\boldsymbol{n}_{l}^{N}\right). & & \displaystyle\end{eqnarray}$$

Using (2.27), $H_{M}(\boldsymbol{B}_{M})=\unicode[STIX]{x1D6F6}(\boldsymbol{B}_{M})$ and $H_{N}(\boldsymbol{B}_{N})=\unicode[STIX]{x1D6F6}(\boldsymbol{B}_{N})$ . Substituting these results into (2.42) and (2.43) and then substituting these into (2.36), the mutual helicity is found to be

(2.44) $$\begin{eqnarray}\displaystyle H(\boldsymbol{B}_{M},\boldsymbol{B}_{N}) & = & \displaystyle \frac{1}{2}\left[\mathop{\sum }_{j=1}^{g_{M}}\left(\oint _{\unicode[STIX]{x1D6FE}_{j}^{M}}\widehat{\boldsymbol{A}}_{|M}\boldsymbol{\cdot }\boldsymbol{t}_{j}^{M}\right)\left(\int _{\unicode[STIX]{x1D6F4}_{j}^{M}}\boldsymbol{B}_{M}\boldsymbol{\cdot }\boldsymbol{n}_{j}^{M}\right)\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\mathop{\sum }_{l=1}^{g_{N}}\left(\oint _{\unicode[STIX]{x1D6FE}_{l}^{N}}\widehat{\boldsymbol{A}}_{|N}\boldsymbol{\cdot }\boldsymbol{t}_{l}^{N}\right)\left(\int _{\unicode[STIX]{x1D6F4}_{l}^{N}}\boldsymbol{B}_{N}\boldsymbol{\cdot }\boldsymbol{n}_{l}^{N}\right)\right].\end{eqnarray}$$

A careful analysis of the values of the linking numbers between $\unicode[STIX]{x1D6FE}_{j}^{M}$ and $\unicode[STIX]{x1D6FE}_{l}^{N}$ would show that this formula is equivalent to that given in Cantarella (Reference Cantarella2000). For simplicity, let us show that this is true in the case of two linked solid tori (Moffatt Reference Moffatt1969; Laurence & Avellaneda Reference Laurence and Avellaneda1993; Cantarella Reference Cantarella2000). We have $g_{M}=1$ and $g_{N}=1$ ; the cycle $\unicode[STIX]{x1D6FE}_{1}^{M}$ is the boundary of a surface $\unicode[STIX]{x1D6F4}_{M}^{\prime }$ , contained in $Q\setminus \overline{M}$ , and the cycle $\unicode[STIX]{x1D6FE}_{1}^{N}$ is the boundary of a surface $\unicode[STIX]{x1D6F4}_{N}^{\prime }$ , contained in $Q\setminus \overline{N}$ . Let us also assume that the unit tangent vector $\boldsymbol{t}_{1}^{M}$ crosses $\unicode[STIX]{x1D6F4}_{M}$ , the cross-section of the torus $M$ , consistently with the direction of normal vector $\boldsymbol{n}_{\unicode[STIX]{x1D6F4}_{M}}$ on it, and similarly for $\boldsymbol{t}_{1}^{N}$ and the cross-section of $N$ . In this way the linking number of $M$ and $N$ has value 1. By Stokes’s theorem,

(2.45) $$\begin{eqnarray}\displaystyle \oint _{\unicode[STIX]{x1D6FE}_{1}^{M}}\widehat{\boldsymbol{A}}_{|M}\boldsymbol{\cdot }\boldsymbol{t}_{1}^{M}=\int _{\unicode[STIX]{x1D6F4}_{M}^{\prime }}\text{curl}\,\widehat{\boldsymbol{A}}\boldsymbol{\cdot }\boldsymbol{n}_{\unicode[STIX]{x1D6F4}_{M}^{\prime }}=\int _{\unicode[STIX]{x1D6F4}_{M}^{\prime }}\widehat{\boldsymbol{B}}\boldsymbol{\cdot }\boldsymbol{n}_{\unicode[STIX]{x1D6F4}_{M}^{\prime }}. & & \displaystyle\end{eqnarray}$$

Since $\widehat{\boldsymbol{B}}=\mathbf{0}$ outside $M\cup N$ , it follows that

(2.46) $$\begin{eqnarray}\displaystyle \int _{\unicode[STIX]{x1D6F4}_{M}^{\prime }}\widehat{\boldsymbol{B}}\boldsymbol{\cdot }\boldsymbol{n}_{\unicode[STIX]{x1D6F4}_{M}^{\prime }}=\int _{\unicode[STIX]{x1D6F4}_{M}^{\prime }\cap N}\boldsymbol{B}_{N}\boldsymbol{\cdot }\boldsymbol{n}_{\unicode[STIX]{x1D6F4}_{M}^{\prime }}=\int _{\unicode[STIX]{x1D6F4}_{N}}\boldsymbol{B}_{N}\boldsymbol{\cdot }\boldsymbol{n}_{\unicode[STIX]{x1D6F4}_{N}}. & & \displaystyle\end{eqnarray}$$

Similarly,

(2.47) $$\begin{eqnarray}\displaystyle \oint _{\unicode[STIX]{x1D6FE}_{1}^{N}}\widehat{\boldsymbol{A}}_{|N}\boldsymbol{\cdot }\boldsymbol{t}_{1}^{N}=\int _{\unicode[STIX]{x1D6F4}_{M}}\boldsymbol{B}_{M}\boldsymbol{\cdot }\boldsymbol{n}_{\unicode[STIX]{x1D6F4}_{M}}, & & \displaystyle\end{eqnarray}$$

and we obtain

(2.48) $$\begin{eqnarray}\displaystyle H(\boldsymbol{B}_{M},\boldsymbol{B}_{N})=\left(\int _{\unicode[STIX]{x1D6F4}_{M}}\boldsymbol{B}_{M}\boldsymbol{\cdot }\boldsymbol{n}_{\unicode[STIX]{x1D6F4}_{M}}\right)\left(\int _{\unicode[STIX]{x1D6F4}_{N}}\boldsymbol{B}_{N}\boldsymbol{\cdot }\boldsymbol{n}_{\unicode[STIX]{x1D6F4}_{N}}\right). & & \displaystyle\end{eqnarray}$$