Abstract
In 1869, Lord Kelvin found that the way vortices are knotted and linked in an ideal fluid remains unchanged in evolution, and consequently hypothesized atoms to be knotted vortices in a ubiquitous ether, different knotting types corresponding to different types of atoms. Even though Kelvin’s atomic theory turned out incorrect, it inspired several important developments, such as the mathematical theory of knots and the investigation of knotted structures that naturally arise in physics. However, in previous studies, knotted and linked structures have been found to untie via local cutandpaste events referred to as reconnections. Here, in contrast, we construct knots and links of nonAbelian vortices that are topologically protected in the sense that they cannot be dissolved employing local reconnections and strand crossings. Importantly, the topologically protected links are supported by a variety of physical systems such as dilute BoseEinstein condensates and liquid crystals. We also propose a classification scheme for topological vortex links, in which two structures are considered equivalent if they differ from each other by a sequence of topologically allowed reconnections and strand crossings, in addition to the typical continuous transformations. Interestingly, this scheme produces a remarkably simple classification.
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Introduction
Mathematically, knots and links are closed loops and configurations thereof in a threedimensional space, respectively^{1,2}. There are numerous physical systems that support knotted structures. Examples include the disclination lines of liquid crystals^{3,4,5}, the cores of vortices in water^{6}, in superfluids^{7}, and in optical^{8,9} and acoustic fields^{10}, strands of DNA (deoxyribonucleic acid)^{11}, Skyrmion cores in classical field theory^{12}, and the quantum knot in the polar phase of a spin1 Bose–Einstein condensate^{13}. In addition, there are deep connections between the mathematical theory of knots and statistical mechanics, topological quantum computing, and quantum field theories^{14,15,16}. In mathematics, knots play a significant role, for example, in the surgery theory of threedimensional manifolds^{17}.
A nontrivially knotted loop tied from a physical string, or more generally a link tied from several loops, cannot be untied without cutting at least one string; the knot and link structures are robust. Similarly, knots and links tied from vortex lines in a frictionless ideal fluid remain forever knotted^{18}, leading to a conserved quantity, helicity^{19}, which measures the total knottedness and linkedness of a vortex configuration. However, it has been observed that even a small amount of dissipation is enough to cause spontaneous untying of knotted vortices owing to local reconnection events^{7,20}. In addition, another local modification, namely strand crossing^{21}, may promote the untying process of vortex knots. These observations naturally lead to the following fundamental question: is it possible to tie a vortex knot, or a link, for which decay through local reconnection events and strand crossings is prohibited by the fundamental properties of the knotsupporting substance?
Here, we propose a class of knotted structures, tied from nonAbelian topological vortices, that have exactly this property. We refer to such a structure as being topologically protected against untying. In order to rigorously investigate this property, we introduce invariants of links colored by the elements of the quaternion group Q_{8} = {±1, ±i, ±j, ±k}, the group law of which is governed by the multiplication rule of quaternions.
Topological vortices are codimensiontwo defects in an ordered medium where the winding of the order parameter field about the vortex core corresponds to a nontrivial element of the fundamental group π_{1}. The elements of the fundamental group π_{1}(M, m) correspond to oriented loops in the order parameter space^{22} M that begin and end at the basepoint m ∈ M, considered up to continuous deformations that keep the endpoints fixed. Here, the group law is provided by the concatenation of loops. The basepoint m is often omitted from the notation.
Topological vortices are nonAbelian if the fundamental group π_{1}(M) is a nonAbelian group, such as the quaternion group Q_{8}. NonAbelian vortices are known to exhibit peculiar behavior, when they interact with each other. For instance, two vortices corresponding to noncommuting elements in π_{1}(M) cannot strand cross, i.e., freely pass through each other. Instead, one concludes on topological grounds alone that a vortex cord, corresponding to the commutator of the crossing vortices in π_{1}(M), forms to connect them^{21}. Precisely such phenomena are responsible for the robustness of the structures discovered in this work.
Knots and links tied from topological vortices have been studied before^{23,24,25}. However, the previous efforts focus on the isotopic evolution of the vortex structure, which does not account for the possibility of the vortex cores crossing or reconnecting. Thus, our approach differs decisively from the previous investigations. By allowing evolution by events, in which the vortex cores cross each other or reconnect, we aim to arrive at an extended description for the stability of the system. Consequently, the results derived from our model, such as the classification of vortex structures, should be physically more relevant than analogous results up to isotopic evolution.
Results
We begin by identifying topological vortex configurations for certain, physically relevant, order parameter spaces, with colored link diagrams (Fig. 1). Then, we identify the rules governing the coretopologyaltering evolution of such structures. In the special case of Q_{8}colored link diagrams, which describe topological vortex configurations in certain physical systems, the colored link diagrams and the evolution rules admit a simple graphical depiction (Figs. 2 and 3). Using invariants of Q_{8}colored links, we identify examples of linked structures that are robust against local reconnections and strand crossings (Fig. 4). Interestingly, we also classify all the Q_{8}colored links up to reconnections and strand crossings, and find a topologically stable knot for a fundamental group of permutations. In contrast to previous classification of topological vortex configurations^{24,25} which do not consider coretopology altering evolution, our classification of Q_{8}colored links has only finitely many classes.
Q _{8}colored links and the evolution of the core topology
A configuration of topological vortices is formalized as follows. The spatial extent of the ordered medium is modeled by the threedimensional euclidean space \({{\mathbb{R}}}^{3}\). Homotopy theoretically, it makes no difference if \({{\mathbb{R}}}^{3}\) is replaced by the unit ball in \({{\mathbb{R}}}^{3}\). The cores of the vortices, i.e., the location where the order parameter is not well defined, form a subset \(L\subset {{\mathbb{R}}}^{3}\) consisting of loops. In other words, the collection of cores forms a link in \({{\mathbb{R}}}^{3}\). The order parameter field is modeled by a continuous map \(\Psi :{{\mathbb{R}}}^{3}\backslash L\to M\), where M is the order parameter space. The induced homomorphism between the fundamental groups \(\psi :{\pi }_{1}({{\mathbb{R}}}^{3}\backslash L,{x}_{0})\to {\pi }_{1}[M,\Psi ({x}_{0})]\) may be described by a π_{1}(M)colored link diagram as illustrated in Fig. 1. If the second homotopy group π_{2}[M, Ψ(x_{0})], i.e., the group defined analogously to the fundamental group but with spheres instead of loops, is trivial, then the group homomorphism ψ retains all the information about the homotopy class^{26,27} of Ψ. In fact, the homotopy classes of continuous maps \({{\mathbb{R}}}^{3}\backslash L\to M\) are in onetoone correspondence with group homomorphisms \({\pi }_{1}({{\mathbb{R}}}^{3}\backslash L)\to {\pi }_{1}(M)\) up to change of coordinates, or conjugacy, defined as follows: homomorphisms \(\psi ,{\psi }^{{\prime} }:{\pi }_{1}({{\mathbb{R}}}^{3}\backslash L)\to {\pi }_{1}(M)\) correspond to the same homotopy class if and only if there exists h ∈ π_{1}(M) such that, for all \(g\in {\pi }_{1}({{\mathbb{R}}}^{3}\backslash L)\), \({\psi }^{{\prime} }(g)=h\psi (g){h}^{1}\). In terms of π_{1}(M)colored link diagrams, this relation considers two colored diagrams equivalent if one of them can be obtained from the other by replacing all g_{i} with hg_{i}h^{−1}. The rules for Q_{8}colored link diagrams admit a purely graphical presentation, as illustrated in Fig. 2a, b.
For the stability of such structures, we make the following assumption: the core topology changes only in topologically allowed local reconnections and topologically allowed strand crossings. Even though such processes require energy^{23}, the cost should be very small due to the local nature of the modifications, and indeed, such events have been observed both in experiments and in simulations^{6,7,28}. Nevertheless, there are several processes which we do not consider in our model, as we explain below.
Firstly, we forbid the possibility of vortex loops disappearing by shrinking into a point, because no structure consisting of vortex loops would be protected against decaying in this fashion. This is not a physically unrealistic assumption: in the polar phase of a spin1 Bose–Einstein condensate, a vortex loop is energetically favorable to a point defect^{29}. Hence, for some physical systems, vortex loops have a nonzero optimal length, and in such systems the loops are protected against decaying by shrinking by an energy barrier.
Secondly, we do not consider vortex crossings between vortices that correspond to noncommuting elements in the fundamental group, as a vortex cord connecting the crossing vortices is formed in such a process^{21}. We justify this assumption by noting that the energy cost associated of the formation of the vortex chord is, roughly speaking, linearly proportional to its length, rendering the formation of vortex cords long enough to be relevant for the evolution of the physical system rare at low temperatures.
Thirdly, in the spirit of the previous assumption, we do not consider the possibility of vortices splitting along a significant distance, because any configuration of topological vortex loops can decay into a simple loop via such processes, as we show in Fig. 5. We justify this restriction by considering systems where the splitting of the vortices of interest is energetically unfavorable, rendering the splitting of the vortex along a meaningful distance rare at low temperatures. This assumption is violated in some systems, such as cholesteric liquid crystals, where the splitting of singularities in the nematic director is energetically favored^{30}. However, for certain biaxial nematic liquid crystal systems, splitting is energetically expensive, and is not observed in simulations^{31}, which indicates that there exists physical systems where spontaneous splitting does not occur.
Verifying the existence of physical systems that satisfy the above three assumptions requires either extensive numerical simulations or experimental efforts, and is left for future work.
A vortex reconnection is topologically allowed if it does not lead to discontinuities in the coloring. A strand crossing is topologically allowed^{21} if the two strands correspond to commuting elements in π_{1}(M). It is a consequence of the Wirtinger relations^{2} that this is equivalent to assuming that the coloring does not change in the undercrossing. Hence, strand crossing is allowed only if no discontinuities in the coloring occur after the crossing has taken place. For Q_{8}colored link diagrams, the local reconnection and strandcrossing rules admit a simple description in terms of the graphical presentation mentioned above (Fig. 2c).
Let us consider physical systems that consist of the cyclic or biaxial nematic phases of a spin2 Bose–Einstein condensate, or of the biaxial nematic phase of a liquid crystal. The corresponding order parameter spaces—M_{C}, M_{BN}, and M_{BNLQ}, respectively—have trivial second homotopy groups; importantly, the fundamental group of M_{BNLQ} is Q_{8}, whereas in π_{1}(M_{C}) and π_{1}(M_{BN}), Q_{8} is the subgroup corresponding to topological vortices with no scalarphase winding about the core (Methods). Hence, in each case, at least some of the topological vortex configurations admit descriptions as Q_{8}colored link diagrams. Moreover, the evolution of these structures under strand crossings and local reconnections may be analyzed using the rules outlined in Fig. 2c.
Topologically protected Q _{8}colored links
Next, we establish the the topological stability of the knotted structures by means of invariants of Q_{8}colored links. The linking invariant l is an element of \({{\mathbb{Z}}}_{2}\) that is the number of times a blue strand crosses over a red strand, modulo 2. Equivalently, it is the total Gauss linking number^{2} between the red and the blue loops, modulo 2. In order for the bicoloring flips to lead to a consistent bicoloring, each red loop is overcrossed an even number of times by a strand of either gray or blue color. Hence, the number of times a red loop is crossed over by a blue strand is equivalent, modulo 2, to the number of times it is crossed over by a gray strand. In fact, one argues in a similar fashion that the linking invariant l is independent of the pair of colors used to compute it. The invariant is conserved in topologically allowed local reconnections and strand crossings (Fig. 2c), since these modifications do not alter crossings that are relevant for l. For a diagram of disjoint and untangled loops, this invariant is clearly zero. Hence, a configuration with a nontrivial linking invariant cannot be unknotted by topologically allowed local reconnections and strand crossings.
There exists a more refined invariant, the Qinvariant, which is computed, roughly speaking, by multiplying all the elements of Q_{8} obtained from the undercrossings of loops of a specified color. The precise definition is explained in Fig. 6. The Qinvariant is valued in \({{\mathbb{Z}}}_{4}\), and it recovers to the linking invariant when reduced modulo 2. Thus, the linking invariant l is redundant, but we have included it in the article due to its simplicity. In Methods, it is verified that the Qinvariant is conserved in topologically allowed strand crossings and local reconnections, and therefore can be used as a marker of topological protection. Remarkably, up to topologically allowed local reconnections and strand crossings, there are only six nontrivial Q_{8}colored links, two for each value Q = [1], [2], and [3], respectively (Fig. 4(a)). Moreover, there are 2^{4} = 16 defects that have Q = [0], each of which is equivalent to a disjoint union of loops of a subset of the four possible colors. Examples of topologically protected and unprotected vortex configurations are illustrated in Fig. 4a, b.
These results on the invariants imply that a system, the fundamental group of which is described by the quaternion group, cannot support topologically protected vortices composed of a single loop, i.e., there exists no topologically protected Q_{8}colored knots. However, this behavior is particular to the group Q_{8}: there is no mathematical obstruction for the existence of topologically protected knots in general, as is illustrated by the tricolored trefoil knot (Fig. 4c). The tricolored trefoil arises as a topologically protected knot colored by the elements of the permutation group S_{3}. The fundamental group of the trefoil complement is the braid group B_{3}, and the homomorphism B_{3} → S_{3} corresponding to the tricoloring coincides with the standard surjective homomorphism. The rigorous justification of its topological protection is left for future work due to the mathematically demanding nature of the argument. To date, we have not found a physically accessible system that supports topological vortices described by the group S_{3}. However, such a system may exist in addition to many other systems supporting topologically protected knots described by other groups such as the tetrahedral group for the cyclic phase of spin2 Bose–Einstein condensates. The discovery of these exciting structures is left for future work.
Discussion
We expect our results to inspire a wide range of theoretical and experimental investigations. In addition to observing experimentally the proposed vortex links in nematic liquid crystals or spin2 Bose–Einstein condensates, analogous topologically protected vortex structures can be studied both theoretically and experimentally in a wide variety of condensedmatter systems, which may stimulate the development of invariants for other types of colored links. Another interesting aspect of our work is the simplicity of the classification: instead of the infinitude of different link types, there is only a small number of Q_{8}colored links up to strand crossings and reconnections. It is appealing to obtain similar classifications for physically relevant groups other than the quaternion group Q_{8}.
In future work, we aim to use numerical simulations to investigate the dynamics of the structures proposed in this article to verify their stability properties and provide insight how they can be prepared and observed in experiments. The preparation of topologically protected vortex structures is a challenging issue, because, due to their topologically protected nature, they cannot be created by local tailoring, which is a method that has been previously employed in the creation of vortex knots^{4}. A potential solution is provided by generating random vortex structures using a rapid phase transition^{5}. However, at least for biaxial nematic liquid crystals, the vortex structures formed in this way seem to be networks rather than links, and these networks are prone to decaying via vortex amalgamation and annihilation of small vortex loops^{31}. Thus it may be that controlled tailoring of the electromagnetic or other fields is needed to bring the knotted structure into the physical system from out side its spatial extent, in the spirit of previously experimentally realized monopole creation in Bose–Einstein condensates^{32,33,34}.
Methods
Order parameter spaces for spin2 biaxial nematic and cyclic phases
Here, we study the order parameter spaces of spin2 biaxial nematic and cyclic phases, and identify the quaternion group Q_{8} as the subgroup of the fundamental group corresponding to topological vortices with no scalarphase winding about the core.
We begin by analyzing the order parameter space of the spin2 biaxial nematic phase M_{BN}. It is known that M_{BN} ≅ [S^{1} × SO(3)]/D_{4}, where the S^{1} accounts for the scalar complex phase, and D_{4} is the symmetry group of a square lying in the xyplane^{35,36}. The dihedral group D_{4} is realized as a subgroup of S^{1} × SO(3) in such a way that the 90^{∘} and the 270^{∘} rotations along the zaxis, and the 180^{∘} rotations along x + y and the x − y axis are supplemented with a phase shift by π. In other words, the elements of the Klein fourgroup K_{4}, corresponding to the subgroup of 180^{∘} rotations along the x, y and the zaxis as well as the identity, are not accompanied by phase shifts, and therefore [S^{1} × SO(3)]/K_{4} ≅ S^{1} × [SO(3)/K_{4}]. Since the inverse image of K_{4} under the twofold covering SU(2) → SO(3) is the quaternion group Q_{8}, we deduce that \({\pi }_{1}({S}^{1}\times [{{{{{{{\rm{SO}}}}}}}}(3)/{K}_{4}])\cong {\mathbb{Z}}\times {Q}_{8}\). Moreover, since \({D}_{4}/{K}_{4}\cong {{\mathbb{Z}}}_{2}\), the order parameter space M_{BN} is homeomorphic to the quotient \(\left\{{S}^{1}\times [{{{{{{{\rm{SO}}}}}}}}(3)/{K}_{4}]\right\}/{{\mathbb{Z}}}_{2}\). Applying the long exact homotopy sequence^{37}, we obtain a short exact sequence of groups
leading us to conclude that \({\mathbb{Z}}\times {Q}_{8}\) is a subgroup of π_{1}(M_{BN}). The elements of π_{1}(M_{BN}) that do not belong to \({\mathbb{Z}}\times {Q}_{8}\) correspond to paths between different points in a \({{\mathbb{Z}}}_{2}\)orbit of S^{1} × [SO(3)/K_{4}]^{37}. As the \({{\mathbb{Z}}}_{2}\)action identifies points that have scalar phase α with points that have scalar phase α + π, no path connecting such a pair has phase winding that is an integer multiple of 2π. Therefore, the subgroup \({\mathbb{Z}}\times {Q}_{8}\subset {\pi }_{1}({M}_{{{{{{{{\rm{BN}}}}}}}}})\) corresponds to those vortices that have integer phase winding, and {0} × Q_{8} ⊂ π_{1}(M_{BN}) corresponds to exactly those vortices that have no scalar phase winding.
Next, we analyze the order parameter space of the spin2 cyclic phase in detail. The order parameter space M_{C} is [S^{1} × SO(3)]/T, where the S^{1}, again, accounts for the complex phase, and where T is the group of rotational symmetries of a regular tetrahedron^{38}. To realize it as a subgroup of S^{1} × SO(3), we employ the presentation of SO(3) as the group of 3 × 3 orthogonal matrices with determinant 1. Then, the elements corresponding to the Klein fourgroup K_{4} ⊂ T, namely I, diag(−1, −1, 1), diag(− 1, 1, −1) and diag(1, −1, −1) are not accompanied by any phase shifts, and the tetrahedral group is generated by the above elements in {0} × SO(3) ⊂ S^{1} × SO(3), and by
As in the previous paragraph, [S^{1} × SO(3)]/K_{4} ≅ S^{1} × [SO(3)/K_{4}] and its fundamental group is isomorphic to \({\mathbb{Z}}\times {Q}_{8}\). The Klein fourgroup is a normal subgroup of T of index 3 and the quotient T/K_{4} is the cyclic group \({{\mathbb{Z}}}_{3}\). Applying the long exact homotopy sequence to the threefold covering S^{1} × [SO(3)/K_{4}] → M_{C}^{37}, we obtain a short exact sequence of groups
The elements of π_{1}(M_{C}) that do not belong to \({\mathbb{Z}}\times {Q}_{8}\) correspond to paths between different points in a \({{\mathbb{Z}}}_{3}\)orbit of S^{1} × [SO(3)/K_{4}]^{37}. As the \({{\mathbb{Z}}}_{3}\)action identifies points that have scalar phase α with points that have scalar phase α + 2π/3 and α + 4π/3, no path connecting such a pair has phase winding that is an integer multiple of 2π. Therefore, the subgroup \({\mathbb{Z}}\times {Q}_{8}\subset {\pi }_{1}({M}_{{{{{{{{\rm{C}}}}}}}}})\) corresponds to those vortices that have integer phase winding, and {0} × Q_{8} ⊂ π_{1}(M_{C}) corresponds to exactly those vortices that have no scalar phase winding.
The Qinvariant and the classification of Q _{8}colored links
Here, we define the Qinvariant of Q_{8}colored links, establish its basic properties, and use it to classify Q_{8}colored links. The invariant is \({{\mathbb{Z}}}_{4}\)valued, and it recovers the linking invariant l when reduced modulo 2. The definition of Q requires focusing on loops of either red, gray, or blue color, and therefore one ends up with three colored invariants Q_{red}, Q_{gray}, and Q_{blue}, the equivalence of which is established later. After defining the colored invariants of a Q_{8}colored link diagram, we establish their independence from the specific diagram chosen to present a Q_{8}colored link, after which we prove that these invariants are conserved in topologically allowed strand crossings and reconnections. Subsequently, we employ the classification of threecomponent links to classify Q_{8}colored links up to strand crossings and reconnections and establish the equivalence of the three colored invariants. The Qinvariant is thus defined as the value of any of the colored invariants. Moreover, we establish a relationship of Q with Milnor’s triple linking number.
Suppose L is a Q_{8}colored link represented by a Q_{8}colored link diagram. Let L_{1}, …, L_{r} be the components of L enumerated in some order. For each nonpurple L_{i} choose a basepointb_{i} at one of the arcs of the loop L_{i} in the diagram. We will use the following orientation convention for bicolored loops with a basepoint: the loop L_{i} is oriented in such a way that when moving from the basepoint b_{i} according to the orientation, the black color of the bicoloring is on the right. It will not be necessary to choose basepoints or specify orientations for the purple loops.
Definition 1
Let (L_{i}, b_{i}) be a pointed loop of color c ∈ {red, gray, blue} in a Q_{8}colored link diagram. We define the αinvariant of (L_{i}, b_{i}) as
where χ_{c} is i, j or k if c is red, gray or blue, respectively; ω is the selfwrithe of the loop L, i.e., the signed count of self crossings of L in the diagram, where the sign of a crossing is decided using the righthand rule, see Fig. 7; and q ∈ Q_{8} is obtained by multiplying the quaternions, corresponding to the crossings of L_{i} under nonpurple strands, in order from right to left, when L_{i} is traversed from the basepoint b_{i} according to the orientation specified above. Concrete examples are presented in Fig. 8.
We record the following useful properties of the αinvariant.
Lemma 2
The αinvariant satisfies the following properties:

1.
α(L_{i}, b_{i}) is a power of χ_{c};

2.
α(L_{i}, b_{i}) does not depend on the basepoint b_{i}.
Proof
Throughout the proof, we ignore all the purple strands since they do not affect the invariant. In order for the bicoloring to be consistent, each loop is overcrossed by a strand of different color an even number of times. For instance, if the loop L_{i} is red, then, in the expression of α(L_{i}, b_{i}), the combined number of occurrences of j and k is even. After reordering, which contributes only a sign, the occurrences of j and k can be replaced by a single power of i, which proves the first claim.
For the proof of the second claim, we investigate the effect on α caused by moving the basepoint through an undercrossing to an adjacent arc in the diagram. As the selfwrithe does not depend on the orientation of L_{i}, we only need to consider how this process affects q. There are two cases to consider, depending on the color \({c}^{{\prime} }\) of the strand crossing over L_{i}.

1.
Case \({c}^{{\prime} }=c\): the basepoint moves, but the orientation is not altered. If the undercrossing through which the basepoint is moved is the first undercrossing according to the orientation, then \(q={q}^{{\prime} }{\chi }_{c}^{\pm \!1}\) is replaced by \({\chi }_{c}^{\pm \!1}{q}^{{\prime} }\). However, since \({q}^{{\prime} }=q{\chi }_{c}^{\mp 1}\) is a power of χ_{c}, it commutes with χ_{c}, so q does not change. The other case is proved in a similar fashion.

2.
Case \({c}^{{\prime} }\ne \,c\): the basepoint moves and the orientation is altered. If the undercrossing is the first undercrossing according to the orientation, then \(q={q}^{{\prime} }{\chi }_{{c}^{{\prime} }}^{\pm 1}\) is replaced by \({({\chi }_{{c}^{{\prime} }}^{\pm 1}{q}^{{\prime} })}^{1}={q}^{{\prime} 1}{\chi }_{{c}^{{\prime} }}^{\mp 1}\). However, since \({q}^{{\prime} }{\chi }_{{c}^{{\prime} }}^{\pm 1}\) is a power of χ_{c}, \({q}^{{\prime} }\notin \{1,1\}\), and therefore \({q}^{{\prime} 1}={q}^{{\prime} }\). Hence, \({q}^{{\prime} 1}{\chi }_{{c}^{{\prime} }}^{\mp 1}={q}^{{\prime} }{\chi }_{{c}^{{\prime} }}^{\pm 1}\), and consequently q does not change in the process. The other case is proved in a similar fashion.□
Below, we define the colored invariants.
Definition 3
Let c ∈ {red, gray, blue}. Then the colored invariant \({Q}_{c}\in {{\mathbb{Z}}}_{4}\) of a Q_{8}colored link diagram is defined as
where l_{c} is the sum of the pairwise linking numbers between loops of color c, modulo 2. The product on the right side is well defined, since α(L_{i}) does not depend on the choice of a basepoint of L_{i} and since the α(L_{i}) commute with each other as they are all powers of χ_{c}. The invariant is well defined since the exponent of χ_{c} is well defined modulo 4. Concrete examples are illustrated in Fig. 8.
The next result establishes a relationship between the invariants Q_{c} and the linking invariant l.
Proposition 4
When reduced modulo 2, the colored invariant Q_{c} is equivalent to the linking invariant l.
Proof
The linking invariant is the number of times a strand of color \({c}^{{\prime} }\) passes over a strand of color c^{″}, modulo 2, where \({c}^{{\prime} }\) and c^{″} are two different nonpurple colors. Let c be color that is different from \({c}^{{\prime} }\) and c^{″}. As there are an even number of crossings between different loops of color c, these contribute an even number of multiplicative factors of ±χ_{c} to \({\prod }_{{L}_{i}}\alpha ({L}_{i})\), where L_{i} ranges over all loops of color c. Moreover, the same is true for each self crossing of a loop of color c, as each self crossing contributes both to terms \({\chi }_{c}^{\omega }\) and q in Eq. (4). In other words, \({\chi }_{c}^{{Q}_{c}}\) is a product of quaternions, an even number of which are ±χ_{c}.
If l = 1, then, in the expression of \({\chi }_{c}^{{Q}_{c}}\), there are an odd number of factors of form \(\pm {\chi }_{{c}^{{\prime} }}\) and an odd number of factors of form \(\pm {\chi }_{{c}^{{\prime\prime} }}\). Reordering the expression, we conclude that \({\chi }_{c}^{{Q}_{c}}=\pm \!{\chi }_{c}\), i.e., Q_{c} ≡ 1 modulo 2. Similarly, if l = 0, then, in the expression of \({\chi }_{c}^{{Q}_{c}}\), there are an even number of factors of form \(\pm {\chi }_{{c}^{{\prime} }}\) and an even number of factors of form \(\pm {\chi }_{{c}^{{\prime\prime} }}\). Reordering the expression, we conclude that \({\chi }_{c}^{{Q}_{c}}=\pm \!1\), i.e., Q_{c} ≡ 0 modulo 2.□
There are various diagrams representing the same abstract link in \({{\mathbb{R}}}^{3}\), and they are related to each other by Reidemeister moves^{2,39,40}. Similarly, there are many Q_{8}colored link diagrams representing the same abstract Q_{8}colored link, and they are related to each other by Reidemeister moves. Given an initial coloring, there exists a unique Q_{8}coloring for the link diagram after a Reidemeister move has taken place; we refer to such moves, endowed with the data of a Q_{8}coloring, as Q_{8}colored Reidemeister moves. Examples are presented in Fig. 9. Next we prove that the invariants Q_{c} are invariants of the Q_{8}colored link rather than the particular link diagram chosen to present it.
Lemma 5
The invariants Q_{c} are conserved in Q_{8}colored Reidemeister moves.
Proof
We consider each type of Reidemeister move separately. The essential cases to consider are presented in Fig. 9. We again ignore the purple strands of the diagram in the proof, because they have no effect on the invariants.

1.
Reidemeister move of type I: such a move has the potential to alter the invariant only if it is applied to a loop L_{i} of color c. Let \({L}_{i}^{{\prime} }\) be the loop after the move has been performed. Choosing a suitable basepoint, one obtains expressions \(\alpha ({L}_{i})={\chi }_{c}^{\omega }q\) and \(\alpha ({L}_{i}^{{\prime} })={\chi }_{c}^{\omega {\prime} }{q}^{{\prime} }={\chi }_{c}^{\omega \mp 1}{\chi }_{c}^{\pm 1}q\), proving that \(\alpha ({L}_{i}^{{\prime} })=\alpha ({L}_{i})\). As the operation does not alter the linking numbers between loops of color c, the invariant Q_{c} remains unchanged.

2.
Reidemeister move of type II: such a move does not alter the selfwrithe, the αinvariant of any loop or the linking numbers between loops, so the invariant remains unchanged.

3.
Reidemeister move of type III: such a move does not alter the selfwrithe, the αinvariant of any loop or the linking numbers between loops, so the invariant remains unchanged.□
Next, we prove the conservation of the colored invariants Q_{c} in topologically allowed strand crossings and local reconnections. As a disjoint union of Q_{8}colored unknotted loops has trivial Q_{c}invariants, they may therefore be regarded as obstructions for the unlinking of a Q_{8}colored link using local reconnections and strand crossings.
Lemma 6
The invariants Q_{c} are conserved in topologically allowed strand crossings.
Proof
The proof is presented in Fig. 10.□
Lemma 7
The invariants Q_{c} are conserved in topologically allowed reconnections.
Proof
If the reconnection takes place between two different loops, they will merge into one loop as a result. As it is enough to establish the conservation of Q_{c} in the inverse process of such an event, we may assume that the reconnection takes place between points x and y on the same loop L_{i}. Moreover, since any knot can be unknotted by crossing changes, Lemma 6 implies that we may assume that L_{i} is an unknot. Hence, we may choose a Q_{8}colored link diagram representing the link L where the loop L_{i} has no selfcrossings, and where no extra arcs appear in the imminent neighborhood of the location where the reconnection takes place. Visualization is provided by Fig. 11.
Let q_{1} and q_{2} be the quaternions defined much like q in Definition 1, but where q_{1} accounts for the undercrossings occurring when traversing from x to y according to the orientation convention at x, and where q_{2} accounts for the undercrossings occurring when continuing from y back to x. By hypothesis, α(L_{i}) = q_{2}q_{1}. There are two cases to consider.

1.
q_{1} is a power of χ_{c}: i.e., on the path from x to y, L_{i} crosses under an even number of strands than of color other than c. In this case, the reconnection splits the loop L_{i} into two loops \({L}_{i}^{{\prime} }\) and \({L}_{i}^{{\prime\prime} }\). Moreover, as \(\alpha ({L}_{i}^{{\prime} })={q}_{2}\) and \(\alpha ({L}_{i}^{{\prime\prime} })={q}_{1}\), and as the process does not affect l_{c} modulo 2, the invariant Q_{c} remains unchanged.

2.
q_{1} is not a power of χ_{c}: by rotating the diagram, by rotating the loop L_{i} around an axis in the plane of the diagram, and by swapping the labels of x and y if necessary, we may assume that the strands are horizontal around the point of reconnection, that x is below y in the picture, and that the orientation at x points to right. There are two possible reconnections which are related by a topologically allowed strand crossing. By Lemma 6, it is enough to investigate only one of these; we focus on the one in which, after traversing the path corresponding to q_{1} in the modified loop \({L}_{i}^{{\prime} }\), the loop crosses under itself. As the self writhe of \({L}_{i}^{{\prime} }\) is −1, by definition \(\alpha ({L}_{i}^{{\prime} })={\chi }_{c}{q}_{2}^{1}{\chi }_{c}^{1}{q}_{1}\). As q_{1} is not a power of χ_{c} but q_{2}q_{1} is, \({q}_{2}^{1}={q}_{2}\) and q_{2} anticommutes with χ_{c}, and therefore \({\chi }_{c}{q}_{2}^{1}{\chi }_{c}^{1}{q}_{1}={q}_{2}{q}_{1}\). In other words, \(\alpha ({L}_{i}^{{\prime} })=\alpha ({L}_{i})\). As the reconnection does not affect the total modulotwo linking number between loops of color c, the invariant Q_{c} remains unchanged.□
Using the previous results, it is possible to classify all the Q_{8}colored links up to strand crossings and local reconnection events (we refer to this, in short, as the classification of Q_{8}colored links), and to prove the equivalence of the colored invariants Q_{red}, Q_{gray} and Q_{blue}. Given a Q_{8}colored link, one can perform local reconnections in order to connect loops of the same color. After doing so and ignoring the potential purple loop, we have a link of at most three components, and each component is labeled with either red, gray or blue color. We will momentarily forget the bicoloring, and choose an orientation for each loop.
Milnor has classified oriented links of at most three components up to link homotopy^{41,42}, i.e., up to such continuous deformations of the link where the different components are not allowed to meet, but where self intersections are allowed. Such deformations may be expressed in terms of Q_{8}colored Reidemeister moves and topologically allowed strand crossings, and therefore Milnor’s classification provides an intermediate step in the classification of Q_{8}colored links. If there is only one loop, then there is only one link up to link homotopy. If there are two loops, L_{1} and L_{2}, then the link is completely characterized by the Gauss linking number μ(12) of L_{1} and L_{2}. In case of three loops, the third loop L_{3} corresponds to a canonical element of form \({\alpha }_{1}{\alpha }_{2}^{\mu (123)}{\alpha }_{1}^{1}{\alpha }_{2}^{\mu (123)}{\alpha }_{2}^{\mu (23)}{\alpha }_{1}^{\mu (13)}\) in the fundamental group \({\pi }_{1}({{\mathbb{R}}}^{3}\backslash ({L}_{1}\cup {L}_{2}))\), where α_{i} is an element of \({\pi }_{1}({{\mathbb{R}}}^{3}\backslash ({L}_{1}\cup {L}_{2}))\) that corresponds to a loop that winds once about L_{i} in the positive direction. A concrete example is provided in Fig. 12. Above, μ(ij) is the linking number between L_{i} and L_{j}, and μ(123) is the triple linking number, which is an integer that is well defined up to the greatest common divisor d of μ(12), μ(13) and μ(23). The link is completely classified by \(\mu (12),\mu (13),\mu (23)\in {\mathbb{Z}}\) and \([\mu (123)]\in {{\mathbb{Z}}}_{d}\).
There are three nontrivial cases to consider.

1.
The link has only two loops. The bicoloring cannot be consistent unless μ(12) is even. Applying the surgery operation depicted in Fig. 13a, it is possible to achieve μ(12) = 0. In other words, such a Q_{8}colored link is in the trivial class.

2.
The link has three loops and μ(12) is odd. In order for the bicoloring to be consistent, also μ(13) and μ(23) have to be odd. Applying the surgery operation depicted in Fig. 13a, it is possible to achieve μ(12) = μ(13) = μ(23) = 1. These numbers classify the link up to link homotopy as their greatest common divisor is 1; such a link is homotopic to a looped chain of length three. There are 8 possible bicolorings for such a link. It is possible to “flip” any two bicolorings simultaneously, as depicted in Fig. 13c, leaving exactly two classes, which correspond to values Q_{c} = [1] and Q_{c} = [3].

3.
The link has three loops and μ(12) is even. In order for the bicoloring to be consistent, also μ(13) and μ(23) have to be even. Applying the surgery operation depicted in Fig. 13a, it is possible to achieve μ(12) = μ(13) = μ(23) = 0 and that μ(123) is either 0 or 1, depending on the parity of μ(123) in the original link. If μ(123) = 0, then the loops are not linked, so this case corresponds to the trivial class. If μ(123) = 1, then the link is homotopic to the Borromean rings. There are again eight possible bicolorings for such a link. Moreover, it is possible to “flip” any single bicoloring, as depicted in Fig. 13d, so all of the possible bicolorings are in the same class which corresponds to Q_{c} = [2].
The above discussion has several immediate consequences. First we consider the equivalence of the colored invariants.
Proposition 8
The invariants Q_{red}, Q_{gray} and Q_{blue} are equivalent.
Proof
According to the above discussion, any nontrivial Q_{8}colored link diagram can be reduced to one of the three links depicted in Fig. 13b by applying topologically allowed strand crossings and reconnections. Hence, one has to check the desired equality Q_{red} = Q_{gray} = Q_{blue} only in these cases.□
The Qinvariant is defined as the common value of the colored invariants Q_{c}. The following result establishes a connection between it and Milnor’s triple linking number.
Proposition 9
If L is a Q_{8}colored link with at most one component of each color and the linking invariant l of L is zero in \({{\mathbb{Z}}}_{2}\), then \(Q=[2\mu (123)]\in {{\mathbb{Z}}}_{4}\), where μ(123) is Milnor’s triple linking number.
Proof
This was established in the third case of the above discussion.□
Finally, we classify Q_{8}colored links.
Theorem 10
(Classification of Q_{8}colored links). Up to topologically allowed strand crossings and reconnections, there are only the following classes of Q_{8}colored links:

1.
16 classes of trivial links, corresponding to untangled disjoint unions of loops of different colors;

2.
2 classes each (with and without a purple loop) for which the invariant Q obtains the values [1], [2] and [3] in \({{\mathbb{Z}}}_{4}\).
Proof
We prove the two cases separately.

1.
A topologically unprotected link is equivalent to a disjoint union of unlinked loops. Since there are four classes of topological vortices, corresponding to the nonidentity conjugacy classes of Q_{8}, there exists exactly 2^{4} = 16 classes of topologically trivial Q_{8}colored links, one of them being the empty link.

2.
According to the above discussion, a nontrivial Q_{8}colored link diagram consisting of loops of classes {±i}, {±j}, and {±k}, can be reduced to one of the three links depicted in Fig. 13b. In the presence of a purple loop, corresponding to the element −1 which commutes with all elements of the group Q_{8}, a nontrivial Q_{8}colored link diagram can be reduced to the disjoint union of one of the three links depicted in Fig. 13b and a purple loop.□
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Acknowledgements
We thank H. Rajamäki for stimulating discussions. We have received funding from the European Research Council under Grant No 681311 (QUESS), from the Academy of Finland Centre of Excellence program (project 336810), and from the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters.
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’The theoretical work was carried out by T.A. with input from M.M. and R.Z.Z. M.M. supervised the work. All authors discussed the theoretical results and commented on the manuscript.
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Annala, T., ZamoraZamora, R. & Möttönen, M. Topologically protected vortex knots and links. Commun Phys 5, 309 (2022). https://doi.org/10.1038/s42005022010712
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DOI: https://doi.org/10.1038/s42005022010712
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