Knotted fields and explicit fibrations for lemniscate knots
We give an explicit construction of complex maps whose nodal line have the form of lemniscate knots. We review the properties of lemniscate knots, defined as closures of braids where all strands follow the same transverse (1, ) Lissajous figure, and are therefore a subfamily of spiral knots generalising the torus knots. We then prove that such maps exist and are in fact fibrations with appropriate choices of parameters. We describe how this may be useful in physics for creating knotted fields, in quantum mechanics, optics and generalising to rational maps with application to the Skyrme-Faddeev model. We also prove how this construction extends to maps with weakly isolated singularities.
A challenging problem in knot theory is writing down an explicit parametrisation of a curve in the form of a given type of knot or link. This problem becomes even more demanding when we ask for an explicit function of three-dimensional space which contains special loci which are knotted and linked, such as a complex-valued function with a zero level set of the form of a given knot, a model knotted field.
The study and construction of knotted fields of this type is of interest in both mathematics and physics. In particular, various kinds of knotted fields find application in areas such as topological fluid dynamics moffatt:1969degree (); lr:2012jones (), nonlinear field theories sutcliffe:2007knots (), nematic liquid crystals ma:2014knotted (), excitable media winfree:1994persistent (), electromagnetic fields ib:2008linked () and optical physics bd:2001knotted (); dkjop:2010isolated (). The physicist’s interest is then usually in the specific conformation of the knot when the function minimises some energy functional or a solution of some linear or nonlinear PDE.
Here, we show how to generate explicit, complex scalar knotted fields for the family of knots we call lemniscate knots, which, in a generalisation of the procedure described in dkjop:2010isolated (), are built up from explicit constructions of certain braids, which are defined by trigonometric functions. Lemniscate knots have not, to our knowledge, been emphasised as a class within knot theory—they are a subclass of so-called spiral knots betvwy:2010spiral ()—and are automatically fibred (i.e. the complement of the knot can be divided into topologically equivalent surfaces parametrised by points on a circle); we prove that the lemniscate knotted fields arising from the construction can be explicit fibrations. Theorems related to the Nash-Tognoli Theorem bcr:1998real () imply that every knot or link type can be realised as the intersection of the zero sets of two real polynomials in the spatial variables , and . However, such theorems are rarely constructive, leaving a physicist or applied mathematician, wishing for an explicit analytic function representation for a knotted field, at a loss on how to proceed.
A construction by Brauner in 1928 brauner:1928geometrie () gives an explicit realisation of such maps for the -torus knot or link. This begins by constructing a complex polynomial of two complex variables and ,
With the restriction , can be used as complex coordinates for the unit three-sphere . Stereographically projecting to , for example with the explicit choice of projection
where , makes in (1) an explicit complex rational function of , and , which indeed has a nodal line in the form of the torus knot (if and are coprime, otherwise it is a torus link). If necessary only the numerator (a polynomial in ) of the rational function can be considered; in both cases the zero level set contains the desired knot.
Brauner’s method can be understood in terms of closing braids to obtain knots or links. The torus knot or link is the closure of the simple braid where with strands forming a helix, undergoing full twists, as the example in Figure 1. This braid maps to a 1-parameter family of complex polynomials, parametrised by real acting as braid height, and the roots sweep out the strands of the braid as increases. This leads directly to the function , where the level set of zero forms the torus link. This can be seen by mapping the complex braid into the complex coordinates of the 3-sphere, which closes the braid.
This approach led Perron to construct a polynomial map for the figure-8 knot , the simplest non-torus knot perron:1982noeud (), by a similar construction involving the pigtail braid shown in Figure 2. Although cannot be represented by a braid that can be drawn on a cylinder, it can be arranged to lie on the surface of the cartesian product of a lemniscate with an interval. This leads naturally to a trigonometric parametrisation of the braid as the lemniscate of Gerono (a -Lissajous figure), and the procedure then essentially follows Brauner’s construction. The braid in Figure 2 consists of three strands with different starting points following the same lemniscate path along the braid (as height increases). Two vertical periods of the characteristic alternating crossing pattern of the pigtail braid are shown. Perron’s construction generalises to all knots and links which can be formed as closures of the braid which consists of copies of this same basic braid ( is , gives the borromean rings , the knot , etc). Similar functions based on this lemniscate were constructed by Rudolph rudolph:1987isolated () and later Dennis et al. dkjop:2010isolated (), of the form , which are holomorphic in one variable (i.e. complex analytic in but not ), which we call semiholomorphic.
Here, we generalise the construction further, explicitly constructing semiholomorphic maps with nodal sets in the form of families of knots based on Lissajous figures. We call these families lemniscate knots, which are based on braids whose strands follow the same generalised lemniscate Lissajous figure (a true lemniscate has ), and are trigonometric functions of braid height parameter . The resulting semiholomorphic complex function , with lemniscate knotted nodal set, gives a complex scalar field of 3-dimensional space by (2), whose nodal lines (phase singularities) have the form of the lemniscate knot. Such functions can then be used for various physical applications, such as holograms to create knotted optical vortices dkjop:2010isolated (), as templates for vector fields whose helicity is determined by the knot kfdi:2016weaving (), or candidate, knotted minimum-energy solutions for the Skyrme-Faddeev model sutcliffe:2007knots ().
The maps constructed by Brauner, Perron and Rudolph in fact satisfy much stronger properties than just having the correct topology on the unit three-sphere, defining neighbourhoods of singularities in 4-dimensional real space, as studied extensively by Milnor milnor:1968singular (). Although we have been unable to extend the mappings based on lemniscate knots to singularities of semiholomorphic type (as Rudolph), nevertheless the symmetries of the lemniscate braids often do allow the explicit construction of polynomial maps with weakly isolated singularities of the type originally described by Akbulut and King ak:1981all ().
As a basis for our later construction, it is helpful to formalise the procedure for torus knots and the figure-8 knot discussed above, generalising to arbitrary lemniscate braids. In the horizontal plane transverse to the braid height, the strands follow the generalised lemniscate curve ( Lissajous figure), parametrised by , given by , where , and
Here, , are stretching factors, set to unity unless otherwise stated. The prefactor in ensures that each ‘lobe’ in the Lissajous figure has aspect ratio approximately unity when . is simply a cosine function, and is independent of . The th point (representing a strand of the braid) moves cyclically to the th point (), and this pattern (equivalent algebraically to a basic braid word) repeats times as . In the 3-dimensional space of the braid, the strands follow the curve parametrised by the height coordinate , with , and increasing upwards,
This braid is represented by the family of complex polynomials , with variable and real, cyclic parameter , that have roots given by the intersection of the parametrised braid with the horizontal plane (now taken to be the complex plane) at height , i.e. , so
The semiholomorphic map with knotted zero line is found by the replacement, in , of with and with , ensuring . This is proved later in Section III.
When , the braid is helical, closing to a torus knot or link. Assuming , each strand follows a circle of radius in the horizontal plane, and the strands are uniformly distributed around this circle. After a increase of , the regular -gon of intersections of strands with the horizontal plane has turned by . Since each root has the form times an th root of unity, the polynomial (5) multiplies out to the form . In this case, the map arises from on identifying , giving, for the torus knot, , equivalent to above. The explicit knotted field of arises from the substitution (2).
The procedure of creating the figure-8 knot uses (following the approach of dkjop:2010isolated ()), and replaces the circular trajectory in the horizontal plane of the braid with the lemniscate (assuming ); the braid whose closure is the figure-8 knot has three strands and two repeats of the basic period, so the figure-8 knot has in (3), and the corresponding polynomial (5) has roots , and, after multiplying out the polynomial and making the identification , , we get the function dkjop:2010isolated ()
Unlike the function for torus knots, this function is semiholomorphic, depending on both and , a consequence of the fact that the trigonometric functions in have different arguments; underlying this is the fact that only braids where all crossings are over-crossings (as in Figure 1 but not Figure 2) can be represented by fully holomorphic polynomials. For closures of the braid with different choices of , and are replaced in (6) with , giving, the borromean rings , etc. as discussed above. Equation (6) gives an explicit figure-8 knotted field in with the identification (2).
The structure of this paper proceeds as follows. In the next section, we consider the mathematical properties of lemniscate knots. In Section III, we prove that the map described above, constructed from the polynomial (5), has the desired knot and link, for appropriate choices of and . Furthermore, with appropriate and , we prove the argument of the resulting function gives a fibration of the knot complement over . Physical applications of the procedure involving knotted fields in quantum mechanics, optics and Skyrme-Faddeev hopfions are discussed in Section IV. The construction of polynomial maps with weakly isolated singularities of the form of Akbulut and King is discussed in Section V, before a concluding discussion in Section VI.
Ii Lemniscate knots: braids and properties
Lemniscate knots and links are defined as the closures of braids whose strands execute the same generalised lemniscate trajectory in the horizontal plane, as given in (3), (4). They are determined by three positive integers: the number of strands , the number of repeats of the basic pattern, and the number of lobes in the generalised lemniscate, i.e. in the Lissajous figure. We will often refer to the functions , suppressing suffixes when the context is clear. Different choices of and give the same braid which is rescaled in horizontal plane. We always assume that . Replacing by gives the braid which is the mirror image and hence the closures of the braids corresponding to and are also mirror images. Unless stated otherwise, we assume . Otherwise and are just scale factors, and do not change the topology.
where is large enough that for all (and upwards increase of corresponds to right-handed increase of azimuthal coordinate in the solid torus). The idea behind this parametrisation can be understood as taking the parametrised braid inside a cylinder of radius and wrapping it around, joining top and bottom of the cylinder. This is illustrated in Figures 1 and 2, and also Figure 3, which involves an generalised lemniscate, with five strands and two repeats, closing to the knot (as discussed later at Table 2). Note that in this process we have identified the braid height coordinate with an azimuthal coordinate of the solid torus in which the knot is embedded. We will revisit this idea in the construction of knotted fields in later sections.
From the point of view of constructing fields with specified zero lines (as for parametrisations of knotted curves) from braids via (7), it is most natural to specify the braid’s strands as parametric curves; however, the usual mathematical framework of braids is via the algebra of crossings of the Artin braid group (described in kt:2008braid ()), which we briefly review before examining the braids closing to lemniscate knots. The group’s generators are the crossings , , with labelling the crossing position in order from the left, with positive power for an overcrossing, and the inverse for an undercrossing. A product of generators is called a braid word, and represents the geometric braid which is formed of strands that perform the crossings specified in the braid word from left to right following the braid downwards. Isotopic braids are equivalent under the braid group relations: generators , commute unless , in which case . This algebraically represents the third Reidemeister move; the fact that is the identity represents the second Reidemeister move.
For knots and links formed by the closure of the braid represented by a word , there are two additional Markov moves: if the generator with power (equivalently, ) occurs exactly once in , then the knot is isotopic to the braid closure of the word with () omitted (this stabilisation move is equivalent to the first Reidemeister move); the conjugation move states that the knot which is the closure of is isotopic to the closure of . A braid defines a permutation on the strand labels; the number of disjoint cycles of the permutation gives the number of disjoint components of the link upon closing the braid.
The convention in braid theory (e.g. kt:2008braid ()) is to read the braid word from a diagram downwards, with a left-handed orientation (i.e. overcrossings of strands from the left are considered positive). The figures of the parametric curves 1, 2, 3 are drawn in a left-handed coordinate system , where increases upwards, so that the figures show braid diagrams that allow to read off the braid word in the usual way. We will attempt to respect both conventions of reading braid words downwards and describing the geometric movements of the strands in the direction of increasing , upwards. This convention means that under braid closure, map, in cylindrical coordinates, to radius, height and azimuth respectively as in (7).
The diagram of the braid comprised of the strands following , , is defined to be the projection of the braid in the -direction, i.e. the braid diagram is made up of the curves , and crossing signs are determined by . Crossings occur when at some and some with . Our convention is that the strand labelled by crosses over at when . From the form of in (3), this requirement is satisfied if and only if or . We choose to place the crossings at at the beginning of the braid word (identical, by Markov conjugation, to considering them at at the end of the word). Note that the crossings at , with odd, are simultaneous in , as are the crossings at with even.
For , this gives the basic braid word, corresponding to the parametrised strands (3),
and for general , we have repeats, i.e. . The signs of the crossings are determined by ; the crossing sign depends on the parity of the lobe of the Lissajous figure in which the crossing occurs. Since the strands cannot intersect, and must be relatively coprime (so there is no for which for some ). We choose , since for any knot with , there is which gives rise to the same braid word. It is straightforward to see that the crossing signs, for , and and coprime, are determined by the following rule:
For and , this implies that the crossings are all positive, as expected for braid representations of torus knots. For , must be odd, and so all crossings are positive, and are negative if . Thus the braid representation of the figure-8 knot is . Equation (9) describes the signs of lemniscate braid with positive . Note that for negative values of the vector is exactly the negative of the for positive . In general, the lemniscate braid representation is not the minimal braid representation of the knot or link as found by gittings:2004minimum ().
The lemniscate knots thus described are in the more general family of spiral knots (or links); a spiral knot (link) is defined as the closure of the th power of a braid word in which every generator appears exactly once, either as positive or negative power betvwy:2010spiral (). We term such braid words isograms. The braid word (8) satisfies this condition and hence all lemniscate knots are spiral, but in general spiral knots are not subject to (9). Following betvwy:2010spiral (), the spiral knot with strands, repeats with signs determined by is denoted , and the lemniscate knot with strands, repeats with lobes by ; therefore for satisfying (9). Spiral knots have several remarkable properties, which do not depend on , summarised in the following.
The spiral knot/link satisfies the following properties.
If , has one component iff it is the unknot;
is an -component link iff , and in particular is a knot when and are relatively coprime;
If , then is a 2-bridge knot (i.e. rational);
Every spiral knot is a periodic knot livingston:1993knot () with period ;
Every spiral knot is fibred stallings:1978constructions ();
If , then the word of can be rearranged (i.e. all anagrams of the same isogram close to the same knot or link) and for arbitrary , is the th power of any such rearrangement;
If is a prime power, , and a 1-component link, then the Alexander polynomial ;
If is a knot, the genus of satisfies ;
If is a prime power and a knot, and the genus of satisfies ;
If is a prime power, the minimal crossing number of satisfies .
The proofs of all of these but parts C and E are given in betvwy:2010spiral () (or are straightforward generalisations; part G is based on Murasugi’s theorem murasugi:1971periodic ()). Part F is what allows a spiral knot to depend only on , and , and not on the specific ordering of the basic word, which justifies the notation . Part C follows from considering a braid diagram as a braid as in (4) as a parametric curve in cylindrical coordinates with , with angle , radius and height ; with as the height function, there are maxima and minima, so is 2-bridge if (equivalent to (7) with and exchanged). More generally, this representation shows the -fold periodicity as a cyclic symmetry generated by a rotation about the axis of cylindrical coordinates. Part E follows from Stalling’s theorem stallings:1978constructions () that a knot is fibred if it has a homogeneous braid representation (i.e. the knot/link can be represented by a word where each generator only appears with the same sign within the word); this follows directly since the braid words of spiral knots are powers of isograms.
Strictly speaking there are two lemniscate knots for every choice of , and , one for a positive value of and one for a negative . The two knots are mirror images, so that some invariants like the Alexander polynomial, the crossing number, the braid index and the genus do not distinguish them. In particular, the statements of Theorem II.1 are valid for both cases.
Lemniscate knots have the additional symmetry that the of (9) is a palindromic vector if is odd, and anti-palindromic if is even, that is . As we show below, this seems to give rise to symmetric tangle representations of rational lemniscate knots, and similarly palindromic minimal braid words (where known), although we do not have a general proof which covers all values of , and . In common with other studies of spiral knots kst:2016sequences (), families of lemniscate knots and links seem to have common properties regarding their Alexander polynomial coefficients (for knots), Jones polynomial coefficients and tangle notation (when ), implying they are worthy of study in general, not simply as the knots simply realizable as nodes of complex scalar functions. We arrange our observations by , principally considering (i.e. the rational knots) and increasing ; knots are recognised from standard tabulations knotatlas (); knotinfo () using polynomial invariants. The limitations of these tables (going no higher than minimal crossing number for knots, and for links) mean that very few lemniscate knots with and can be identified (although invariants can be calculated for others).
When , we have the torus knots, with being the torus knot (which is isotopic to the torus knot). Since all crossings in the braid words for these have the same sign, the braid words generating the knots are not only homogeneous, but strictly positive. The properties of torus knots are well-known kawauchi:1996survey (), and we do not consider them further here.
The next case are the ‘figure-8 family’ of lemniscate knots with . Since and must be coprime, the braids have an odd number of strands (starting at ), and must be knots as and are coprime. We have the following Theorem:
The period 2, figure-8 lemniscate knot has minimal braid word , and has Alexander polynomial given by
The proof is given in the Appendix. Combining the results of Theorem II.1 and king:2010thesis (), we see that these knots are rational, with minimal crossing number (whereas the original generating braid has crossings), braid index (i.e. number of strands of minimal braid word), and genus . Properties of the first few members of this family are given in Table 1.
The tangle notation for , always has the symmetric form . The symmetries of the braid word for even imply that these knots are achiral, and hence their Jones polynomials have a similar form to the Alexander polynomial (i.e. with alternating signs of coefficients), with the coefficient of the constant term always positive. Since 2-bridge knots are alternating, the span of the Jones polynomial is equal to the crossing number. Furthermore, as increases, the sequence of coefficients appears to settle to the odd integers including 2, i.e. (although we have no general proof).
We previously listed the figure-8 family of knots and links with : again, these are (), (), (), (), . This suggests that for these knots ; some properties of this sequence, such as the values of the determinants , have been described in betvwy:2010spiral (); kst:2016sequences (). The only other lemniscate knot (i.e. and ) appearing in tabulations is , which is isotopic to .
For lemniscate knots and links with , the cases for the lowest numbers of strands (coprime to 3) are given in Table 2. Both knots and links appear in the list, so Alexander polynomials are not considered. For odd the lemniscate knots are in general not achiral, so the closures of the braids with are not mutually isotopic. In Table 2 we consider positive values of , as it seems to give rise to a tangle notation where all entries are positive. Constructing a similar table for negative values of from 2 is trivial. As in the case considered above, the tangle representations are all symmetric; for they follow the same pattern where (and ). The pattern of crossing numbers in the table suggests that . The absolute values of the coefficients of the Jones polynomials form a triangular arrangement of integers, with a maximum at the constant coefficient; as increases, pattern seems to settle down to sequence (as negative powers decrease to ), which for is given by the formula . For decreasing positive powers, the coefficients form the sequence which for is given by . Minimum braid words have been found for these cases up to , being for , for and for . The braid index in all these cases is equal to . Increasing the period with gives the sequence of knots and links beginning (), (), then an untabulated 12-crossing link. These appear to have , and the sequence of determinants of any minor of the crossing matrix is found in kst:2016sequences ().
Increasing further gives families which have similar features discussed in king:2010thesis (). These include, for , similar patterns in tangle notation and Jones polynomial coefficients as discussed here for and .
The patterns in crossing numbers, braid index and Conway tangle notation indicated by Tables 2 and 3 generalise to the following result, which holds in general for spiral knots and links.
Let be a spiral link with . Then it is rational and if we write the vector
with for all , and , then the Conway tangle notation of is
We also have that the minimal crossing number is and if is a knot, then the braid index is .
The proof is given in the Appendix. Recall that a lemniscate knot is just a special case of a spiral knot and that the number of loops is equal to the number defined implicitly in the theorem. Since in Table 2 we consider the and cases, Theorem II.3 confirms the patterns indicated in the tables. While the proof of Theorem II.3 gives a formula for the braid index , it does not provide a form of a braid on strands. From the known braid words for we expect the minimal braid word of to be of the form if and if .
We speculate that any spiral knot with small (hence for any such lemniscate knot), , which would generalise known for torus knots (with ) and in Theorem II.3. If the knot is alternating and is a prime power, using murasugi:1991braid () and betvwy:2010spiral (), the braid index then should be of the form . While examples for low are consistent with these formulas, a theorem by Lee and Seo ls:2010formula () implies that if , then , where is defined as in Theorem II.3. Hence if is a prime power and is alternating, the crossing number is of the form . This means that in this case the braid diagram in its spiral form minimises both the braid index and the crossing number. A theorem by Lee lee:2004alexander () gives an upper bound for where is genus. Combining this with Theorem II.1I and Theorem II.3 it can be shown that (the closure of is spiral, but not lemniscate and (the closure of is the closure of a homogeneous braid, but not spiral.
This concludes our discussion of the properties of the lemniscate knots. We now show how the closures of the corresponding parameteric braids leads to complex maps of which the lemniscate knots are nodal lines.
Iii Complex maps and fibrations for lemniscate knots
The construction of polynomial maps where the nodal lines form lemniscate knots was outlined at the end of Section I. In this section, we prove that these maps indeed have the desired knotted zeros and are fibrations (for appropriate values of and in (3)), and then briefly explore some generalisations of the lemniscate knot construction. With , and be positive integers with and coprime, (3) gives a parametrisation , where and of the lemniscate braid with loops, strands and repeats of the basic braid word. These give rise to the family of complex polynomials as in (5) with roots . By construction, the lemniscate braid in its parametrisation is the preimage of zero of as a map from .
It can be shown, for example by using induction on the number of factors and elementary arithmetic of roots of unity, that due to the trigonometric form of the roots the map , can be written as a polynomial in the variables , and (where, of course, ). Hence can be seen as the restriction of a complex map to the set (where necessary, is now the complex unit circle ) and is a polynomial in complex variables , and . is derived from by writing for every instance of and for every instance of in the polynomial expression of (from the construction defined above, no term in the polynomial has any occurrence of ). In doing so we identify and , which closes the ends of the braid. Thus we have , where is the lemniscate knot .
We have not so far considered the positive stretching parameters in (3). Although the construction is valid for any , in order to guarantee that we get the desired knot, i.e. , we need to consider specific choices for these parameters. We consider the one-parameter family of parametrisations where , is a positive parameter, and and are fixed positive real numbers. To indicate the dependence of , and on , we will write these maps respectively as , and (suppressing other suffixes).
We want to show that, for small enough , is isotopic to the desired knot , using the complex coordinates of the unit 3-sphere, . Note in particular that for sufficiently small , , i.e. for all , . Thus the image of under the map (where ),
is ambient isotopic to (the image of is independent of the modulus of the second argument). This result can be easily seen when is given in the parametrisation (7). We now outline a proof of the ambient isotopy from the image under of , known to be , to for small enough .
For all positive integers , and with and coprime and all choices of and , there exists an such that for all , where is the lemniscate knot.
First note that by definition . Furthermore, for all fixed and the function is a polynomial in of degree . We extend the definition of with to be the roots of . This allows us to choose small enough such that .
We consider the different as functions of . These are are smooth at , since for every , the roots of are disjoint and disjoint roots of polynomials depend smoothly on coefficients. Hence there is a independent of such that for all and . Now, we can choose such that for any , for every and every the curve intersects at a unique point and . Then define to be
By construction is a smooth isotopy from to if . By the isotopy extension theorem, it extends to an ambient isotopy ek:1971imbeddings () showing that . ∎
Theorem III.1 shows that the zero set of restricted to has the desired knot type provided is small enough. The proof does not specify how small has to be, since it does not give a value for . However, values can be calculated using bounds on the modulus of roots of polynomials and the implicit function theorem. Note in particular that does not depend on . We have checked numerically for the explicit examples in Section II, with ; for the (Table 1), is sufficient, and for the (Table 2), is sufficient (from the numerical behaviour, we suspect these will suffice for higher ).
Since is a polynomial in and for all , , and , the points on the intersection of and are regular points of , so has full rank (here and below, denotes the gradient map on a manifold ). Since the intersection of with is transverse, the knot is in fact a set of regular points of the restriction of to the unit 3-sphere, i.e. . This allows small smooth perturbations of the coefficients without altering the link type of the nodal set. This is particularly advantageous when additional physical constraints have to be taken into account.
The transversality of the intersection also builds a connection to the notion of transverse -links. These were defined by Rudolph rudolph:2005knot () to be the links that arise as transverse intersections of a complex plane curve and the unit 3-sphere. In our case, we do not deal with complex plane curves, but with zero sets of semiholomorphic polynomials, a significantly weaker notion.
Recall from Theorem II.1E that lemniscate knots are fibred. Having constructed a polynomial with , one might ask whether the map is a fibration. By the Ehresmann Fibration Theorem ehresmann:1950connexions () it is sufficient to check that the phase function does not have any critical points, i.e. . We have the following result.
If are such that does not have any phase-critical points , i.e. no points at which (for one value of and equivalently for all ), then there is no point with such that at , for all small enough . Hence induces a fibration over .
Note that the derivative converges uniformly to on as . In particular, at some , when evaluated at all with and . This means that for with the same statement holds for all .
Since does not have any phase-critical points and , it follows from the continuity of away from the zeros of that there is a such that for all with . Now choose such that and it follows that everywhere, as either or . Hence for all sufficiently small , does not have any phase-critical points on . ∎
Theorem III.2 gives a sufficient condition for an explicit fibration of the knot complement over as the argument of a semiholomorphic polynomial. We are not aware of any procedure that would find values for and , such that this condition is satisfied for . In fact, it is not even clear if such values always exist. We have checked numerically for the explicit lemniscate knots identified in Section II, is sufficient, meaning the maps constructed for them (with the previous values of ) are, indeed, fibrations.
We have proved, therefore, that the nodal lines of the functions , for small enough , indeed form the lemniscate knot or link, and for the explicit knots considered in Section II, these are in fact explicit fibrations. In spite of these technical details, the procedure for constructing complex functions with zeros in the form of lemniscate knots (or indeed fibrations of lemniscate knots) is just that in Section I with appropriate choices of and . For low , we found that it is sufficient that if , and if . The braid polynomial is constructed as from Equation 5 and then multiplied out and simplified. Then all occurrences of are replaced by , and by . Rewriting and in terms of according to (2) gives an explicit complex polynomial of three-dimensional real space with a nodal knot or link. If and are rational, all coefficients are rational, so by multiplying by a constant, we can make sure that has integer coefficients. This generalises the procedure for the figure-8 knot whose function was given in (6). Other examples are