Derivation of the HOMFLYPT knot polynomial via helicity and geometric quantization
DDerivation of the HOMFLYPT knot polynomial via helicity andgeometric quantization
Antonio Michele MITI †,‡ and Mauro SPERA † † Dipartimento di Matematica e Fisica “Niccolò Tartaglia", Università Cattolica del Sacro CuoreVia dei Musei 41, 25121 Brescia, Italia ‡ Departement Wiskunde, KU-LeuvenCelestijnenlaan 200B, B-3001 Leuven (Heverlee), België
Abstract
In this Letter we propose a semiclassical interpretation of the HOMFLYPT polynomialbuilding on the Liu-Ricca hydrodynamical approach to the latter and on the Besana-S.symplectic approach to framing via Brylinski’s manifold of mildly singular links.MSC 2010: 53D50, 58D10, 53D12, 53D20, 57M25, 76B47, 81S10.
Keywords : Knot polynomials, symplectic geometry, Lagrangian submanifolds, hydrodynamics,geometric quantization, Maslov index.
In this article, building on the Maslov-type methods developed in [2], we present a novel interpretationof the HOMFLYPT (and hence of the Jones) polynomial ([6, 19]) as a WKB-wave function via geometricquantization of the so-called Brylinski manifold of singular knots (and links), taking inspiration fromthe ad hoc helicity-based hydrodynamical procedures devised in [14, 15]. Our approach can be com-pared with the Jeffrey-Weitsman one ([9, 10]), providing a rigorous framework for the Jones-Witten the-ory ([24, 12]). The latter, though again based on geometric quantization, is much more sophisticated.In our setting, no reference to Lie groups (other than U (1)) is made and, as in Liu-Ricca, everythingis based on helicity only, at the cost of relying on the Maslov-Hörmander techniques of [2], togetherwith an appropriate semiclassical interpretation of the skein relation. The present note is an improvedversion of part of the preprint [17]. In this section we concisely review some basic notions related to geometric quantization, tailored toour needs (see e.g. [26] for background). First recall that a submanifold Λ of a symplectic manifold( M , ω ) is Lagrangian when the symplectic form ω vanishes thereon and it is of maximal dimensionwith respect to this property. If Q is a smooth manifold, then its cotangent space T ∗ Q is a symplecticmanifold (equipped with a canonical symplectic form). A Lagrangian submanifold Λ ⊂ T ∗ Q in generalposition can be described in the following way (Maslov-Hörmander Morse family theorem , see e.g. [16,8, 7]): there exists (locally) a smooth function φ = φ ( q , a ), ( q , a ) ∈ Q × (cid:82) k ( (cid:82) k being a space of auxiliaryparameters) and a submanifold C φ = {( q , a ) ∈ Q × (cid:82) k | d a φ = d ( d a ) of maximal rank thereon (here d = d q + d a ) such that the map C φ (cid:51) ( q , a ) (cid:55)→ ( q , d q φ ) ∈ T ∗ Q a r X i v : . [ m a t h . S G ] O c t s an immersion with image Λ . If the Hessian H a (with respect to the auxiliary variables a ) is non degen-erate, one can solve a = a ( q ) and define the phase function F = F ( q ) : = φ ( q , a ( q )), with ( q , d F ( q )) ∈ Λ .The covector d F ( q ) = : p ( q ) is the momentum at q . This fails at the singular points of the obviousprojection Λ → Q , but the singular locus Z (the Maslov cycle ) turns out to be orientable and of codi-mension 1 in Λ with ∂ Z of codimension ≥
3. Taking a good open cover { V i } i ∈ I of Λ , and letting σ i be the signature of the Hessian H a on V i \ Z , one readily manufactures the so-called Maslov cocycle { h i j = ( σ i − σ j )} yielding a class M ∈ H ( Λ , (cid:90) ), dual to the Maslov cycle Z , see e.g. [7, Ch.II, ¢g7]. Thissituation holds for a general symplectic manifold, as a consequence of a result of Weinstein ([23]).Now, given a prequantizable symplectic manifold ( M , ω ), i.e. [ ω ] ∈ H ( M , (cid:90) ), then, by Weil-Kostant(see e.g. [26]), there exists a complex line bundle L → M (prequantum bundle), equipped with aHermitian metric and compatible connection ∇ with curvature Ω ∇ = − π i ω . Since the symplectic 2-form ω vanishes on any Lagrangian submanifold Λ ⊂ M , any (local) symplectic potential ϑ ( d ϑ = ω )becomes a closed form thereon, giving a (local) connection form pertaining to the restriction of theprequantum connection ∇ , denoted by the same symbol. The latter is a flat connection, and a globalcovariantly constant section ( ∇ s =
0) of the (restriction of ) the prequantum line bundle - called
WKBwave function - exists if and only if it has trivial holonomy. A WKB wave function is subject to suddenphase changes upon crossing the Maslov cycle Z (“passage through a caustic"), governed by the Maslovcocycle, see e.g. [7, 26, 16]. The theory developed in [2], see also [21], was aimed at placing the construction of the (Abelian) Witteninvariant in [24] on firm ground by avoiding the use of path integrals and it was strongly inspired bythe constructions recalled in the preceding section, albeit with modifications dictated by the infinitedimensional environment. We resume it by closely following these papers with appropriate en route modifications and referring, for the symplectic, hydrodynamical and knot theoretical background, to[4, 7, 16, 8, 1, 22, 11, 13].We shall act within the generalized Brylinski symplectic manifold of oriented mildly singular links in (cid:82) , (cid:98) Y (allowing a finite number of crossings and finite order tangencies), whose symplectic structurereads, at a generic link L with components L j , j =
1, . . . n , represented up to orientation-preservingreparametrizations by smooth embeddings γ j ∈ C ∞ ( S , (cid:82) ) ≡ L (cid:82) with velocities ˙ γ j : Ω L ( · , · ) : = n (cid:88) j = (cid:90) L j ν ( ˙ γ i , · , · )(with ν = d x ∧ d y ∧ d z being the standard volume form of (cid:82) ). The manifold consisting of all bona fide oriented links in (cid:82) will be denoted by Y , and it is clearly non-connected.The volume form ν can be portrayed as ν = d x ∧ d y ∧ d z = d ( z d x ∧ d y ) ≡ d ˆ θ in terms of the (multisymplectic) potential ˆ θ ; the latter transgresses to a (symplectic) potential θ for Ω ,which vanishes identically when restricted on the plane z =
0. The submanifold Λ ⊂ (cid:98) Y consisting ofthe links on a plane (with indentations keeping track of crossings) is a Lagrangian one, see [2].Now observe, again following [2], that the links in (cid:82) can be interpreted as solutions of the Euler-Lagrange equation pertaining to a
Chern-Simons Lagrangian ( helicity , in hydrodynamical parlance)with source T L given by the link itself Φ = Φ ( A , L ) : = k π (cid:90) (cid:82) A ∧ d A + (cid:90) L A ≡ k π H ( A ) + T L ( A )2ith A denoting an Abelian connection (form) on (cid:82) with curvature F A = d A - rapidly decaying atinfinity to ensure convergence of the integral - and k a non-zero integer or real number.This CS Lagrangian is then taken, as in [2], as a Morse family , with the auxiliary parameters (also cf.[7]) given by the Abelian connections. The Euler-Lagrange equation reads: k π F A + T L = k π d A + T L = de Rham current ([5])) whose curvature is concentrated(i.e. δ -like) on L . The solution can be given in standard vector calculus terms with a so-called Coulombgauge fixing, div A =
0, or, Hodge theoretically, δ A =
0. The (singular) connection A L such that d A L = T L and δ A L = A L = − π k ∆ − δ T L where ∆ is the Hodge Laplacian on 1-forms, acting component-wise as the ordinary Laplacian (up toa negative constant) since we operate in flat space. Existence, in the sense of currents, follows fromthe Hörmander-Łojasiewicz theorem, see e.g. [25]. Notice that if we want to insert A L into Φ to get alocal phase φ in accordance with the general portrait depicted above (see e.g. [7, 2]), we are forced toconsider ordinary links . Proceeding as in [2] (cf. Theorem 3.1 therein) we get, for the local phase φ , theexpression (involving the helicity H ( L ) = H ( A L ) of a framed link) φ ( L ) = Φ ( A L , L ) = − π k H ( L ) ≡ πλ n (cid:88) i , j = (cid:96) ( i , j ),where λ : = − k and with (cid:96) ( i , j ) = (cid:96) ( j , i ) being the Gauss linking number of components L i and L j if i (cid:54)= j and where (cid:96) ( j , j ) is the framing of L j , equal to (cid:96) ( L j , L (cid:48) j ) with L (cid:48) j being a section of the normal bun-dle of L j , see e.g. [20, 18, 2, 21]. A regular projection of a link L onto a plane produces a natural framingcalled the blackboard framing , and H ( L ) = w ( L ), the writhe of L . The helicity can be interpreted, asin [2], as a regularised signature (cf. Section 2 above) and, as such, it enters the Maslov theory devel-oped therein, cf. Theorem 4.1. Since the symplectic potential of Brylinski’s form can be taken equalto zero, the phase, i.e. the helicity, is (locally) constant, being a topological invariant. The Lagrangiansubmanifold Λ is thence locally given by the graph( L , d H ( L )) = ( L , 0)( d H ( L ) = eikonal equation , see [2, 21]). We point out that one could equivalentlyemploy, mutatis mutandis , the Lagrangian submanifold manufactured via the cone construction of [2].In our context the assumptions of the Weil-Kostant theorem are fulfilled ( Ω is exact) and a covari-antly constant section (also called WKB wave function) is just a locally constant function on Y since,as in [2], we neglect the so-called “half-form" correction (see e.g. [26]). One must then accommo-date passage through the corresponding Maslov cycle Z , given in our case by the (mildly) singular linkspossessing exactly one singular point causing a sudden jump of writhe (helicity) by ± removal of a crossing changes the number of components of a given link and thus places the new link ina different connected component of the space Y . To this aim, let us consider the following provisional wave function (for genuine links L ) ψ = ψ ( L ) : = e π i λ H ( L ) (3.1)which is a regular isotopy link invariant (i.e. up to the first Reidemeister move), cf. [2], Theorems 3.1and 5.1. The generic value taken by λ (in particular, it can be taken equal to a root of ±
1) avoids trivial-ities. 3 L - L + Figure 1: CrossingsDenote, as usual, by L + , L − and L three links(regularly projected onto a plane, z =
0, say) dif-fering at a single crossing (( ± E ± , that is trivial knotswith ( ± H ( E ± ) =
1. Starting, for in-stance, from L , one can “add" E + to the two co-herently oriented parallel strands of L in such away that E + comes with the opposite orientation: apartial cancellation occurs and the net result is L + .Conversely, proceeding backwards we can, by adding appropriately an E − , produce L from L + and soon. Therefore, addition of E ± allows one to pass from one local configuration to the other, see Figure 2. E + L L + Figure 2: Surgery via E + Now set: α : = e π i λ H ( E + ) = e π i λ , α − = e − π i λ = e π i λ H ( E − ) so that, trivially, ψ ( L ± ) = α ± ψ ( L ), ψ ( L ± ) = α ± ψ ( L ∓ )and α − ψ ( L + ) − αψ ( L − ) =
0. (3.2)Thus we see that α ± arises as the local contributionto the WKB wave function ψ upon addition (surgery)of an eight figure (or “curl") - which can be applied toa single branch as well (first Reidemeister move) - and α ± as the corresponding contribution upon crossing the Maslov cycle Z . We now wish to modify ψ so as to produce a genuine ambient isotopy link invariant, keeping theabove interpretation . For this purpose, let Ψ be a covariantly constant wave function stemming fromapplication of the GQ-procedure, normalised in such a way that Ψ ( (cid:176) ) = (cid:176) being the unknot): assuch it is not uniquely determined, since Y is not connected, but Ψ can be made to depend naturally ontwo parameters, the above α and z , below. We require that, upon replacement of ψ by Ψ , the modifiedl.h.s. of (3.2) becomes proportional to Ψ ( L ) (for a suitable constant z which is assumed to be universal,i.e. independent of the specific link at hand. Consequently, the sought wave function Ψ must satisfythe skein relation (and normalization) for the HOMFLYPT polynomial P ([6, 19] - here α − is LR’s a ) α − Ψ ( L + ) − α Ψ ( L − ) = z Ψ ( L ) , Ψ ( (cid:176) ) =
1, (3.3)this assuring its existence. The trivial wave function Ψ ≡ α = z =
0. The procedure isstill partially ad hoc , this depending on the non-connectedness of the manifold Y . The skein relation(3.3) can be equivalently written in the form Ψ ( L − ) = α − Ψ ( L + ) − z α − Ψ ( L )which tells us that Ψ ( L − ) can be obtained by suitably adding Ψ ( L + ), corrected by a Maslov type tran-sition (local surgery via α − - one has the same number of link components) and Ψ ( L ), corrected bya “component transition" α − (and multiplied by an extra coefficient z ). The latter contribution wasabsent in [2] since that paper dealt with knots only. Notice that upon setting z = α − − α and letting α →
1, we get the trivial invariant Ψ ≡ Remarks.
1. In this way we essentially recover the hydrodynamical portrait of Liu and Ricca [14, 15],essentially stating that “ P = t H " albeit with a different (and more conceptual) interpretation. Inparticular, the two parameters used in HOMFLYPT are not quite the same. The local surgery operation4nvolves helicity, as in LR, but we portray the latter as a local phase function, governing a componenttransition or Maslov, upon squaring it, as in [2].2. Passage from L ± to L (and conversely) in (cid:98) Y - abutting, as already remarked, at a change in thenumber of the link components - involves coalescence of two crossings into one and correspondingtangent alignment. This is a sort of “higher order" contribution beyond the Maslov one.The upshot of the previous discussion is the following: Theorem 3.1.
The HOMFLYPT polynomial P = P ( α , z ) can be recovered from the geometric quantizationprocedure applied to the Brylinski manifold (cid:98) Y and to its Lagrangian subspace Λ , namely, it coincides(after normalization) with a suitable covariantly constant section Ψ = Ψ ( α , z ) thereby obtained. Thecoefficient α of P is a phase factor related to the helicity of a standard “eight-figure" and z comes fromaccounting for the variation of the number of components of a link. Acknowledgements.
The authors, both members of the GNSAGA group of INDAM, acknowledge sup-port from Unicatt local D1-funds (ex MIUR 60% funds). They are grateful to Marcello Spera for helpwith graphics.
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