Motivations Knot polynomials in terms of helicity Examples Outlook Topological Dynamics in the Physical and Biological Sciences Knot polynomial invariants in terms of helicity for tackling topology of fluid knots Xin LIU School of Mathematics and Statistics University of Sydney, NSW 2006, Australia [email protected] Joint work with: Renzo L. Ricca (University of Milano-Bicocca, Italy) 1 November 2012, Isaac Newton Institute, Cambridge Motivations Knot polynomials in terms of helicity Examples Outlook CONTENTS • Motivations • Determining topology of fluid knots • Heuristics of quantum Chern-Simons theory • Exponential form of helicity • Knot polynomial invariants in terms of helicity • Main Theorem • Revisit to knot polynomials and realization of skein relations • Proof of theorem • Examples: computing the Jones polynomials of typical knots/links • Trefoil knots • Figure-of-eight knot • Whitehead link • Borromean rings • Outlook Motivations Knot polynomials in terms of helicity Examples Outlook Motivations • Determination of fluid knot topology needs powerful tools investigations on =⇒ knot polynomial invariants • Heuristics of quantum Chern-Simons eHelicity theory Motivations Knot polynomials in terms of helicity Examples Outlook Topology is important: how tangled is a fluid knot? TRACE (NASA) Uddin et al. (2009) Magnetic fields and vorticity localization occur in classical and quantum fluids. Topology of fluid knots is crucial, as it is related to energy and dynamical properties. How to characterise topology of fluid knots? Barenghi et al. (2001) Raymer & Smith (2007) Motivations Knot polynomials in terms of helicity Examples Outlook Current approach: linking numbers • Helicity of vortex flows (Woltjer, 1958) Z H = u · ω d3x, u — velocity, ω — vorticity Ω 3 3 ω = ∇ × u on the support Ω ⊂ R ; ∇ · u = 0 in R ; ω · nˆ = 0 on ∂Ω. • For a link K of N vortex knots of circulation κi, we have (Moffatt 1969; Ricca & Moffatt 1992) X 2 X H (K) = κi Sli + 2 κiκj Lkij i ij Sli —Calug˘ areanu-White˘ self-linking number Lkij — Gauss linking number • However, sometimes linking numbers fail to detect essential topology: Motivations Knot polynomials in terms of helicity Examples Outlook Why helicity? To solve this problem, our idea is to focus on helicity: • Helicity is the most important topological invariant in fluid mechanics. Thanks to this Newton Institute program (— IUTAM & Edinburgh workshops): • Helicity plays an central role in the research of fluid mechanics: theoretical and experimental • Higher order linking numbers and Kontsevich integrals: • Cohomological class: Generalized helicity based on generalized Gauss mapping • Helicity has remarkable geometric significance — Abelian Chern-Simons 3-form (or abelian Chern-Simons action) Motivations Knot polynomials in terms of helicity Examples Outlook Chern-Simons theory • Generic Chern-Simons action with a non-abelian gauge group G: k Z 2 S = A ∧ dA + A ∧ A ∧ A, M 3— 3-manifold. 4π M 3 3 a where A is the gauge potential, A = A Ta, with Ta denoting the generators of G. The cubic term A ∧ A ∧ A labels the non-abelianity. • Fluid mechanics is treated as an abelian case, namely, the gauge group is U(1). • Helicity, as an isotopy invariant, is defined as Z H = u · ω d3x, ω = ∇ × u. Ω In the language of differential forms and wedge products: i u =⇒ 1-form u = uidx , ω =⇒ 2-form ω, ω = du. The helicity becomes a 3-form integral: Z H = u ∧ du. Ω • Hence, helicity is an abelian Chern-Simons action. Motivations Knot polynomials in terms of helicity Examples Outlook Extraordinary success of quantum Chern-Simons theory • The past two decades have witnessed the remarkable progress of quantum CS theory as the most important 3-dimensional topological quantum field theory. It provides a field theoretical justification for knot and 3-manifold topological invariants beyond algebraic quantum groups. • Witten’s seminal work (1989; Fields Medal 1990) =⇒ Two major research approaches in 3D TQFT: • algebra-geometric description for conformal field theory; • Reshetikhin-Turaev constructions from tensor category. • The third major approach in 3D TQFT is perturbative expansions of CS integrals. • Bar-Natan, Birman-Lin, Guadagnini, Labastida, Le-Murakami-Ohtsuki, et al. • Exponential expansions =⇒ Vassiliev invariants, via the Kontsevich integral. • Birman & Lin (1993): The Jones, HOMFLY and Kauffman polynomials exponential form −→ Vassiliev invariants (finite-type invariants) Motivations Knot polynomials in terms of helicity Examples Outlook Powerful tools of knot theory: knot polynomial invariants • Knot polynomials: Alexander-Conway, Jones, Kauffman, HOMFLY, . • Vassiliev finite type invariants, . Why no such tools in fluid mechanics? — complex flux tube structures (bundle of vortex lines)? — thousands of crossing sites? Motivations Knot polynomials in terms of helicity Examples Outlook Heuristics • In quantum Chern-Simons theory: Topological invariants ⇐= Vacuum expectation value of Wilson loops * N H + Z P H Y i A 1 i A iS e γk = [DA] e k γk e Z — partition function Z k=1 • In the case of superfluid, helicity turns to be line integrals Z X I H (K) = u ∧ du = κi u · dl, Ω i γi K = {γi}, γi — knotted thin filaments; κi — circulation of γi; u — induced velocity arising from the Biot-Savart integral * This also holds for the cases of thin vortex filaments and thin magnetic flux tubes (Barenghi, Ricca & Samuels, 2001): ω = ω0ˆt, where: ω0 — constant, ˆt — the unit tangent to the vortex axis Motivations Knot polynomials in terms of helicity Examples Outlook Heuristics (. continued) i P H u·dl • Thus, eiS ≡ eiH =⇒ e k γk , and the functional integral turns to be Z i P H u·dl2 [Du] e k γk . i P H u·dl This means the exponential e k γk is at the center of the problem. • Inspiration! This provides us a totally new starting point for studying this problem — the exponential form of the line integral P H i u·dl iH e k γk = e [Remark]: in this classical mechanical case we don’t do functional integrals. • Recall the Edinburgh workshop: • Significance of Line Integral • Phases in the generalized Aharonov-Bohm effect Motivations Knot polynomials in terms of helicity Examples Outlook Knot polynomial invariants in terms of helicity • Main theorem • Revisit to knot polynomials • Realization of knot polynomials in a physical system • Preliminary for proof • Proof of theorem (2 steps) Motivations Knot polynomials in terms of helicity Examples Outlook Result: knot polynomials in terms of helicity For simplicity, let us refer to superfluid vortex lines: • Circulations of vortices are quantized and usually equal, κi = κ, in superfluid. • For thin filaments in quantum fluids, since ω = ω0ˆt, helicity can be reduced to X I H (K) = κi u · dl, u — velocity induced from the Biot-Savart integral. i γi By rescaling eiH(K) → tH(K), and taking κ = 1 for simplicity, we have Theorem (Liu & Ricca, 2012) Let K be a fluid knot or link. If the helicity of K is H = H (K), then H(K) H u·dl t = t K , appropriately re-scaled, satisfies the skein relations of the Jones polynomial V = V (K). Motivations Knot polynomials in terms of helicity Examples Outlook Revisit to knot polynomials • Reidemeister moves: Two isotopic knots can be transformed into each other through a sequence of Reidemeister moves. Figure: Type-I, II & III Reidemeister moves. Ambient-isotopic: invariant under all the three types of moves Regular-isotopic: invariant under only the type-II & III moves • Crossing states at a site: The method of knot theory to determine topological equivalence among knots is to examine the different states at a crossing site. For instance, for oriented links, we consider They represent three links which are almost the same but differ only at the crossing site. Motivations Knot polynomials in terms of helicity Examples Outlook Revisit to knot polynomials (. continued) • A knot polynomial is defined by the skein relations it satisfies. The skein relations can be regarded as the recurrence relations for computing the knot polynomial (see the examples in the next section). • Skein relations of typical knot polynomials • Jones polynomial (for oriented links; ambient-isotopic) V ( ) = 1 −1 1 − 1 τ Vf I − τV @I@ = τ 2 − τ 2 V ] • Kauffman bracket polynomial (for unoriented links; regular-isotopic) h i = 1, f = a h i + a−1 h i , = a−1 h i + a h i , h t i = − a2 + a−2 h i , • HOMFLY polynomialc (for oriented links; ambient-isotopic) P ( ) = 1 t−1Pf I − tP @I@ = zP ] • Alexander-Conway polynomial, R-polynomial, . Motivations Knot polynomials in terms of helicity Examples Outlook Realizing skein relations • In a physical system, to find a knot polynomial means to realize its skein relations. • Example: Guadagnini, et al. (Nucl. Phys. B, 1990) proposed a method to derive skein relations of knot polynomials in a non-abelian Chern-Simons study. Consider the difference between an over- and under-crossing: This is represented by the difference between two VEVs of Wilson loops * !+ * "I # !+ W I − W @I@ = T r ··· A ··· U (3, 4) ··· loop Z 1 = T r ··· i F ··· U (3, 4) ··· area 2 δS δ Then, using a property of Chern-Simons, F = δA , the field tensor F is replaced by the operator δA , and then lead to the desired skein relations. • This is just one way to realize skein relations. For your problems, you can develop a different method to realize the skein relations of a knot polynomial. Motivations Knot polynomials in terms of helicity Examples Outlook Preliminary remark: reduction techniques on crossing sites • Our strategy: P H u·dl To take benefit of the exponential form of the line integral, tH(K) = t i γi , by reforming the strands of K — technique of local path-addition. • Take L+ and L− for instance: We add local imaginary paths to the strands, which have opposite orientations and can cancel each other. This is an identical operation from the fluid mechanical point of view.