AA Cable Knot and BPS-Series
John ChaeDepartment of Physics and QMAP, UC Davis, 1 Shields Ave, Davis, CA, 95616, [email protected]
Abstract
A series invariant of a complement of a knot was introduced recently. The invariant forseveral prime knots up to ten crossings have been explicitly computed. We present the firstexample of a satellite knot, namely, a cable of the figure eight knot, which has more than tencrossings. This cable knot result provides nontrivial evidence for the conjectures for the seriesinvariant and demonstrates the robustness of integrality of the quantum invariant under thecabling operation. Furthermore, we find interesting effects of the cabling on the series invariant.
CONTENTS a r X i v : . [ m a t h . G T ] F e b Introduction
Inspired by the categorification of the Witten-Reshitikhin-Turaev invariant of a closed oriented3-manifold [37, 30, 29] in [15, 16], a two variable series invariant F K ( x, q ) for a complement of aknot M K was introduced in [13]. Although its rigorous definition is yet to be found, it possessesvarious properties such as the Dehn surgery formula and the gluing formula. This knot invariant F K takes the form F K ( x, q ) = 12 ∞ (cid:88) m ≥ m odd (cid:0) x m/ − x − m/ (cid:1) f m ( q ) ∈ c q ∆ Z (cid:2) x ± / (cid:3)(cid:2)(cid:2) q ± (cid:3)(cid:3) , (1)where f m ( q ) are Laurent series with integer coefficients , c ∈ Z + and ∆ ∈ Q . Moreover, x -variable is associated to the relative Spin c (cid:0) M K , T (cid:1) -structures, which is affinely isomorphic to H (cid:0) M K , T ; Z (cid:1) ∼ = H (cid:0) M K ; Z (cid:1) ; it has an infinite order, which is reflected as a series in F K . Therational constant ∆ was investigated in [14], which elucidated its intimate connection to the d-invariant (or the correction term) in certain versions of the Heegaard Floer homology ( HF ± ) forrational homology spheres. The physical interpretation of the integer coefficients in f m ( q ) arenumber of BPS states of 3d N = 2 supersymmetric quantum field theory on M K together withboundary conditions on ∂M K . Furthermore, it was conjectured that F K also satisfies the Melvin–Morton–Rozansky conjecture [24, 31, 32] (proven in [1]): Conjecture 1.1 ([13, Conjecture 1.5]) . For a knot K ⊂ S , the asymptotic expansion of theknot invariant F K (cid:0) x, q = e (cid:126) (cid:1) about (cid:126) = 0 coincides with the Melvin–Morton–Rozansky (MMR)expansion of the colored Jones polynomial in the large color limit: F K (cid:0) x, q = e (cid:126) (cid:1) x / − x − / = ∞ (cid:88) r =0 P r ( x )∆ K ( x ) r +1 (cid:126) r , (2) where x = e n (cid:126) is fixed, n is the color of K , P r ( x ) ∈ Q (cid:2) x ± (cid:3) , P ( x ) = 1 and ∆ K ( x ) is the (sym-metrized) Alexander polynomial of K . Additionally, motivated by the quantum volume conjecture/AJ-conjecture [7, 11] (explained inSection 2.2), it was conjectured that F K -series is q-holonomic: Conjecture 1.2 ([13, Conjecture 1.6]) . For any knot K ⊂ S , the normalized series f K ( x, q ) satisfies a linear recursion relation generated by the quantum A-polynomial of K ˆ A K ( q, ˆ x, ˆ y ) : ˆ A K ( q, ˆ x, ˆ y ) f K ( x, q ) = 0 , (3) where f K := F K ( x, q ) / (cid:0) x / − x − / (cid:1) . The actions of ˆ x and ˆ y areˆ xf K ( x, q ) = xf K ( x, q ) ˆ yf K ( x, q ) = f K ( xq, q ) . Implicitly, there is a choice of group; originally, the group used is SU(2). They can be polynomials for monic Alexander polynomial of K (See Section 3.2) F K -series, several prime knots up to ten crossings have been analyzed [13,27, 28]. They include the torus knots, the figure eight knot in [13], and in [28]. Positive braidknots ( , ), strongly quasipositive braids knots (m( ) , , ), double twistknots (m( ), m( ), m( )), and a few more prime knots (m( ), m( )) were examined in [27].In this paper we verify the above conjectures by computing the F K -series for (9 , F K -series for a family of a cable knot of the figure eight.The rest of the paper is organized as follows. In Section 2 we review the satellite operation ona knot and the recursion ideal of the quantum torus. In Section 3 we analyze knot polynomials ofthe cable knot of the figure eight. In Section 4 we derive the recursion relation for the cable knot.Then we deduce (cid:126) expansion from the recursion in Section 5. Finally in Section 6 consequences ofthe cabling operation are discussed and we propose conjectures about a family of a cable knot. Acknowledgments.
I would like to thank Sergei Gukov, Thang Lˆe and Laura Starkston forhelpful conversations. I am grateful to Ciprian Manolescu for valuable suggestions on a draft ofthis paper. I am also grateful to Colin Adams for valuable comments.
The satellite operation consists of a pattern knot P in the interior of the solid torus S × D , acompanion knot K (cid:48) in the S and an canonical identification h K (cid:48) h K (cid:48) : S × D −→ ν ( K (cid:48) ) ⊂ S , (4)where ν ( K (cid:48) ) is the tubular neighborhood of K (cid:48) .Figure 1: A pattern knot P (left), companion K (cid:48) (center) and satellite knot P ( K (cid:48) ) (right).A well-known example of satellite knots is a cable knot h K (cid:48) ( P ) = C ( r,s ) ( K (cid:48) ) that is obtained bychoosing P to be the ( r, s )-torus knot pushed into the interior of the S × D . This map h K (cid:48) hasbeen investigated in [23, 25, 26]. 3 .2 Quantum Torus and Recursion Ideal Let T be a quantum torus T := C [ t ± ] (cid:10) M ± , L ± (cid:11) / ( LM − t M L ) . The generators of the noncommutative ring T acts on a set of discrete functions, which are coloredJones polynomials J K,n ∈ Z [ t ± ] in our context, as M J
K,n = t n J K,n LJ K,n = J K,n +1 . The recursion(annihilator) ideal A K of J K,n is the left ideal A K in T consisting of operators thatannihilates J K,n : A J K,n := { α K ∈ T | α K J K,n = 0 } . As it turns out that A K is not a principal ideal in general. However, by adding inverse polynomialsof t and M to T [7], we obtain a principal ideal domain ˜ T ˜ T := (cid:88) j ∈ Z a j ( M ) L j (cid:12)(cid:12)(cid:12) a j ( M ) ∈ C [ t ± ]( M ) , a j = almost always 0 Using ˜ T we get a principal ideal ˜ A K := ˜ T A K generated by a single polynomial ˆ A K ˆ A K ( t, M, L ) = d (cid:88) j =0 a j ( t, M ) L j . This ˆ A K polynomial is a noncommutative deformation of a classical A-polynomial of a knot [3] (seealso [4]). Alternative approaches to obtain ˆ A K ( t, M, L ) are by quantizing the classical A-polynomialcurve using a twisted Alexander polynomial or applying the topological recursion [17]. A conjecturecalled AJ conjecture/quantum volume conjecture was proposed in [7, 11] via different approaches: Conjecture 2.1.
For any knot K ⊂ S , ˆ A K ( t = − , L, M ) reduces to the (classical) A-polynomialcurve A K ( L, M ) up to a solely M-dependent overall factor. In other words, J K,n ( t ) satisfies a linear recursion relation generated by ˆ A K ( t, M, L ). This propertyof J K,n is often called q-holonomic [9]. The conjecture was confirmed for a variety of knots [5, 7,8, 10, 19, 22, 36, 34].
In this section we will analyze the colored Jones polynomial and the Alexander polynomialof a cable knot to show that the former satisfies the MMR expansion and the latter is monic.Furthermore, the MMR expansion enables us to read off the initial condition that is needed inSection 5. 4 .1 The Colored Jones Polynomial
For (r,2)-cabling of the figure eight knot , we set P = T ( r,
2) and K (cid:48) = in (4). The cablingformula for an unnormalized sl ( C ) colored Jones polynomial of a ( r, K (cid:48) in S is [35] ˜ J C ( r, ( K (cid:48) ) ,n ( q ) = q − r ( n − ) n (cid:88) w =1 ( − r ( n − w ) q r w ( w − ˜ J K (cid:48) , (2 w − ( q ) , | r | > . Figure 2: (r,2)-cable of the figure eight knot.Its application to K = C (9 , ( ) , whose diagram has 25 crossings, is˜ J K,n ( q ) = q − ( n − ) n (cid:88) w =1 (cid:34) ( − ( n − w ) q w ( w − [2 w − w − (cid:88) r =0 r (cid:89) k =1 (cid:16) − q − k − q k + q − w + q w − (cid:17)(cid:35) . Using the (0-framed) unknot U value together with q = t J U,n ( t ) = t n − t − n t − t − , the first few unknot normalized polynomials J K,n ( q ) are J K, ( q ) = 1 J K, ( q ) = q − q + 1 q + 1 q − q + 1 q − q + 1 q − q J K, ( q ) = q − q − q + q − q + q + q − q + q − q + 1 q − q + 1 q − q + 1 q − q − q + 2 q − q + 1 q − q + 1 q + 1 q − q − q + 1 q − q + 2 q − q − q + 1 q This cabling parameters correspond to for the pattern knot. We assume 0-framing for . (cid:126) series are J K,n ( e (cid:126) ) = 1 + (cid:0) − n (cid:1) (cid:126) + (cid:0) −
42 + 42 n (cid:1) (cid:126) + (cid:18) − n + 1232 n (cid:19) (cid:126) + (cid:18) − n − n (cid:19) (cid:126) + (cid:18) − n + 14986 n − n (cid:19) (cid:126) + (cid:18) − n − n + 84009760 n (cid:19) (cid:126) + (cid:18) − n + 5458551712 n − n + 132737633360 n (cid:19) (cid:126) + · · · (5)We see that, at each (cid:126) order, the degree of the polynomial in n is at most the order of (cid:126) , which isan equivalent characterization of the MMR expansion of the colored Jones polynomial of a knot.Secondly, as a consequence of the cabling, odd powers of (cid:126) appear in the expansion, which areabsent in the case of the figure eight knot [13]. Moreover, the coefficient polynomials for the odd (cid:126) -powers have one lower degree whereas the degree of the polynomials are the same for the even (cid:126) -powers. Hence they are unaffected by the cabling operation. The cabling formula for the Alexander Polynomial of a knot K is [18]∆ C ( p,q ) ( K ) ( t ) = ∆ K ( t p )∆ T ( p,q ) ( t ) , ≤ p < | q | gcd( p, q ) = 1 , where ∆( t ) is the symmetrized Alexander polynomial and T ( p,q ) is the (p,q) torus knot. Note thatour convention for the parameters of the torus knot are switched (i.e. p ≡ , q ≡ r ). Applying theabove formula to C (9 , ( ), we get∆ C (9 , ( ) ( x ) = ∆ ( x )∆ T (2 , ( x )= − x − x + x + 1 x + 2 x + 2 x − x − x + x + 1 x − x − x + 1 . From this Alexander polynomial its symmetric expansion about x = 0 (in x) and x = ∞ (in 1 /x )in the limit of (cid:126) → q → F K ( x, q ) = 2 s.e (cid:32) x / − x − / ∆ K ( x ) (cid:33) = x / − x / + 2 x / − x / + 5 x / − x / + 13 x / − x / + 34 x / − x / − x / + 1 x / + 89 x / − x / − x / + 2 x / + 233 x / − x / − x / + 5 x / + 610 x / − x / + · · · ∈ Z (cid:104)(cid:104) x ± / (cid:105)(cid:105) (6)The coefficients in the expansions are integers and hence the Alexander polynomial is monic, whichis a necessary condition for f m ( q )’s in (1) to be polynomials.6 The Recursion Relation
The quantum (or noncommutative) A-polynomial of a class of cable knot C ( r, ( ) in S havingminimal L-degree is given by [33]ˆ A K ( t, M, L ) = ( L − B ( t, M ) − Q ( t, M, L ) (cid:0) M r L + t − r M − r (cid:1) ∈ ˜ A K (7)where Q ( t, M, L ) = Q ( t, M ) L + Q ( t, M ) L + Q ( t, M ) , B ( t, M ) := (cid:88) j =0 c j b ( t, t j +2 M ) b ( t, M ) = M (1 + t M )( − t M )( − t + t M ) t − t − c = ˆ P ( t, t M ) ˆ P ( t, t M ) , c = − ˆ P ( t, t M ) ˆ P ( t, t M ) , c = ˆ P ( t, t M ) ˆ P ( t, t M ) . The definitions of the operators ˆ P i are written in Appendix A. For K = C (9 , ( ), applying (7) to f K ( x, q ) together with x = q n yields α ( x, q ) F K ( x, q ) + β ( x, q ) F K ( xq, q ) + γ ( x, q ) F K ( xq , q ) + δ ( x, q ) F K ( xq , q ) + F K ( xq , q ) = 0 , (8)where α, β, γ, δ functions and their (cid:126) series are documented in [2]. From (8) we find the recursionrelation for f m . f m +98 ( q ) = − q m (cid:16) − q m (cid:17) (cid:104) t f m +94 + t f m +90 + t f m +86 + t f m +82 + t f m +80 + t f m +78 + t f m +76 + t f m +74 + t f m +72 + t f m +70 + t f m +68 + t f m +66 + t f m +64 + t f m +62 + t f m +60 + t f m +58 + t f m +56 + t f m +54 + t f m +52 + t f m +50 + t f m +48 + t f m +46 + t f m +44 + t f m +42 + t f m +40 + t f m +38 + t f m +36 + t f m +34 + t f m +32 + t f m +30 + t f m +28 + t f m +26 + t f m +24 + t f m +22 + t f m +20 + t f m +18 + t f m +16 + t f m +12 + t f m +8 + t f m +4 + t f m (cid:105) ∈ Z [ q ± ](9)where t v = t v ( q, q m )’s are listed in [2]. Using this recursion and the initial data documented in [2], F K ( x, q ) can be obtained to any desired order in x. We next compute a series expansion of the F K of complement of the cable knot K . Specifically,a straightforward computation from (8) yields an ordinary differential equation(ODE) for P m ( x )at each (cid:126) order. Using the initial conditions for the ODEs obtained from (5) P (1) = 0 , P (1) = 6 , P (1) = − , P (1) = 8012 , P (1) = − , · · ·
7e find that P ( x ) = 5 x + 5 x − x − x − x − x + 36 x + 36 x − x − x − x − x + 15 x + 15 x − x − x + 16 x + 16 x − x − x + 19 x + 19 x − x − x + 20 P ( x ) = 25 x x − x − x − x − x + 306 x + 306 x − x − x − x − x + 2011 x x − x − x − x − x + 612 x + 612 x − x − x + 508 x + 508 x − x − x + 1648 x + 1648 x + 1538 x + 1538 x − x − x + 1574 x + 1574 x + 1670 x + 1670 x − x − x + 396 x + 396 x − x − x + 4082 x + 4082 x − x − x − x − x + 16831 . The subsequent P m ’s are documented in [2]. Substituting them into (2) results in2 F ( x, e (cid:126) ) = (cid:16) x / − x / + 2 x / − x / + 5 x / − x / + 13 x / − x / + 34 x / − x / − x / + 1 x / + 89 x / − x / − x / + 2 x / + 233 x / − x / − x / + 5 x / + · · · (cid:17) + (cid:126) (cid:16) x / − x / + 12 x / − x / + 35 x / − x / + 104 x / − x / + 306 x / − x / − x / + 15 x / + 890 x / − x / − x / + 36 x / + 2563 x / − x / − x / + 105 x / + · · · (cid:17) + (cid:126) (cid:16) x / −
252 1 x / + 36 x / − x / + 2472 x / − x / + 426 x / − x / + 1441 x / − x / − x / + 2252 1 x / + 4781 x / − x / − x / + 324 x / + 2563 x / − x / − x / + 22072 1 x / + · · · (cid:17) . Comparing to the series of the figure eight knot [13], we notice that every order of (cid:126) appears in theabove series whereas the series corresponding to the figure eight knot consists of only even powersof (cid:126) (i.e. P i ( x ) = 0 for i odd). This difference is an effect of the torus knot whose expansioninvolve all powers of (cid:126) [13]. Furthermore, the x-terms begin from m = 11 instead of m = 1 andthere are gaps in their powers. Specifically, x ± / , x ± / , x ± / and x ± / are absent. This is aconsequence of the structure of (6). A distinctive feature of the cable knot is that from x ± / thecoefficients are negative. Moreover, the positive and the negative coefficients alternate from thatx-power for all (cid:126) powers. These differences persist in the higher (cid:126) -orders, which are recorded in [2].We will see these differences in a manifest way in the next section.8 Effects of the Cabling
Since the initial data plays a core role in the recursion relation method, we discuss their featuresfor the cable knot and then propose conjectures about them, which can be a useful guide for findinginitial data for a family of the cable knots.In the initial data (see [2]) for the recursion relation (9), we notice several differences fromthat of the figure eight knot [13]. Before discussing them, let us begin with the properties of the F K that are preserved by the cabling. The initial data consists of an odd number of terms andpower of q increases by one between every consecutive terms in a fixed f m for all m’s, which arealso true for f and f . Additionally, the reflection symmetry of coefficients is retained up to f for positive coefficients and up to f for the negative ones but of course, those f m ’s do nothave the complete amphichiral structure. These invariant properties are a remnant feature of theamphichiral property of the figure eight knot.A difference is that the nonzero initial data begins from f and the gaps between the powersof x is four up to x / , which is in the accordance with f m ’s. These features are direct conse-quences of the symmetric expansion of the Alexander polynomial of the cable knot (6). In thecase of the figure eight its coefficient functions start from f and there are no such gaps. Anotherdistinctive difference is that f m ’s containing negative coefficients appear from m = 29. Moreover,the positive and negative coefficient f m ’s alternate from f (i.e. positive coefficients for f , f , . . . and negative coefficients for f , f , . . . ). Furthermore, from f the reflection symmetry ofthe positive coefficients in the appropriate f m ’s is broken. This phenomenon also occurs for thenegative coefficient f m ’s from m = 65. Breaking of the symmetry is expected since the cable knotof the figure eight is not amphichiral. The next difference is that the largest power of q in the pos-itive coefficient f m ’s for m ≥
15, the powers increase by 2 , , , , , , . . . , ,
11. For the negativecoefficient case, the changes are 4 , , , , , , . . . , ,
11 from m = 33. The smallest powers of f m ’shaving positive coefficients exhibit their changes as 0 , , − , − , − , − , − , − , . . . and for thosewith negative coefficients the pattern is 2 , , , , , , − , − , − , − , . . . .An universal feature of the negative coefficient f m ’s in the initial data is that their coefficientmodulo sign is determined by the positive coefficient f m ’s. For example, the absolute value of thecoefficients of f is same as that of f ; f ’s coefficients come from that of f up to sign andso forth. Hence coefficients of f m having negative coefficients are determined by f m − . In fact,this peculiar coefficient correlation also exists in the non-initial data f whose coefficients arecorrelated with that of f . Conjecture 6.1.
For a class of a cable knot of the figure eight K r = C ( r, ( ) ⊂ S , r > andodd having monic Alexander polynomial, the coefficient functions (cid:8) f m ( q ) ∈ Z [ q ± ] (cid:9) in F K r ( x, q ) can be classified into two (disjoint) subsets: one of them consists of elements having all positivecoefficients (cid:8) f + t ( q ) (cid:9) t ∈ I + and the other subset contains elements whose coefficients are all negative { f − w ( q ) } w ∈ I − . Furthermore, for every element in { f − w ( q ) } , its coefficients coincide with that of anelement in (cid:8) f + t ( q ) (cid:9) up to sign. Conjecture 6.2.
For the family of knots in Conjecture 6.1, every nonzero element in { f m ( q ) } consists of an odd number of terms and power of q increases(decreases) by one between every onsecutive terms of the element. Moreover, coefficients of an element f − v ( q ) ∈ { f − w ( q ) } agree withcoefficients of f + v − r ( q ) ∈ (cid:8) f + t ( q ) (cid:9) up to sign for all v ∈ I − . A The Definitions of the Operators
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