Quadratic integer programming and the slope conjecture
QQUADRATIC INTEGER PROGRAMMING AND THE SLOPECONJECTURE
STAVROS GAROUFALIDIS AND ROLAND VAN DER VEEN
Abstract.
The Slope Conjecture relates a quantum knot invariant, (the degree of thecolored Jones polynomial of a knot) with a classical one (boundary slopes of incompressiblesurfaces in the knot complement).The degree of the colored Jones polynomial can be computed by a suitable (almost tight)state sum and the solution of a corresponding quadratic integer programming problem. Weillustrate this principle for a 2-parameter family of 2-fusion knots. Combined with the resultsof Dunfield and the first author, this confirms the Slope Conjecture for the 2-fusion knotsof one sector.
Contents
1. Introduction 21.1. The Slope Conjecture 21.2. Boundary slopes 21.3. Jones slopes, state sums and quadratic integer programming 21.4. 2-fusion knots 41.5. Our results 42. The colored Jones polynomial of 2-fusion knots 62.1. A state sum for the colored Jones polynomial 62.2. The leading term of the building blocks 82.3. The leading term of the summand 93. Proof of Theorem 1.1 103.1. Case 1: m , m ≥ m ≤ , m ≥ m ≤ , m ≤ − k -seed links and k -fusion knots 155.1. Seeds and fusion 155.2. 1 and 2-fusion knots 165.3. The topology and geometry of the 2-fusion knots K ( m , m ) 17Acknowledgment 18Appendix A. Sample values of the colored Jones function of K ( m , m ) 19 Date : August 1, 2016.1991
Mathematics Classification.
Primary 57N10. Secondary 57M25.
Key words and phrases: knot, link, Jones polynomial, Jones slope, quasi-polynomial, pretzel knots, fusion,fusion number of a knot, polytopes, incompressible surfaces, slope, tropicalization, state sums, tight statesums, almost tight state sums, regular ideal octahedron, quadratic integer programming. a r X i v : . [ m a t h . G T ] A ug STAVROS GAROUFALIDIS AND ROLAND VAN DER VEEN
References 20 Introduction
The Slope Conjecture.
The Slope Conjecture of [Gar11b] relates a quantum knotinvariant, (the degree of the colored Jones polynomial of a knot) with a classical one (bound-ary slopes of incompressible surfaces in the knot complement). The aim of our paper is tocompute the degree of the colored Jones polynomial of a 2-parameter family of 2-fusion knotsusing methods of tropical geometry and quadratic integer programming, and combined withthe results of [DG12], to confirm the Slope Conjecture for a large class of 2-fusion knots.Although the results of our paper concern an identification of a classical and a quantumknot invariant they require no prior knowledge of knot theory nor familiarity with incom-pressible surfaces or the colored Jones polynomial of a knot or link. As a result, we will notrecall the definition of an incompressible surface of a 3-manifold with torus boundary, nordefinition of the
Jones polynomial J L ( q ) ∈ Z [ q ± / ] of a knot or link L in 3-space. Thesedefinitions may be found in several texts [Hat82, HO89] and [Jon87, Tur88, Tur94, Kau87],respectively. A stronger quantum invariant is the colored Jones polynomial J L,n ( q ) ∈ Z [ q ± / ],where n ∈ N , which is a linear combination of the Jones polynomial of a link and its parallels[KM91, Cor.2.15].To formulate the Slope Conjecture, let δ K ( n ) denote the q -degree of the colored Jonespolynomial J K,n ( q ). It is known that δ K : N −→ Q is a quadratic quasi-polynomial [Gar11a]for large enough n . In other words, for large enough n we have δ K ( n ) = c K, ( n ) n + c K, ( n ) n + c K, ( n )where c K,j : N −→ Q are periodic functions. The Slope Conjecture states that the finite setof values of 4 c K, is a subset of the set bs K of slopes of boundary incompressible surfaces inthe knot complement. The set of values of c K, is referred to as the Jones slopes of the knot K . In case c K, is constant, as often the case, it is called the Jones slope, abbreviated js K .At the time of writing no knots with more than one Jones slope are known to the authors.1.2. Boundary slopes.
In general there are infinitely many non-isotopic boundary incom-pressible surfaces in the complement of a knot K . However, the set bs K of their boundaryslopes is always a nonempty finite subset of Q ∪ {∞} [Hat82]. The set of boundary slopesis algorithmically computable for the case of Montesinos knots (by an algorithm of Hatcher-Oertel [HO89]; see also [Dun01]) and for the case of alternating knots (by Menasco [Men85])where incompressible surfaces can often be read from an alternating planar projection. The A -polynomial of a knot determines some boundary slopes [CCG + A -polynomial is difficult to compute, for instance it is unknown for the alternating Montesinosknot 9 [Cul09]. Other than this, it is unknown how to produce a single non-zero boundaryslope for a general knot, or for a family of them.1.3. Jones slopes, state sums and quadratic integer programming.
There are closerelations between linear programming, normal surfaces and their boundary slopes. It is lessknown that that the degree of the colored Jones polynomial is closely related to tropical
UADRATIC INTEGER PROGRAMMING AND THE SLOPE CONJECTURE 3 geometry and quadratic integer programming . The key to this relation is a state sum formulafor the colored Jones polynomial. State sum formulas although perhaps unappreciated,are abundant in quantum topology. A main point of [GL05b] is that state sums imply q -holonomicity. Our main point is that under some fortunate circumstances, state sums giveeffective formulas for their q -degree. To produce state sums in quantum topology, one mayuse (a) a planar projection of a knot and an R -matrix [Tur88, Tur94],(b) a shadow presentation of a knot and quantum 6 j -symbols and R -matrices [Tur92,Cos09, CT08],(c) a fusion presentation of a knot and quantum 6 j -symbols [Thu02, vdV09, GvdV12].All those state sum formulas are obtained by contractions of tensors and in the case of thecolored Jones polynomial, lead to an expression of the form:(1) J K,n ( q ) = (cid:88) k ∈ nP ∩ Z r S ( n, k )( q )where • n is a natural number, the color of the knot, • P is a rational convex polytope such that the lattice points k of nP are the admissiblestates of the state sum, • the summand S ( n, k ) is a product of weights of building blocks. The weight of abuilding block is a rational function of q / and its q -degree is a piece-wise quadraticfunction of ( n, k ).Let δ ( f ( q )) denote the q -degree of a rational function f ( q ) ∈ Q ( q / ). This is definedas follows: if f ( q ) = a ( q ) /b ( q ) where a ( q ) , b ( q ) ∈ Q [ q / ] with b ( q ) (cid:54) = 0, then δ ( f ( q )) = δ ( a ( q )) − δ ( b ( q )), with the understanding that when a ( q ) = 0, then δ ( a ( q )) = −∞ . It is easyto see that the q -degree of a rational function f ( q ) ∈ Q ( q / ) is well-defined and satisfies theelementary properties δ ( f ( q ) g ( q )) = δ ( f ( q )) + δ ( g ( q ))(2a) δ ( f ( q ) + g ( q )) ≤ max { δ ( f ( q )) , δ ( g ( q )) } (2b)The state sum (1) together with the above identities implies that the degree δ ( n, k ) of S ( n, k )( q ) is a piece-wise quadratic polynomial in ( n, k ). Moreover, if there is no cancellationin the leading term of Equation (1) (we will call such formulas tight ), it follows that the degree δ K ( n ) of the colored Jones polynomial J K,n ( q ) equals to ˆ δ ( n ) where(3) ˆ δ ( n ) = max { δ ( n, k ) | k ∈ nP ∩ Z r } Computing ˆ δ ( n ) is a problem in quadratic integer programming (in short, QIP) [LORW12,Onn10, DLHO +
09, KP00].The answer is given by a quadratic quasi-polynomial of n , whose coefficient of n is inde-pendent of n , for all but finitely many n . If we are interested in the quadratic part of ˆ δ ( n ),then we can use state sums for which the degree of the sum drops by the maximum degreeof the summand by at most a linear function of n . We will call such state sums almost tight . STAVROS GAROUFALIDIS AND ROLAND VAN DER VEEN
A related and simpler real optimization problem is the following(4) ˆ δ R ( n ) = max { δ ( n, x ) | x ∈ nP } Using a change of variables x = ny , it is easy to see that ˆ δ R ( n ) is a quadratic polynomial of n , for all but finitely many n .Thus, an almost tight state sum for the colored Jones polynomial a knot (of even more, ofa family of knots) allows us to compute the degree of their colored Jones polynomial usingQIP. Our main point is that it is easy to produce tight state sums using fusion, and in thecase they are almost tight, it is possible to analyze ties and cancellations. We illustrate inTheorem 1.1 below for the 2-parameter family of 2-fusion knots.1.4. 2 -fusion knots. Consider the 3-component seed link K as in Figure 1 and the knot K ( m , m ) obtained by ( − /m , − /m ) filling on K for two integers m , m . K ( m , m ) isthe 2-parameter family of 2-fusion knots. This terminology is explained in detail in Section 5. Figure 1.
Left: The seed link K and the 2-fusion knot K ( m , m ). As anexample K (2 ,
1) is the ( − , ,
7) pretzel knot.The 2-parameter family of 2-fusion knots includes the 2-strand torus knots, the ( − , , p )pretzel knots and some knots that appear in the work of Gordon-Wu related to exceptionalDehn surgery [GW08]. The non-Montesinos, non-alternating knot K ( − ,
3) = K was thefocus of [GL05a] regarding a numerical confirmation of the volume conjecture. The topologyand geometry of 2-fusion knots is explained in detail in Section 5.3.1.5. Our results.
Our main Theorem 1.1 gives an explicit formula for the Jones slope forall 2-fusion knots K ( m , m ). Recall that the Jones slope(s) js K of a knot K is the setof values of the periodic function c K, : N → Q that governs the leading order of the q -degree of J K,n ( q ). In our case set of Jones slopes is a singleton for each pair m , m so wedenote by js( m , m ) ∈ Q the unique element of the set of Jones slopes of K ( m , m ). Theformula for js is a piece-wise rational function of m , m defined on the lattice points Z ofthe plane, which are partitioned into five sectors shown in color-coded fashion in Figure 2.The reader may observe that the 5 branches of the function js : Z → Q do not agree whenextrapolated. For example for m < m = 0 the formula 2 m + from the red regiondoes not agree (when extrapolated) with the actual value 0 for the Jones slope at m = 0.This disagreement disappears when we study the corresponding real optimization problemin Section 4 below. The branches given there actually fit together continuously. UADRATIC INTEGER PROGRAMMING AND THE SLOPE CONJECTURE 5
Figure 2.
The formula for the Jones slope of K ( m , m ). Theorem 1.1.
For any m , m there is only one Jones slope. Moreover, if we dividethe ( m , m ) -plane into regions as shown in Figure 2 then the Jones slope js( m , m ) of K ( m , m ) is given by: (5)js( m , m ) = ( m − m + m − + m +9 m +34 if m ≥ , m ≥ m m + m +1) + m +9 m +34 if m ≤ , m ≥ − − m , m ≥ m + if < m , m < − − m if m ≤ , m ≤ − m , or ( m , m ) = (2 , − (2 m +3 m ) m + m − ) if m > − m , m ≤ − with js(1 ,
0) = 3 / . Combining the work of [DG12, Thm.1.9] we obtain a proof for the slope conjecture for alarge class of 2-fusion knots.
Corollary 1.2.
The slope conjecture is true for all 2-fusion knots K ( m , m ) with m > , m > STAVROS GAROUFALIDIS AND ROLAND VAN DER VEEN
We should remark that the incompressibility criterion of [DG12] can also be applied toprove the slope conjecture for the remaining 2-fusion knots. However, this is not the focusof the present paper, and we will not provide any further details on this separate matter.
Remark 1.3.
Using the involution(6) K ( m , m ) = − K (1 − m , − − m ) , K ( − , m ) = K ( − , − m )Theorem 1.1 computes the Jones slopes of the mirror of the family of 2-fusion knots. Hence,for every 2-fusion knot, we obtain two Jones slopes. Remark 1.4.
The proof of Theorem 1.1 also gives a formula for the degree of the coloredJones polynomial. This formula is valid for all n , and it is manifestly a quadratic quasi-polynomial. See Section 4. Remark 1.5.
Theorem 1.1 has a companion Theorem 4.2 which is the solution to a realquadratic optimization problem. Theorem 4.1 implies the existence of a function js R : R → R with the following properties:(a) js R is continuous and piece-wise rational, with corner locus (i.e., locus of pointswhere js R is not differentiable) given by quadratic equalities and inequalities whosecomplement divides the plane R into 9 sectors, shown in Figure 5.(b) js R is a real interpolation of js in the sense that it satisfiesjs R ( m , m ) = js( m , m )for all integers m , m except those of the form ( m ,
0) with m ≤ , − R (after multiplication by 4) becomes a boundary slopeof K ( m , m ) valid in the corresponding region, detected by the incompressibilitycriterion of [DG12, Sec.8].2. The colored Jones polynomial of 2-fusion knots
A state sum for the colored Jones polynomial.
The cut-and-paste axioms ofTQFT allow to compute the quantum invariants of knotted objects in terms of a few build-ing blocks, using a combinatorial presentation of the knotted objects. In our case, we areinterested in state sum formulas for the colored Jones function J K,n ( q ) of a knot K . Ofthe several state sum formulas available in the literature, we will use the fusion formulas that appear in [CFS95, Cos09, MV94, GvdV12, KL94, Tur88]. Fusion of knots are knottedtrivalent graphs. There are five building blocks of fusion (the functions µ, ν, U , Θ , Tet below),expressed in terms of quantum factorials. Recall the quantum integer [ n ] and the quantumfactorial [ n ]! of a natural number n are defined by[ n ] = q n/ − q − n/ q / − q − / , [ n ]! = n (cid:89) k =1 [ k ]!with the convention that [0]! = 1. Let (cid:20) aa , a , . . . , a r (cid:21) = [ a ]![ a ]! . . . [ a r ]! UADRATIC INTEGER PROGRAMMING AND THE SLOPE CONJECTURE 7 denote the multinomial coefficient of natural numbers a i such that a + . . . a r = a . We saythat a triple ( a, b, c ) of natural numbers is admissible if a + b + c is even and the triangleinequalities hold. In the formulas below, we use the following basic trivalent graphs U , Θ , Tetcolored by one, three and six natural numbers (one in each edge of the corresponding graph)such that the colors at every vertex form an admissible triple shown in Figure 2.1. c ae b fdabca
Let us define the following functions. µ ( a ) = ( − a q − a ( a +2)4 ν ( c, a, b ) = ( − a + b − c q a ( a +2)+ b ( b +2) − c ( c +2)8 U( a ) = ( − a [ a + 1]Θ( a, b, c ) = ( − a + b + c [ a + b + c (cid:20) a + b + c − a + b + c , a − b + c , a + b − c (cid:21) Tet( a, b, c, d, e, f ) = min S j (cid:88) k =max T i ( − k [ k + 1] (cid:20) kS − k, S − k, S − k, k − T , k − T , k − T , k − T (cid:21) where(7) S = 12 ( a + d + b + c ) S = 12 ( a + d + e + f ) S = 12 ( b + c + e + f )(8) T = 12 ( a + b + e ) T = 12 ( a + c + f ) T = 12 ( c + d + e ) T = 12 ( b + d + f ) . An assembly of the five building blocks can compute the colored Jones function of any knot.The next theorem is an exercise in fusion following word for word the proof of [GL05a,Thm.1]. An elementary and self-contained introduction to fusion is available in [GL05a,Sec.3.2]. In particular, the calculation of the colored Jones polynomial of the 2-fusion knot K ( − ,
3) (generalized verbatim to all 2-fusion knots) is given in [GL05a, Sec.3.3, p.390].Consider the function S ( m , m , n , k , k )( q ) = µ ( n ) − w ( m ,m ) U( n ) ν (2 k , n, n ) m +2 m ν ( n + 2 k , k , n ) m +1 (9) · U(2 k )U( n + 2 k )Θ( n, n, k )Θ( n, k , n + 2 k ) Tet( n, k , k , n, n, n + 2 k ) . Theorem 2.1.
For every m , m ∈ Z and n ∈ N , we have: (10) J K ( m ,m ) ,n ( q ) = (cid:88) ( k ,k ) ∈ nP ∩ Z S ( m , m , n , k , k )( q ) , STAVROS GAROUFALIDIS AND ROLAND VAN DER VEEN where P is the polytope from Figure 3 and the writhe of K ( m , m ) is given by w ( m , m ) =2 m + 6 m + 2 . Figure 3.
The polygon P on the left and its decomposition into three regions P , P , P on the right. Remark 2.2.
Notice that for every n ∈ N , we have: { ( k , k ) ∈ Z | ≤ k ≤ n, | n − k | ≤ n + 2 k ≤ n + 2 k } = nP ∩ Z . For the purpose of visualization, we show the lattice points in 4 P and 5 P in Figure 4. Figure 4.
The lattice points in 4 P and 5 P .2.2. The leading term of the building blocks.
In this section we compute the leadingterm of the five building blocks of our state sum.
Definition 2.3. If f ( q ) ∈ Q ( q / ) is a rational function, let δ ( f ) and lt( f ) the minimal degree and the leading coefficient of the Laurent expansion of f ( q ) ∈ Q (( q / )) with respectto q / . Let(11) (cid:98) f ( q ) = lt( f ) q δ ( f ) denote the leading term of f ( q ).We may call (cid:98) f ( q ) the tropicalization of f ( q ). Observe the trivial but useful identity:(12) (cid:99) f g = ˆ f ˆ g for nonzero functions f, g . UADRATIC INTEGER PROGRAMMING AND THE SLOPE CONJECTURE 9
Lemma 2.4.
For all admissible colorings we have:lt( µ )( a ) = ( − a lt( ν )( c, a, b ) = ( − a + b − c lt(U)( a ) = ( − a lt(Θ)( a, b, c ) = ( − a + b + c lt(Tet)( a, b, c, d, e, f ) = ( − k ∗ where k ∗ = min S j and δ ( µ )( a ) = − a ( a + 2)4 δ ( ν )( c, a, b ) = a ( a + 2) + b ( b + 2) − c ( c + 2)8 δ (U)( a ) = a δ (Θ)( a, b, c ) = −
18 ( a + b + c ) + 14 ( ab + ac + bc ) + 14 ( a + b + c ) δ (Tet)( a, b, c, d, e, f ) = δ ( b )( S − k ∗ , S − k ∗ , S − k ∗ , k ∗ − T , k ∗ − T , k ∗ − T , k ∗ − T ) + k ∗ S j and T i are given in Equations (7) and (8), b ( a , . . . , a ) = (cid:2) aa ,a ,...,a (cid:3) is the 7-binomial coefficient and δ ( b )( a , . . . , a ) = 14 (cid:32) (cid:88) i =1 a i (cid:33) − (cid:88) i =1 a i . Proof.
Use the fact that (cid:99) [ a ] = q a − and (cid:99) [ a ]! = q a − a This computes the leading term of Θ and of the quantum multinomial coefficients. NowTet( a, b, c, d, e, f ) is given by a 1-dimensional sum of a variable k . It is easy to see that theleading term comes the maximum value k ∗ of k . The result follows. (cid:3) The leading term of the summand.
Consider the function Q defined by Q ( m , m , n, k , k ) = k − k − k k − k − k m − k m − k m − k m − k n (13) − k n + 2 m n + 4 m n − k m n − n + m n + 2 m n + 12 (cid:0) (1 + 8 k + 4 k + 8 n ) min { l , l , l } − { l , l , l } (cid:1) where l = 2 k + n, l = 2 k + k + n, l = k + 2 n. Notice that for fixed m , m and n , the function k = ( k , k ) (cid:55)→ Q ( m , m , n, k ) is piece-wisequadratic function. Moreover, for all m , m and n the restriction of the above function toeach region of nP is a quadratic function of ( k , k ). Lemma 2.5.
For all ( m , m , n , k , k ) admissible, we haveˆ S ( m , m , n , k , k ) = ( − k + n +min { k , k + k ,k + n } q Q ( m ,m ,n , k ,k ) Proof.
It follows easily from Section 2.2 and Equation (12). (cid:3) Proof of Theorem 1.1
The proof involves four cases:Case 0 Case 1 Case 2 Case 3 m ∈ { , − } m , m ≥ m ≤ , m ≥ m ≤ − K ( m ,
0) = T (2 , m +1) and K ( m , −
1) = T (2 , m −
3) for which the Jones slopes were already known [Gar11b].In the remaining three cases we will take the following steps:(1) Estimate partial derivatives of Q in the various regions P i to narrow down the locationof the lattice points that achieve the maximum of Q on nP ∩ Z . In all cases theywill be on a single boundary line of Q .(2) Since the restriction of Q to a boundary line is an explicit quadratic function in onevariable, there can be at most 2 maximizers and we can readily compute them.(3) If there are two maximizers, compute the leading term of the corresponding summandto see if they cancel out.(4) If there is no cancellation, then we can evaluate Q ( m , m , n, k ) /n at either of themaximizers k to get the slope.(5) If there is cancellation we first have to explicitly take together all the canceling termsuntil no more cancellation occurs at the top degree. This happens only in the difficultCase 3.3.1. Case 1: m , m ≥ . Recall that Q i is Q restricted to the region nP i defined in Figure3. We have: ∂Q ∂k < ∂Q ∂k , ∂Q ∂k < ∂Q ∂k < . (14)Before we may conclude that the maximum of Q on nP ∩ Z is on the line k = − k wehave to check the following. For odd n there could be a jump across the line k = n betweenregions nP and nP . We therefore set n = 2 N + 1 explicitly check that Q ( m , m , N + 1 , N, − N ) − Q ( m , m , N + 1 , N + 1 , − N ) > k = − k , Q is a negative definite quadratic in k with critical point c = 1 − m + m + m n − m + m ) UADRATIC INTEGER PROGRAMMING AND THE SLOPE CONJECTURE 11
For m > c ∈ ( − , n ] and for m = 1 we have c = n +12 . In both cases themaximizers are the lattice points in the diagonal closest to c satisfying k ≤ n . Theremay be a tie for the maximum between two adjacent points. To rule out the possibility ofcancellation we take a look at the leading term restricted to the line k = − k . The leadingterm is ( − n . Since the sign of the leading term is independent of k along the diagonal,there cannot be cancellation. We may conclude that the slope is given by the constant termof Q ( m , m , n, c , − c ) /n . This gives the slope ( m − m + m − + m +9 m +14 indicated in the blueregion of Figure 2.3.2. Case 2: m ≤ , m ≥ . We have: ∂Q ∂k > , ∂Q ∂k < ∂Q ∂k < ∂Q ∂k < . (15)Before we may conclude that the maximum of Q on nP ∩ Z is on the line k = k − n wehave to check the following. For odd n there could be a jump across the line k = n betweenregions nP and nP . We therefore set n = 2 N + 1 explicitly check that Q ( m , m , N + 1 , N + 1 , − N ) − Q ( m , m , N + 1 , N, − N ) > k = k − n the coefficient of k in Q is a = − − m − m . If a > c is given by c = 1 − m + m + m n − m + m )Since c < n the maximizer is given by k = n and so the slope is: 2 m + as shown in redin Figure 2. If a = 0 we have the same conclusion because along the diagonal Q is now anincreasing linear function in k . Finally if a < c ∈ [ n − , n + ].We always have c > n − , and if in addition 1 + 2 m + m < c > n − /
2. Thismeans the maximizer is k = n and the slope is + 2 m as shown in red in Figure 2.If 1 + 2 m + m ≥ c ∈ [ n − , n + ] and the maximizers are the lattice points onthe line closest to c . There may of course be cancellation if there is a tie. To rule this outwe check that along the line the sign of the leading term is independent of k . Indeed theleading term on this line is ( − n .We may conclude that the slope is given by the constant term of Q ( m , m , n, c , c − n ) /n This gives the slope m m + m +1) + m +9 m +14 indicated in the purple region of Figure 2.3.3. Case 3: m ≤ , m ≤ − . One can check that: ∂Q ∂k > ∂Q ∂k > ∂Q ∂k > . (16)This means that the lattice maximizers of Q will be on the diagonal k = k . Here therestriction of Q is a quadratic and the coefficient of k is − m − m . If m ≤ − m thenit is positive definite with critical point given by c = − m + 2 m + 2 n + 2 m n − m − m ) We have c < k = n giving a slope of 0 as shown in yellowin Figure 2.If m > − m the quadratic Q is negative definite on the diagonal and the critical point c satisfies c > − . Furthermore c ≥ n − if and only if − m ≥ m and this case weget again the maximizer k = n and slope 0.The only remaining case is 2 m > − m , which means c ∈ ( − , n − ]. Here we have tocheck for cancellation and indeed, there will be cancellation along a subsequence since theleading term alternates along the diagonal, it is ( − k + n .To finish the proof we must rule out the possibility of a new slope occurring when thedegree drops dramatically due to cancellation. Below we will deal with the cancellationand show the drop in degree is at most linear in n so that no new slope can appear. Ourconclusion then is that the slope is given by the constant term of Q ( m , m , n, c , c ) /n which is: (2 m +3 m ) m + m − ) as shown in green in Figure 2.3.4. Analysis of the cancellation in Case 3.
Cancellation happens exactly when thecritical point on the diagonal is a half integer c ∈ + Z . Note also that not just the twoterms tying for the maximum cancel out. All the terms along the diagonal corresponding to k = c ± b +12 cancel out to some extent. Here b = 0 . . . min( c , n − c ) − .Along the diagonal the Tet consists of a single term so that the summand S simplifiesconsiderably, call it D : D ( k ) := S ( m , m , n, k, k ) = ( − (2 m +1) n/ n q − (2 m +1) n / [ n ]! × ( − k q − ( m + m ) k ( k +1) − (2 m +1) n (2 k +1) / [ n + 2 k + 1][2 k + 1]![ k ]![ n + k + 1]!To see how far the degree drops when taking together the canceling terms in pairs andtake together D ( k ) and D ( k − a ). For a ∈ N the result is: D ( k ) + D ( k − a ) = C (cid:18) q α { n + 2 k − a + 1 } { k } ! { n + k + 1 } ! { k − a } ! { n + k − a + 1 } !+ ( − s q β { n + 2 k + 1 } { k + 1 } ! { k − a + 1 } ! (cid:19) . Here C is an irrelevant common factor and in case of cancellation the monomials q α and (1) s q β are determined to make the leading terms of equal degree and opposite sign.Lastly we have taken out all denominators of the quantum numbers and factorials anddefine { k } = [ k ]( q − q − ).Since we assume the leading terms cancel we investigate the next degree term in both partsof the above formula. For this we can ignore C and the monomials and restrict ourselves tothe two products of terms of the form { x } . Both products can be simplified to remove thedenominator. The difference in degree between the two terms of { x } is exactly x . If { x } isthe least integer that occurs in the product then the difference in degree between the leadingterm and the highest subleading term is exactly x . For the first term x is k − a + 1 and forthe second term it is x = 2 k − a + 2. In conclusion the highest subleading term does notcancel out and has degree exactly k − a + 1 lower than the leading term. UADRATIC INTEGER PROGRAMMING AND THE SLOPE CONJECTURE 13
To finish the argument we would like to show that the b = 0 terms k = c ± still producethe highest degree term after cancellation. This is not obvious since the degree drops byexactly c − b + . In other words after cancellation the degree of the terms correspondingto b gains exactly b relative to the b = 0 terms. To settle this matter we show that thedifference in degree before cancellation was more than b . Q ( m , m , n, c + 12 , c + 12 ) − Q ( m , m , n, c − b − , c − b −
12 ) = b (1 + b )2 ( − m +2 m ) > b Because b ≥ m > − m so − m + 2 m > − − m ≥ b > min( c , n − c ) − that did not suffer any cancellation because their symmetric partner was outside of nP . We need to show that the difference in degree before cancellation is at least c + . Sofor b = min( c , n − c ) − check explicitly that b (1+ b )2 ( − m + 2 m ) > c + . This istrue provided that n > m .Finally we check that the degree of the b = 0 terms before cancellation is greater than c + plus the degree of any off-diagonal term. For this we only need to consider the terms( k , k ) = ( k , k − Real versus lattice quadratic optimization
Real quadratic optimization with parameters.
In this section we study the realquadratic optimization problem of Equation (4) and compare it with the lattice quadraticoptimization problem of Theorem 1.1.Fix a rational convex polytope P in R r and a piece-wise quadratic function δ in thevariables n, x where x = ( x , . . . , x r ). Then, we have:ˆ δ R ( n ) := max { δ ( n, x ) | x ∈ nP } = max { δ ( n, nx ) | x ∈ P } . Observe that δ ( n, nx ) is a quadratic polynomial in n with coefficients piece-wise quadraticpolynomial in x . it follows that for n large enough, ˆ δ R ( n ) is given by a quadratic polynomialin n . If js R denote the coefficient of n in ˆ δ R ( n ), and δ ( x ) denotes the coefficient of n in δ ( n, nx ) then we have: js R = max { δ ( x ) | x ∈ P } . If δ depends on some additional parameters m ∈ R r , then we get a function(17) js R : R r (cid:55)→ R . Assume that dependence of δ on m is polynomial with real coefficients. To compute js R ( m ),consider the piece-wise quadratic polynomial (in the x variable) δ ( m, x ), which achieves amaximum at some point of the compact set P . Subdividing P if necessary, we may assumethat δ ( m, x ) is a polynomial in x . If the maximum ˆ x is at the interior of P , since δ ( m, x )is quadratic, its gradient is an affine linear function of x , hence it has a unique zero. Inthat case, it follows that ˆ x is the unique critical point of δ ( m, x ) and δ ( m, x ) has negativedefinite quadratic part. Since the coefficients of the quadratic function δ ( m, x ) of x arepolynomials in m , it follows that in the above case the coefficients of ˆ x are rational functionsof m . The condition that ˆ x is a maximum point in the interior of P can be expressed bypolynomial equalities and inequalities on m . This defines a semi-algebraic set [BPR03]. On the other hand, if ˆ x lies in the boundary of P , then either ˆ x is a vertex of P or there existsa face F of P such that ˆ x lies in the relative interior of F . Restricting δ ( m, x ) and usinginduction on r , or evaluating at ˆ x a vertex of P implies the following. Theorem 4.1.
With the above assumptions, js R : R r (cid:55)→ R is a piece-wise rational functionof m , defined on finitely many sectors whose corner locus is a closed semi-algebraic set ofdimension at most r − . Moreover, js R is continuous. Recall that the corner locus of a piece-wise function on R r is the set of points where thefunction is not differentiable. Note that the proof of Theorem 4.1 is constructive, and easierthan the corresponding lattice optimization problem, since we do not have to worry aboutties. Moreover, since we are doing doing a sum, we do not have to worry about cancellations.4.2. The case of -fusion knots. We now illustrate Theorem 4.1 for the case of 2-fusionknots, where δ ( m , m , n, x , x ) is given by Equation (13). Notice that δ ( m, n, x ) is an affinelinear function of m = ( m , m ) ∈ R . A case analysis (similar but easier than the one ofSection 3 shows the following. Figure 5.
The nine regions of js R of Theorem 4.2.Define js R ( m , m ) to be the real maximum of the summand for the fusion state sum of K ( m , m ). Theorem 4.2.
If we divide the ( m , m ) -plane into regions as shown in Figure 5 then js R ( m , m ) is given by: js R ( m , m ) = UADRATIC INTEGER PROGRAMMING AND THE SLOPE CONJECTURE 15 (18) ( m − m + m − + m +9 m +34 if m > , m ≥ m +9 m +34 if ≤ m ≤ , m + 3 m ≥ , − m + m ≥ m m + m +1) + m +9 m +34 if m ≤ , m ≥ , m ≥ − − m m + if m > , m + m ≥ (3 m +1) m + ) if − ≤ m ≤ , m + 3 m + 4 m m ≤ if m ≤ − , m + 3 m ≤ , m + 4 m ≤ , m ≤ − m m +3 m ) m + m − ) if m > − m , m ≤ − m + 2 m + if − ≤ m ≤ , − m + m ≤ , m + 4 m ≥ I ( m , m ) if m + 3 m + 4 m m ≥ , − ≤ m ≤ , − ≤ m ≤ where I ( m , m ) = 3 + 6 m + 4 m + 18 m + 24 m m + 8 m m + 27 m + 18 m m m + 3 m + 2 m m ) Corollary 4.3.
An comparison between Theorems 1.1 and 4.2 reveals that js( m , m ) =js R ( m , m ) for all pairs of integers ( m , m ) ∈ Z except those of the form ( m ,
0) with m ≤ , − K ( m , m ) is a torus knot.5. k -seed links and k -fusion knots Seeds and fusion.
There are several ways to tabulate and classify knots, and amongthem(a) by crossing number as was done by Rolfsen [Rol90],(b) by the number of ideal tetrahedra (for hyperbolic knots) as is the standard in hyper-bolic geometry [Thu77, CDW],(c) by arborescent planar projections, studied by Conway and Bonahon-Siebenmann [Cos09,BS11],(d) by fusion [Thu02],(e) by shadows [Tur92].Here we review the fusion construction of knots (and more generally, knotted trivalentgraphs) which originates from cut and paste axioms in quantum topology. The constructionwas introduced by Bar-Natan and Thurston, appeared in [Thu02] and further studied by thesecond author [vdV09]. Our definition of fusion is reminiscent to W. Thurston’s hyperbolicDehn filling [Thu77], and differs from a construction of knots by the same name (fusion)that appears in Kawauchi’s book [Kaw96, p.171].
Definition 5.1. A seed link is a link that can be produced from the theta graph by applyingthe moves A, U, X shown in figure 6. The additional components created by U and X arecalled belts . A k -seed link is a seed link with k belts.Note that the sign of the crossing introduced by the X -move is does not affect the com-plement of the seed link. If desired we may always perform all the A moves first. Figure 6.
The moves A , U , X and the theta graph (upper right). Definition 5.2.
Let L be a k -seed link together with an ordering of its belts. Define the k -fusion link L ( m , . . . , m k ) to be the link obtained by − m j Dehn filling on the j -th belt of L for all j = 1 , . . . , k .Recall that the result of − /m Dehn filling along an unknot C which bounds a disk D replaces a string that meets D with m full twists, right-handed if m > m <
0; see Figure 7 and [Kir78].
Figure 7.
The effect of Dehn filling on a link. In the picture we have taken m = 2.In a picture of a seed link the belts will always be enumerated from bottom to top. So forexample the first belt of K is the smallest one.As suggested above, fusion is not just a way to produce a special class of knots. All knotsand links can be presented this way although not in a unique way. Theorem 5.3.
Any link is a k -fusion link for some k . The number of fusions is at most thenumber of twist regions of a diagram. This theorem has its roots in Turaev’s theory of shadows. A self-contained proof can befound in [vdV09].5.2. 1 and -fusion knots. We now specialize the discussion of k -fusion knots to the case k = 1 ,
2. Figure 8 lists the sets of 1-seed and 2-seed links. Since we are interested in knots,let S k denote the finite set of seed links with k belts and k + 1 components. Lemma 5.4.
Up to mirror image, we have S = { T } , S = { K , K , K } where T, K i , K are the links shown in Figure 8. UADRATIC INTEGER PROGRAMMING AND THE SLOPE CONJECTURE 17
Figure 8.
The seed links T = L a = T (2 ,
4) torus link, K = L a = t K = L n
84 = 10 = t K = L n = t Proof.
The seed link T is obtained from the theta graph by a single X move. The links K and K are obtained by first doing an A move to get a tetrahedron graph and thenapplying two U (cid:48) s or a U and an X on a pair of disjoint edges. Finally K is obtained fromthe tetrahedron by doing one X move and then a U move on one of the edges newly createdby the X . One checks that all other sequences with at most one A move either give linkswith homeomorphic complement or links including two components that are not belts. (cid:3) T ( m ) is the well-understood torus knot T (2 , m + 1). Observe that K is the seed link ofthe fusion knots K ( m , m ). K ( m , m ) and K ( m , m ) are alternating double-twist knots(with an even or odd number of half-twists) that appear in [HS04]. The Slope Conjecture isknown for alternating knots [Gar11b]. In particular, the Jones slopes are integers.The next lemma which can be proved using [CDW] summarizes the hyperbolic geometryof the seed links K and K . Lemma 5.5.
Each of the links K and K is obtained by face-pairings of two regular idealoctahedra. K and K are scissors congruent with volume 7 . . . . , commensurablewith a common 4-fold cover, and have a common orbifold quotient, the Picard orbifod H / PSL(2 , Z [ i ]).5.3. The topology and geometry of the 2-fusion knots K ( m , m ) . In this section wesummarize what is known about the topology and geometry of 2-fusion knots. The sectionis independent of the results of our paper, and we include it for completeness.The 2-parameter family of 2-fusion knots specializes to • The 2-strand torus knots by K ( m ,
0) = T (2 , m + 1). • The non-alternating pretzel knots by K ( m ,
1) = ( − , , m + 3) pretzel. In partic-ular, we have: K (2 ,
1) = ( − , , K (1 ,
1) = ( − , ,
5) = 10 K (0 ,
1) = ( − , ,
3) = 8 K ( − ,
1) = ( − , ,
1) = 5 K ( − ,
1) = ( − , , −
1) = 5 K ( − ,
1) = ( − , , −
3) = 8 . • Gordon’s knots that appear in exceptional Dehn surgery [GW08]. More precisely, if L GW and L GW denote the two 2-component links that appear in [GW08, Fig.24.1],then L GW ( n ) = K ( − , n ). These two families intersect at the ( − , ,
7) pretzel knot;see also [EM97, Fig.26]. Moreover, the knot K ( − ,
3) = K (following the notationof the census [CDW]) was the focus of [GL05a].We thank Cameron Gordon for pointing out to us these specializations.The next lemma summarizes some topological properties of the family K ( m , m ). Lemma 5.6. (a) K ( m , m ) is the closure of the 3-string braid β m ,m , where β m ,m = ba m +1 ( ab ) m where s = a, s = b are the standard generators of the braid group.(b) K ( m , m ) is a twisted torus knot obtained from the torus knot T (3 , m +1) by applying m full twists on two strings.(c) K ( m , m ) is a tunnel number 1 knot, hence it is strongly invertible. See [Lee11] andalso [MSY96, Fact 1.2].(d) We have involutions(19) K ( m , m ) = − K (1 − m , − − m ) , K ( − , m ) = K ( − , − m )(e) K ( m , m ) is hyperbolic when m (cid:54) = 0 , m (cid:54) = 0 , − K ( m , m ) are not always Montesinos, noralternating, nor adequate. So, it is a bit of a surprise that one can compute some boundaryslopes using the incompressibility criterion of [DG12] (this can be done for all integer values of m , m ), and even more, that we can compute the Jones slope in Theorem 1.1 and verify theSlope Conjecture. Thus, our methods apply beyond the class of Montesinos or alternatingknots. Remark 5.7. K ( m , m ) is not always a Montesinos knot. Indeed, recall that the 2-foldbranched cover of a Montesinos knot is a Seifert manifold [Mon73], in particular not hy-perbolic. On the other hand, SnapPy [CDW] confirms that the 2-fold branched cover of K ( − , −
3) (which appears in [GL05a]) is a hyperbolic manifold, obtained by ( − ,
3) fillingof the sister m003 of the 4 knot. Acknowledgment.
S.G. was supported in part by NSF. R.V. was supported by the Nether-lands Organization for Scientific Research. An early version of a manuscript by the firstauthor was presented in the Hahei Conference in New Zealand, January 2010. The first au-thor wishes to thank Vaughan Jones for his kind invitation and hospitality and Marc Culler,Nathan Dunfield and Cameron Gordon for many enlightening conversations.
UADRATIC INTEGER PROGRAMMING AND THE SLOPE CONJECTURE 19
Appendix A. Sample values of the colored Jones function of K ( m , m )In this section we give some sample values of the colored Jones function J K ( m ,m ) ,n ( q )which were computed using Theorem 2.1 after a global change of q to 1 /q . These valuesagree with independent calculations of the colored Jones function using the ColouredJones function of the
KnotAtlas program of [BN05], confirming the consistency of our formulaswith KnotAtlas. This is a highly non-trivial check since KnotAtlas and Theorem 2.1 arecompletely different formulas of the same colored Jones polynomial. Here, J K,n ( q ) is nor-malized to be 1 for the unknot (and all n ) and J K, ( q ) is the usual Jones polynomial of K . n J K (2 , ,n ( q )0 11 q + q − q + q − q q + q + q − q + q − q − q + q − q + q − q − q + 2 q − q − q + 3 q − q − q + 2 q q + q + q − q + q − q − q + q − q + q + q − q + q + q − q − q + q + q − q − q + q + 2 q − q − q + 2 q + 3 q − q − q + 3 q + 3 q − q − q + q + 3 q − q − q q + q + q − q + q − q + q − q − q + 2 q − q − q − q + 2 q − q + q − q − q + 2 q − q + q − q + 3 q − q − q + 3 q − q − q + 2 q + q + q − q − q + 2 q + 2 q − q − q + 2 q + 2 q + 2 q − q − q + 2 q +2 q + 3 q − q − q + 3 q + 2 q + 4 q − q − q + 4 q + 2 q + 4 q − q − q + 2 q + q + 4 q − q − q + q + q n J K (1 , ,n ( q )0 11 q + q − q q + q + q − q + q − q + q − q + q − q + q − q + q − q − q + q − q − q + q − q + q − q + q − q + q − q + q − q + q − q + q q + q + q − q + q − q + q − q + q − q + q − q + q − q − q + q − q − q + q − q + q − q + q − q + q − q + q − q + q − q + q q + q + q − q + q − q + q − q + q − q + q − q + q − q + q − q − q − q + q − q + q − q + q − q + q − q + q − q + q − q + q − q + q − q + q − q + q − q + q − q + q + q − q + q − q + q − q + q − q − q + q − q + q − q + q − q + q n J K ( − , ,n ( q )0 11 q + q − q + q − q + q − q q + q + q − q + q − q + q − q + 2 q − q − q + 3 q − q − q + 2 q − q − q + q − q + q − q + q q + q + q − q + q − q + q − q − q + q − q + q − q + 2 q − q − q − q + 2 q − q + q − q − q + 2 q + 3 q − q − q + 2 q + 6 q − q − q + q + 6 q + q − q − q + 7 q + 2 q − q − q + 7 q + 2 q − q − q + 7 q +3 q − q − q + 3 q + 2 q − q − q + q − q q + q + q − q + q − q + q − q − q + q − q + q + q + q − q + q − q + q − q + 2 q − q + 2 q − q + 2 q − q − q + 3 q − q + 4 q − q − q + q − q + 5 q + q − q − q + 3 q + q + 2 q − q + 2 q − q − q + q + q + 6 q + q − q − q + 2 q + 11 q + 6 q − q − q − q +13 q + 11 q − q − q − q + 15 q + 14 q − q − q − q +15 q + 15 q − q − q − q + 15 q + 15 q − q − q − q +14 q + 15 q − q − q − q + 7 q + 9 q − q − q + q + 3 q + q − q − q + q References [Ago00] Ian Agol,
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