arXiv:2105.13897v1 [math.GT] 28 May 2021 eeecsteen swl s[] aeou[]cntutdaver a constructed knots [9] an 3-bridge Kanenobu [3] of mutation [8]. family Conway as nite well that polynomial, as known Jones therein, identical is with references It knots generate crossings. generalisations 22 ous to up evidence oniecssol ecniee oeterl hnteexception the w than unclear rule still the is It more considered [10]. be in are should examples coincidences Further polynomial. Jones same on only h oevral)Jnsplnma a endi 6 sa min is ambient an as [6] in in defined links was of invariant polynomial Jones (one-variable) The Introduction 1 2020: tang Classification rational Subject knots, Mathematics rational polynomial, The se Jones essentially ra us. an Keywords: gives a to and thesis of known Ph.D. formulae account. polynomial author’s ra other second Jones associated the than the the on based terms of for expansion fewer formula fraction significantly new continued has a a on give based poly we knot HOMFLYPT the process checking the by c verified These In is coincidences. as mutants, rational not Jones determinant generically all with generate knots to sufficient rational (based are conjecture all and from ratio polynomial evidence associated Jones tal the the change of not expansion do coincidences fraction prove rational continued ‘Jones the call on we moves what knots, rational amongst Abstract: htis that ai rprisrmi nnw.Freape ti tl nnw whe unknown still is it that example, For holds, conjecture unknown. Jones’ remain properties basic etdwt ayitrsigohrmteaia tutrs(Te structures group mathematical quantum other interesting algebra, many with nected ∗ isenIsiueo ahmtc,Hbe nvriyo Je of University Hebrew Mathematics, of Institute Einstein V p K + ≡ eivsiaecicdne fte(n-aibe oe polynomial Jones (one-variable) the of coincidences investigate We q hti,a nnt aiyo nnt aiisec hrn the sharing each families infinite of family infinite an is, that ; o knot a for 1 uhLawrence Ruth oe ainlcoincidences rational Jones S 3 lhuhi a usqetybe on ob con- be to found been subsequently has it Although . nmmr fVuhnJones Vaughan of memory In U q K sl K 2 V nyif only oooia unu edter t.,many etc.), theory field quantum topological , a 1 2021 31, May p,q K itnuse h nntfo te knots, other from unknot the distinguishes ( ∗ ,q p, n r Rosenstein Ori and K 1 ∈ steuko;se[8 o numerical for [18] see unknot; the is 57K10 Z hs oe oyoildepends polynomial Jones whose ) rusalem < 0)ta hs moves these that 900) les ∗ icdne are oincidences nexperimen- on .W provide We ’. pca infi- special y mperley-Leib a hc we which nal inl which tional, lf-contained e 1]and [15] see . ae is paper nomial. vari- d hether tional otopy ther In this paper we concentrate on coincidences amongst Jones’ polynomials of rational knots. Since the determinant of a knot is a special value of the Jones polynomial and there are only a finite number of rational knots with given determinant, the number of coincidences is limited and can be checked by direct computation. In sections 2,3,4 we give an elementary and self-contained account of the evaluation of the Jones polynomial of rational knots as a matrix product (The- orems 4.2 and 4.6) via the Kauffman bracket vector of (2, 2)-tangles. Similar matrix formulae were used in [4] and [11] for computer computations of the Jones polynomial and other polynomial invariants of rational knots. By a careful analysis of cancellations of terms in the evaluation of the matrix product, we obtain in section 5 a seemingly new combinatorial formula (Theorem 5.4) for the Jones polynomial of rational knots in terms of a continued fraction expansion, with significantly fewer terms than other formulae we found in the literature, such as in [4]. By analyzing properties of products of the particular family of matrices ap- pearing, we prove in section 6 the invariance of the Jones polynomial of rational knots under a family of moves, Templates I, II (Theorems 6.5, 6.8). A third move, a generalization of Template I, is introduced in section 7 (Theorem 7.10) using the notion of pivoting pairs defined there. On the way a new structural property of Jones polynomials of rational knots is found (Corollary 7.5). In [17] it was found that out of 80,317 rational knots with determinant < 900, there are 223 Jones rational coincidences, three of which are triplets; these are listed in §8. All of these coincidences were accounted for via the moves proved in sections 6,7 and we conjecture that they are indeed generating moves for Jones rational coincidences.

2 Kauffman bracket

Let A be an indeterminate and R = Z[A, A−1] be the ring of polynomials in A, A−1 with integer coefficients. Set d = A2 A−2 R, u = iA2 and t = A−4. − − ∈ − The usual Kauffman bracket algorithm [14] to calculate the Jones polynomial of an oriented link L in S3 works as follows. Choose a link diagram D of L and evaluate it (as an unoriented diagram) by replacing every crossing by a combination of diagrams in which the crossing has been smoothed in the two possible ways ∼> A +A−1 (1) A diagram with n crossings is thus replaced by a combination of 2n diagrams, each of which has no crossings, that is, all resulting diagrams are isotopic to a disjoint collection of loops; the coefficients are powers of A. Evaluating a diagram which is a disjoint collection of k loops to dk−1, the combined evaluation of D is its Kauffman bracket, written D R Z[A, A−1]. h i ∈ ≡

2 Equivalently, one can consider the appropriate skein modules. Namely in the R-module generated by all link diagrams, one imposes the skein relation −1 identifying the two sides of (1) considered as a local move D AD∞ + A D0 and identifies any disjoint union with an unknot, D U with∼ d D where U is the diagram of the unknot consisting of a single loop. The quotient· is one- dimensional, any link diagram will be equivalent to` some multiple of U, this multiple being the Kauffman bracket. This is a regular isotopy invariant (in- variant under Reidemeister moves II, III) and its renormalisation

3 −w(D) VL = ( A ) D (2) − h i by the writhe w(D) is an ambient isotopy invariant, the (one-variable) Jones −1 −4 polynomial VL of L [6]. For knots, VL is a polynomial in t,t where t = A . Recall that the writhe of a diagram is defined as the sum over all crossings of the the sign of the crossing (determined by the relative orientations of the segments on D involved in the crossing); we use the sign convention shown below. + −

3 Calculus of (2, 2)-tangles

Let denote the monoid of framed unoriented (2, 2)-tangles considered up to ambientT isotopy equivalence, under vertical composition, or equivalently (using blackboard framing) of diagrams of (2, 2)-tangles up to Reidemeister moves II and III. The identity is the tangle T∞ (see below). Then R defines an associative unital algebra over R, whose elements are finite linear combinationsT of (2, 2)-tangles with coefficients in R. Call a tangle loopless if every component contains two boundary points, that is, it is an embedding of the disjoint union of two intervals in R2 [0, 1]. Tangles can be manipulated× in various ways to produce new tangles. Ele- ments of can be rotated in the plane of the diagram (T becomes T~ under T π ′ clockwise rotation through 2 ) as well as combined side-by-side (T and T com- bine to give T +T ′). They can also be closed horizontally or vertically to obtain (unoriented) link diagrams; we denote these by T N (numerator closure) and T D (denominator closure) respectively.

T 0

T TT 0 T T

T T~ T D T + T 0 T ∗ T 0 T N

3 −→ These operations are related, for example (T T ′)= T~ + T~′ while (T~)N = T D. The operation + also makes into a monoid∗ (a different structure from ); the T ∗ identity here is T0,

T1 = T0 =

Tangle type. Any (2, 2)-tangle T provides a subdivision of the four boundary points into two pairs, via the connections it provides (ignoring which paths pass over or under). There are three possible subdivisions of four points into two pairs and we define the tangle type of T accordingly, to be 0, 1 or as in the diagram below. We denote tangle type by ι(T ) 0, 1, . ∞ ∈{ ∞}

Type 0 Type 1 Type 1

Under rotation by π , types 0 and interchange while type 1 stays fixed, so 2 ∞ we write ι(T~) = ι(T )−1. Under the tangle operation +, the tangle types add, where addition is modulo 2 enhanced by the rule x + = + x = for all x, ∞ ∞ ∞ ι(T + T ′)= ι(T )+ ι(T ′)

The numerator closure T N of a loopless tangle T will be a knot so long as ι(T ) = 0. 6 Writhe vector. The definition of writhe uses an orientation on the link dia- gram. In the case of a knot, the writhe is independent of the choice of orientation while this is not true for links. For any loopless (2, 2)-tangle T , the numerator closure T N is a knot when ι(T ) = 0 and is a 2-component link otherwise, while the denominator closure T D is a6 knot when ι(T ) = and is a 2-component link otherwise. Define T N , T D as oriented link diagrams6 ∞ by placing orientations as indicated, according to the tangle type of T .

T N

T D

Type 0 Type 1 Type 1

w(T N ) Define the writhe vector of T to be w(T )= . The operations of closure w(T D) introduce no additional crossings and the orientations  have been chosen so that

4 in every case contributions to w(T N ) and w(T D) from crossings involving both ‘strands’ of T will be opposite. Under the operations ~ and + we have 0 1 w(T~)= w(T ) (3) 1 0   while T + T ′ is loopless assuming that T,T ′ are loopless and not both of tangle type , ∞ w(T )+ w(T ′) if ι(T ) = ι(T ′) 6 ∞ w(T + T ′)= 0 1 0 1 (4)  w(T )+ w(T ′) if ι(T )=  1 0 0 1 ∞     This is obtained by verifying each possible combination of types; see the figure below where, for compactness we omit the closing arcs (where no crossings occur) and only include relevant orientations on T + T ′ considered as a subdiagram of (T + T ′)N or (T + T ′)D.

(T + T 0)N

(T + T 0)D

ι T ι T ι T 1 ι T 1 ι(T ) = 0 ι(T ) = 0 ( ) = 1 ( ) = 1 ( ) = ( ) = 0 0 0 0 0 0 ι T ι T ι T ι T ι(T ) = 0 ι(T ) = 1 ( ) = 0 ( ) = 1 ( ) = 0 ( ) = 1

Rational tangles. A tangle which is obtained as a sum of copies of the tangle

T1 =

π or of its rotation (denoted T−1) through 2 , is called a horizontal twist box Tn, n Z. A tangle which can be obtained by alternating use of the operations ∈ +Tn and ~ from the tangle T0 is called a rational tangle; let denote the set of rational tangles. By [3] (see also [12]), a rational tangle isRT determined up to isotopy equivalence by the corresponding rational number; equivalently there is a well-defined bijection r : Q for which RT → ∪ {∞} r(T + T ′)= r(T )+ r(T ′), r(T~)= r(T )−1 while r(T )=1 − 1 For example, r(T ) = 0 and r(T ) = ; and so it is consistent to denote by 0 ∞ ∞ Tr the rational tangle with fraction r for arbitrary r Q . The fact that ∈ ∪ {∞} the tangle type of T1 is 1 combined with the transformation properties above for tangle type under ~ and + show that the tangle type of Tr is the ‘modulo p 2 evaluation of r’, that is, if r = q where p, q have no common factors then 0 if p is even ι(T )= 1 if p, q are both odd r  if q is even  ∞  5 In particular, for r N, Tr = T + + T (r times) will be a horizontal twist ∈ 1 ··· 1 box with r twists while T−r is a horizontal twist box with r twists of opposite orientation (which we will call a horizontal twist box with r twists). For 1 1 −1 1 r Q with continued fraction expansion r = nk where ∈ − nk−1− nk−2− ··· n1 n ,...,nk Z, the rational tangle Tr can be obtained by k iterations, starting 1 ∈ from the twist box Tn1 , rotating, adding the twist box Tn2 , rotating and so on, up until the last addition of the twist box Tnk . This rational tangle will also be denoted R(nk,nk−1,...,n1). Remark 3.1. Note that it is more conventional to use the operation of reflection in a diagonal, as the second operation in the construction of rational tangles, π rather than rotation through 2 . This is the cause of the minus signs in the continued fraction expansions that we obtain.

4 2

−2

−4 1

49 1 1 1 1 T49 = R(2; 4; 1; −4; −2); = 2 − − −− −− 29 29 4 1 4 2

Note that the continued fraction expansion of a rational is far from unique. For example m 1 1 1 = (m + n) 1 while x 1 1 1 1 = x. − 0− n− x − x − 0− −2− −1− −2 p Lemma 3.2. Any rational in reduced form q with p odd and q even can be written as a continued fraction of even integers and of even length, that is 1 1 1 r = nk − nk nk ··· n −1− −2− 1 where k is even and n ,...,nk 2Z. This expression is unique when the further 1 ∈ condition that n ,...,nk =0 is imposed. 1 −1 6 p Proof. Suppose r = q Q has odd numerator and even denominator. It is certainly not an odd integer∈ and therefore there exists a unique closest even integer, say 2m. Then 2m r ( 1, 1) is a rational with odd numerator (with − ∈ − 1 absolute value less than q ) and even denominator (still q). Put s = 2m−r ; this is a rational with even| | numerator q and odd denominator (with absolute value less than q ) whose absolute value is strictly greater than 1 and so it cannot be an odd| | integer. Let 2n = 0 be the unique closest even integer to s so that 2n s ( 1, 1) is a rational6 (possibly zero) with even numerator and odd − ∈ − 1To fix notations, this continued fraction is the rational obtained from zero by alternately applying the operations x 7→ ni − x, i = 1,...,k and reciprocation; that is, starting from zero, subtract from n1, reciprocate, subtract from n2, reciprocate, and so on, until the final operation of subtraction from nk.

6 1 denominator (both of absolute value less than q ). If2n s = 0 then r =2m 2n 1| | − 1 1 − and we are done. If not, then setting x = 2n−s we have r =2m 2n− x where x is a rational, x > 1 with odd numerator and even denominator− (both less than q ) to which| | we can apply the same argument again with x replacing r. This iterative| | process is finite since the denominators q are of strictly decreasing absolute value. Uniqueness follows from the fact that a continued fraction with all non-zero even entries must have absolute value strictly greater than 1.

Kauffman bracket for tangles. If one starts with a framed (2, 2)-tangle and then chooses a planar representation (using blackboard framing), one can perform the same replacement (1) of crossings by a combination of smoothings as in the Kauffman bracket description of the previous section. The result will be an R-combination of (2, 2)-tangle diagrams without crossings, each of which must be a disjoint union of some number of loops and one of the two basic (2, 2)-tangles T , T . Identifying D U d D results in an evaluation of D 0 ∞ ∼ · as an R-linear combination of T0, T∞, which is known as the Kauffman bracket for (2, 2)-tangles, now a vector-valued` regular isotopy invariant, see [1]. Equivalently, consider R as an R-module and its submodule T I = D AD A−1D ,D U d D − ∞ − 0 − · D E where D runs over all (2, 2)-tangle diagrams witha local smoothing moves as in the last section. The quotient is the relevant skein module, which is two- dimensional, generated by T0, T∞, and the map R R /I = R2 T −→ T ∼ is the Kauffman bracket of (2, 2)-tangles, again denoted . Thus for a (2, 2)- f h·i tangle T , its Kauffman bracket T = where f,g R are such that T h i g ∈ ∼   fT0 + gT∞. By applying the corresponding operations to the equivalent tangle, it follows immediately that T~ fT + gT , T N (df + g)U and T D (f + dg)U so that ∼ ∞ 0 ∼ ∼ g T~ = , T N = d f + g, T D = f + d g (5) h i f h i · h i ·   f ′ while when combined with another tangle T ′ for which T ′ = , h i g′   dff ′ + fg′ + gf ′ ff ′ T T ′ = , T + T ′ = (6) h ∗ i gg′ h i fg′ + f ′g + dgg′    

Lemma 3.3. The Kauffman bracket vector of rational tangle diagrams is given by

P n −1 s k 0 1 1 R(nk,nk ,...,n ) = (iA ) s ( i) Bn ... Bn h −1 1 i − 1 0 k 1 0    

7 −n 2 un−u−n [n] iu where n ,...,nk Z, u = iA , [n]= −1 and Bn . 1 ∈ − u−u ≡ iun 0   Remark 3.4. In the cases u = 1, [n] must be replaced by its limiting values, n for u = 1 and ( 1)n−1n for u =± 1. − − A Proof. By (1), T = and so by (6), h 1i A−1   Af A 0 u 0 T + T = = T =iA−1 T h 1i A−1f +Ag+dA−1g A−1 A−3 h i i u−1 h i    −   −  Applying this n times gives

n n −1 n u 0 −1 n u 0 T + Tn = (iA ) T = (iA ) T h i i u−1 h i i[n] u−n h i  −   −  1 0 Meanwhile by (5), T~ = T . Since T is the identity with respect to h i 0 1 h i 0   + and is obtained by rotating T∞, thus R(nk,nk−1,...,n1) may be obtained by starting with T∞ and making k iterations, each step being first a rotation 0 and then adding Tn, where in turn n = n ,...,nk. By definition T = 1 h ∞i 1 and so it follows that  

R(nk,nk ,...,n ) h −1 1 i nk n −1 n1+···+nk u 0 0 1 u1 0 0 1 0 = (iA ) −nk −n1 i[nk] u 1 0 ··· i[n1] u 1 0 1  −    −    1 P ns ns −1 s 0 1 0 1 u 0 1 = (iA ) −ns 1 0 1 0 i[ns] u 0 s k !   Y=    −    The term inside the product can be identified as iBn and the lemma follows. − s

4 The Jones polynomial of a rational knot

A rational knot is a knot which can be expressed as the closure of a rational N tangle. We will use numerator closure and write Kr (Tr) to denote the ≡ numerator closure of the rational tangle Tr for r Q . This will be a knot ∈ ∪{∞} so long as ι(Tr) = 0, that is, so long as r has odd numerator. 6

Theorem 4.1 (Schubert’s Theorem [19], [13]). Two rational knots K p and K p′ q q′ p p′ ′ corresponding to reduced fractions q and q′ (with p,p > 0) are ambient isotopic if, and only if, p = p′ and q q′±1 (mod p). ≡

8 To calculate the Jones polynomial of Kr, pick a continued fraction expansion of r, say p 1 1 1 r = = nk q − nk nk ··· n −1− −2− 1 where n1,...,nk Z, so that Tr R(nk,...,n1) defines a tangle diagram for ∈ ∼ N Tr. The Kauffman bracket of the knot diagram R(nk,...,n1) of Kr is obtained by combining Lemma 3.3 with T N = d 1 T from (5), to give h i h i P n N −1 s
k 1 R(nk,nk ,...,n ) = (iA ) s ( i) 1 d Bn ... Bn (7) h −1 1 i − k 1 0  
By (2), the Jones polynomial VL(Kr) is obtained from this Kauffman bracket evaluation by normalising using the writhe of the knot diagram, namely the first component of w(R(nk,...,n1)). This writhe vector can be computed iter- atively from (3) and (4) starting from the writhe vector of a horizontal twist box n w(T )= . An explicit formula is given in [17] but we will not need it here. n n −  Writing everything as a function of u, noting that d = A2 A−2 = i(u−1 u) we get the following lemma. − − −

Theorem 4.2. The Jones polynomial of the knot obtained by numerator closure of the rational tangle R(nk,...,n1) is

1 N 1 k 3 N 1 w + P ns− − w − P ns 1 VK (t) = ( 1) 4 4 s 2 u 2 2 s 1 d Bn ... Bn r − k 1 0  
where wN is the writhe of the numerator closure of the knot diagram while −n −2 −1 un−u−n [n] iu t = u , d = i(u u), [n] = −1 and Bn . The − − u−u ≡ iun 0 exponents of 1 and u in the normalisation are integral.   − Remark 4.3. Integrality of the exponents of 1,u follows from the fact that the matrix product is a polynomial in u,u−1−with integral coefficients, while the Jones polynomial of a knot is always a polynomial in t,t−1 with integer coefficients.

Remark 4.4. Since VL is a topological invariant, the expression given above for VKr (t) should depend on the integers k,n1,...,nk only via the continued p fraction r = q . However there is no known direct formula for it in terms of p, q. Remark 4.5. When A = √i, then u = 1, t = 1 and the Jones polynomial evaluates (up to sign) to the determinant of the− knot 1 VK ( 1) = 1 0 Bn ... Bn r − ± k 1 0  
n i 1 where Bn , see Remark 3.4. The product Bn ... Bn evaluates ≡ i 0 k 1 0     p via the continued fraction to and so we get VK ( 1) = p, as expected. iq | r − |  

9 Any rational knot can be written as K p with p odd. By Schubert’s theorem, q changing q by a multiple of p does not alter the knot and so q may be chosen to be even. By Lemma 3.2, we can now choose a continued fraction expansion of even length in even integers, that is, assume that k and n1,...,nk are even. In this case, the tangle type of the intermediate rational tangles R(ns,...,n1) (s = 1,...,k) alternate between 0 (s odd) and (s even). Setting ws = ∞ w(R(ns,...,n1)) then by (3) and (4)

0 1 ns −→ ws−1 + s odd 1 0 ns ws = w R(ns−1,...,n1)+ Tns =    −  0 1 ns    ws−1 + − s even  1 0 ns   −    k N from which it follows that the first entry in wk is w = ns. The formula − s=1 of Theorem 4.2 now simplifies, while still enabling the the JonesP polynomial of an arbitrary rational knot to be computed.

Theorem 4.6. For even k and n ,...,nk 2Z, the Jones polynomial of the 1 ∈ knot obtained by numerator closure of the rational tangle R(nk,...,n1) is

k P ns −1 1 VK (t) = ( 1) 2 u s 1 i(u u) Bn ... Bn r − − k 1 0  
−n −2 −1 un−u−n [n] iu where t = u , d = i(u u), [n]= −1 and Bn . − − u−u ≡ iun 0   5 A combinatorial formula for Jones polynomi- als of rational knots

In this section we evaluate the matrix product in Theorem 4.6 and carefully investigate cancellations, to arrive at the compact formula for the Jones poly- nomial of rational knots given in Theorem 5.4. N Let K = R(2m2k,..., 2m1) be a rational knot. Write [k]= 1,...,k , not to be confused with the q-number of the last section. { }

Definition 5.1. For k N, denote by k the set of all subsets S [k] (possibly empty) for which ∈ S ⊂ (i) S has the same parity as k; | | (ii) for S = , when the elements of S are listed in ascending order, the parities alternate6 ∅ with the first (smallest) element odd.

For a set S k, the complementary set [k] S has an even number of elements, ∈ S \ which when written in ascending order will be denoted a1,b1,a2,b2,....

10 k−|S| P (na −nb ) i u j j j [nr] 1 S∈Sk · r∈S Lemma 5.2. Bnk ... Bn1 = k−|S| nk+P (na −nb ) 0  P i u j j jQ [nr]   S · r S  ∈Sk−1 ∈   P Q  Proof. We expand the matrix product by definition as a sum over all interme- diate indices of products of matrix entries. Thus the upper entry of the matrix product in the lemma is

(Bn )u ,u ... (Bn )u ,u ( ) k k k−1 1 1 0 ∗ u1,...,uXk−1∈{1,2} where the indices ui are extended so that u0 = uk = 1. Similarly the lower entry is obtained from the same sum but where the indices are extended so that u0 = 1, uk = 2. −n n Recall that (Bn)11 = [n], (Bn)12 = iu , (Bn)21 = iu while (Bn)22 = 0. Sequences u0,...,uk making non-zero contributions to (*) will therefore be ones in which all adjacent pairs are 11, 12 or 21; that is they consist of isolated 2’s separated by strings of 1’s. Such sequences starting and beginning with 1 are in bijection with elements S k where S is the set of j [k] for which ∈ S ∈ uj = uj = 1. Then the complementary set [k] S when written in ascending −1 \ order will have form a1,b1,a2,b2,... where the aj denote the positions (index values) of the isolated 2’s in the sequence ui and bj = aj +1. The matrix element product in (*) corresponding to this particular sequence ui is the summand in the upper element given in the lemma. Similarly for the lower entry in the matrix product, where now uk = 2. Removing this last element, relevant sequences (uj ) are seen to be in bijection with k . S −1 13 Example 5.3. The sequence (uj )j=0 given by 11121211121121 is a relevant se- quence for k = 13 and corresponds to S = 1, 2, 7, 8, 11 13 which has complement in [13] which reads in ascending order{ 3, 4, 5, 6,}9, ∈10 S, 12, 13 making the integers aj into 3, 5, 9, 12 with bj = aj + 1. Applying this lemma to the matrix product in Theorem 4.6 with 2k replacing k and 2ms replacing ns gives

2 P(maj −mbj ) −|S| j i u [2mr] 2 P ms −1 S∈S2k · r∈S VK (t)=u s 1 i(u u)   2m2k+P(2maj −2mbj ) − P −|S| j Q i u [2mr]
 ·  S∈S2k−1 r∈S   P Q  4 P maj −|S| j 2mr = i u (u [2mr]) · S r S X∈S2k Y∈ 4m2k+P 4maj −1 −|S| j 2mr + i(u u) i u (u [2mr]) − · S r S ∈SX2k−1 Y∈

11 2 P ms where in the last line we distribute the factor u s as factors u2ms amongst s 2 −1 either in S or of form aj ,bj (or, in the second line, 2k). Recall that u = t 2m t−2m−1 − and so u [2m]= u−u−1 . Thus

|S| 2 −2 P maj ( 1) j −2mr VK (t)= − t (t 1) (u u−1)|S| · − S r S X∈S2k − Y∈ |S|−1 −2m −2 m ( 1) 2 2k P aj − t j (t−2mr 1) − (u u−1)|S|−1 · − S r S ∈SX2k−1 − Y∈ Note that in the first sum S is always even while in the second sum S is | | −2ma| | always odd. Furthermore (u u−1)2 = t +2+ t−1 while the factor t j can −2ma − − be split up as (t j 1) + 1 and more generally, −

−2 P maj −2ma m t j = t j = (t−2 a 1) − j A a A Y X Y∈ where the sum is over all (not necessarily proper) subsets A a1,a2,... . Thus ⊂ { }

|S| −1 − −2mr VK (t)= (t +2+ t ) 2 (t 1) S∈S2k · A⊂{a1,...} r∈S∪A − S (8) −1 − | |−1 −2m P (t +2+ t ) 2 P Q (t r 1) −S∈S2k−1 · A⊂{a1,... 2k} r∈S∪A − P P Q Next observe that in the first line, the possible subsets S A [2k] which can occur are: the empty set and every subset whose smallest∪ element⊂ is odd. Each such set T appears precisely once and the corresponding sets S, A can be recov- ered by the following algorithm. If T = then S = A = . Otherwise, reading ∅ ∅ T as a sequence in ascending order, split T into subsequences T1,T2,...,Tp of consecutive elements of T of the same parity; the first subsequence T1 will be of odd parity, the second T2 of even parity and so on. The largest element of T will have the same parity as p. Then

max Tj 1 j p if p is even S = ≤ ≤ max Tj 1 j p 1 if p is odd   ≤ ≤ −

By definition S has an even number of elements, 2l where l p , while, if it 2 is non-empty then its smallest element is odd and the parities≡ ⌊ of⌋ successively increasing elements alternate. In either case, it is easy to see that A = T S is \ a subset of the aj ’s associated with the set S. Meanwhile a similar analysis in the second line shows that the possible sets S A [2k] which can occur are all those subsets T with odd minimum element. To∪ reconstruct⊂ S and A, read T in ascending order, subdivide into subsequences T1,...,Tp by parity as in the previous case. Then

max Tj 1 j p 1 if p is even S = ≤ ≤ − max Tj 1 j p if p is odd   ≤ ≤

 12 p−1 Now S has an odd number of elements, 2l + 1 where l 2 , their parities alternating when placed in ascending order, the first being≡ ⌊ odd.⌋ Changing the order of the summations in (8) now gives

p(T ) −1 −⌊ ⌋ −2mr VK (t)=1+ (t +2+ t ) 2 (t 1) − T r T X Y∈ −1 −⌊ p(T )−1 ⌋ −2m (t +2+ t ) 2 (t r 1) − − T r T X Y∈ where the summations are over all (non-empty) subsets T [2k] whose smallest element is odd. The first term comes from the empty set. Here⊂ p(T ) denotes the number of subsequences into which T is split as above (consecutive elements of the same parity), equivalently it is one more than the number of parity changes between adjacent elements when T is listed in ascending order. When p(T ) is odd, the contributions of T to the two sums cancel. When p(T ) is even the contributions differ by a factor of (t +2+ t−1). In conclusion we have the following formulation of the Jones polynomial.− Theorem 5.4. Suppose K is a rational knot expressed as the numerator closure of a rational tangle made out of an even number of even-length twist boxes R(2m2k,..., 2m1). Then the Jones polynomial of K is given by

p(T ) −1 −1 − −2mr VK (t)=1 (t +1+ t ) (t +2+ t ) 2 (t 1) − − T r T X Y∈ where the sum is over all non-empty subsets T [2k] whose minimal element is odd and maximal element is even and p(T ) is⊂ one more than the number of parity changes between consecutive elements when T is listed in ascending order.

Remark 5.5. Note that (t−2mr 1) is divisible by t2 1 and thus the innermost − − 2 |T | product in the formula for VK (t) in the above theorem is divisible by (t 1) . Meanwhile t +2+ t−1 = t−1(t + 1)2 and so since p(T ) T , the summand− is a polynomial in t,t−1 divisible by (1 t)|T |. ≤ | | − 6 Templates generating Jones coincidences

In this section we use Theorem 4.2 and matrix identities amongst the matrices Bn to prove invariance of the Jones polynomial of rational knots under certain moves. Lemma 6.1. If K and K′ are two knots whose Jones polynomials differ by at most a sign and multiplication by a power of t, then their Jones polynomials are identical.

Proof. Since VK (1) = 1, thus there can be no sign difference. Suppose otherwise n that VK VK′ ; without loss of generality, assume that VK′ = t VK with n N. 6≡ 3 ∈ By Proposition 12.5 in [7], VK (t) 1 is divisible by (t 1)(t 1). Hence − − −

13 n 3 VK′ VK = (t 1)VK (t) is divisible by (t 1)(t 1). Dividing through by − − n−1 − − 3 t 1 we see that (1+ t + + t )VK (t) is divisible by t 1 which contradicts the− fact that its evaluation··· at t = 1 is n. − Combining with Theorem 4.2, this means that for identification of Jones polynomials of two rational knots, it is sufficient to prove that the corresponding matrix product agrees (even up to a sign and power of u). [n] iu−n Lemma 6.2. The matrices Bn SL have the following ≡ iun 0 ∈ 2   elementary properties. Here d = i(u−1 u). − ∗ ∗ (i) Bn = B−n where is conjugate transpose (conjugate being defined by i −i, u u). 7−→ − 7−→ α iβ (ii) Products of B matrices are matrices of the form C = where n iγ δ −1 −1  ∗ α,β,γ,δ Z[u,u ] with αδ + βγ = 1; in particular C = B0C B0, ∈ 1 1 1 − 1 1 0 C∗ = 1 0 C and 1 d C∗ = 1 d C . 0 0 d d     −  − 



(iii) BnB Bm = Bn m. 0 − +

n 1 −n 1 (iv) id Bn = u 1 0 u 1 d . · d − 0 −   

k 1 1 1 (v) 1 d B = 1 d B . nj 0 nj 0 j=1   j=k  
Q
Q k 1 1 2 1 (vi) 1 d Bn = (1 d ) 1 0 Bn . j d − · j 0 j=1 −  j=k  
Q
Q 1 1 1 (vii) 1 d C = 1 d C∗ = 1 0 C for any matrix 0 0 d   u→u−1   −  product
C of matrices Bn.

Proof. The first four properties are immediate from the definition of Bn. Prop- erties (v), (vi) follow by induction on k; the inductive step uses (iv). Thus, for k = 0, an empty product is the identity matrix and (v),(vi) hold. Meanwhile, k ←− 1 assuming both (v) and (vi) for k and writing C = Bnj , C = Bnj while j=1 j=k Q Q

14 m = nk+1 we have by (iv) 1 id 1 d CBm · 0   (iv)
1 1 1 1 = um 1 d C 1 0 u−m 1 d C 1 d d 0 − 0 0 −       
1
1

= um 1 d C u−m 1 d C d − 0 −    (v),(vi)
←− 1
←− 1 = um(1 d2) 1 0 C u−m 1 d C − 0 − 0     1
←− 1
1 ←− 1 = um 1 d 1 0 C u−m 1 d 1 d C d 0 − 0 0 −        (iv)
←− 1


= id 1 d Bm C · 0  
which is (v) with k + 1 replacing k. Similarly for (vi). ←− T Finally for (vii), note that Bn = (Bn)u→u−1 from which it follows that C = T C . Since d = d = d −1 , by (v) u→u−1 − u→u 1 ←− 1 1 T 1 1 d C = 1 d C = 1 d CT = 1 0 C 0 −1 0 −1 − 0 d   u→u   u→u   − 

Lemma 6.3 (Key Lemma). Suppose that Uj , Vj (0 j k) are products of ≤ ≤ matrices Bn and n1,...,nk Z. (1) If ∈ 1 1 1 d Uj = 1 d Vj j 0 0 ∀    

1 1 1 1 1 d Uj 1 0 Ul = 1 d Vj 1 0 Vl j = l d 0 d 0 ∀ 6 −    −    then



1 1 1 d U Bn U Uk Bn Uk = 1 d V Bn V Vk Bn Vk 0 1 1 ··· −1 k 0 0 1 1 ··· −1 k 0    

(2) If 1 1 1 d Uj = 1 d Vj j 0 0 ∀    

1 1 1 1 1 d Ul 1 0 Uj = 1 d Vj 1 0 Vl j = l d 0 d 0 ∀ 6 −    −    then



1 1 1 d U Bn U Uk Bn Uk = 1 d VkBn Vk V Bn V 0 1 1 ··· −1 k 0 k −1 ··· 1 1 0 0    

15 Proof. The proof proceeds by replacing each Bnj in the two sides of the equality to be proved by a combination of matrix products using (iv). This results k 1 k in (id) 1 d U Bn U Uk Bn Uk being replaced by a sum of 2 · 0 1 1 ··· −1 k 0 Pǫj nj terms, each of
which is of the form ǫj u j (for ǫj 1, 1 ) times a j · ∈ { − } product of (k +1) scalars, each of whichQ is of one of the forms 1 1 1 1 1 d U , 1 d U , 1 0 U , 1 0 U j 0 j d j 0 j d   −    − 



according as (ǫj ,ǫj ) = ( 1, 1), ( 1, 1), (1, 1), (1, 1), respectively, where we +1 − − − − extend ǫj to j =0, k + 1 by setting ǫ0 = ǫk+1 = 1. Similarly for the right hand side of the equality to be proved. On comparison,− the first condition ensures that 1 1 −1 1 d Uj = 1 d Vj . By Lemma 6.2(vii), replacing u u gives 0 0 →    
1
1 also 1 0 U = 1 0 V . Since ǫ = ǫ , adjacent unequal j d j d 0 k+1 −  −  terms in the
sequence ǫj come
in pairs ( 1, 1) followed (possibly after a string of 1’s) by (1, 1); the second condition− allows the product of a pair of scalars − associated with Uj ’s to be equated with the corresponding product of scalars associated with Vj ’s (with order reversed in the case of (2)). Definition 6.4. Suppose that n Zr is an integer sequence. Denote by n ← ∈ − and n respectively, the sequences obtained from n by reversing all the signs and ← by reversing order of the sequence, respectively. Define n∗ = n. − Theorem 6.5 (Template I). If two rational knots have integer sequence pre- sentations related by the move

ǫ0 ǫ1 ǫk ǫk ǫk−1 ǫ0 (n , d , n ,...,dk, n ) (n , dk, n ,...,d , n ) 1 −→ 1 for some d1,...,dk Z, ǫ0,...,ǫk 1, and integer sequence n, then their Jones polynomials coincide.∈ ∈ { ∗} ∗ Proof. Let C = Bn1 Bnr where n = (n1,...,nr). By Lemma 6.2(i), C = r ··· ( 1) B−nr B−n1 . The theorem follows from the Key Lemma (2) with Uj = − ǫj ··· Vj = C . The first condition is automatically satisfied since Uj = Vj . The ∗ second condition follows from the fact that in any case Uj , Vj C, C so ∈ { } 1 1 1 1 that 1 d U = 1 d V and 1 0 U = 1 0 V l d j d j 0 l 0 −  −      by Lemma
6.2(ii).

Example 6.6. Using n = (2, 3), k = 2, ǫ0 = ǫ1 = ǫ2 = 1, d1 = 0, d2 = 1 in the first template leads to a Jones rational coincidence between the integer− sequences (2, 3, 0, 2, 3, 1, 2, 3) and (2, 3, 1, 2, 3, 0, 2, 3). The associated contin- ued fractions are − − 1 1 1 1 1 1 1 245 1 1 1 1 1 1 1 245 2 = , 2 = − 3 0 2 3 1 2 3 137 − 3 1 2 3 0 2 3 142 − − − − − − − − − − − − − − 16 giving coincidence of the Jones polynomials of K 245 and K 245 . 137 142 Example 6.7. Using n = (2, 2), k = 2, ǫ = (1, 1, 1), d1 = 1, d2 = 4 in the first template generates the Jones− rational coincidence given by the integer sequences

(2, 2, 1, 2, 2, 4, 2, 2) (2, 2, 4, 2, 2, 1, 2 , 2) − − − ↔ − − − that is between K 495 and K 495 . On the other hand, the first template with 218 203 n = (2, 5, 1, 0), k = 2, ǫ = (1, 1, ), d1 = 0, d2 = 3 generates the Jones rational coincidence given by the integer∗ sequences −

(2, 5, 1, 0, 0, 2, 5, 1, 0, 3, 0, 1, 5, 2) (0, 1, 5, 2, 3, 2, 5, 1, 0, 0, 2, 5, 1, 0) − − − − ↔ − − − − −1 that is between the rational knots K 495 and K 495 . Since 302 218 (mod 302 383 ≡ 495), this gives a Jones rational triple coincidence K 495 = K 495 , K 495 and K 495 . 218 302 203 383

← ǫ Let n, n be denoted by n where ǫ = , , respectively. ← → Theorem 6.8 (Template II). If two rational knots have integer sequence presentations related by the move

ǫ0 φ0 ǫk φk ǫ0 φ0 ǫk φk ∗ ∗ ∗ ∗ (n,m , n, d ,...,dk, n,mk, n ) (n ,m , n , d ,...,dk, n ,mk, n ) 0 − 1 − −→ 0 − 1 − for some m0,...,mk, d1,...,dk Z, ǫ0,...,ǫk, φ0,...,φk , and integer sequence n, then their Jones polynomials∈ coincide. ∈ {← →}

Proof. The template follows from the Key Lemma (1) where Uj and Vj are the ǫj φj products of Bn matrices associated with the integer sequences (n,mj , n) and ǫj φj − ∗ ∗ (n ,mj , n ). It remains to verify the two conditions in Lemma 6.3(1). For − the first one, either ǫj = φj in which case

1 ← 1 1 d CB C− = 1 d C∗B C m 0 m 0    

← − ∗ follows from Lemma 6.2(v), since the sequences CBnC , C BnC are exact reversals of each other; or ǫj , φj are in opposite directions, in which case,

1 1 1 d CB C∗ = 1 d C∗B C m 0 m 0    

follows from Theorem 6.5 with k = 1. The second condition to be verified is that

ǫ φ 1 ǫ′ φ′ 1 1 d B(n)BmB( n) 1 0 B(n)BkB( n) − d · − 0 −   
ǫ φ
ǫ′ φ′ ∗ ∗ 1 ∗ ∗ 1 = 1 d B(n )BmB( n ) 1 0 B(n )BkB( n ) − d · − 0 −   

17 where B(n) Bn1 Bnr . This is checked using Lemma 6.2(iv) to expand out ≡ ··· 2 the Bm and Bk terms on both sides. Thus ( d ) times the expression on either side of the equality to be proved becomes− a sum of four terms with external factors of u±m±k. The four matching individual terms are found to be equal, each being± a product of four scalars. For example, the um−k term on the left hand side is −

ǫ 1 φ 1 ǫ′ 1 φ′ 1 1 d B(n) 1 0 B( n) 1 0 B(n) 1 d B( n) d · − d · 0 · − 0 −  −     



while on the right hand side it is the same with n∗ replacing n. The matching of first and third factors is via 1 1 1 1 1 d C = 1 d C∗ , 1 0 C = 1 0 C∗ d d 0 0 −  −     



ǫ ǫ′ from Lemma 6.2(ii) (where C = B(n), B(n) respectively). The second factor can be transformed into a type similar to the fourth using Lemma 6.2(vii); the product of the second and fourth factors is now

φ 1 φ′ 1 1 d B( n∗) 1 d B( n) − 0 · − 0    

which by Lemma 6.2 is independent of φ, φ′ and is thus unchanged when n is replaced by n∗ (the two factors interchanging).

Example 6.9. Take n = (4, 2), k = 1, d1 = 0, m = ( 1, 0) with ǫ = φ = ( , ). The second template now gives a Jones rational coincidence− between the→ integer→ sequences (4, 2, 1, 4, 2, 0, 4, 2, 0, 4, 2), ( 2, 4, 1, 2, 4, 0, 2, 4, 0, 2, 4), − − − − − − − − − − that is between the rational knots K 329 and K 329 . 89 −193 Remark 6.10. All the explicit examples of Jones rational equivalences given in the text are not mutants as can be verified by computing their HOMFLYPT polynomial ([5], [16]). By using the freedoms in the Templates of Theorems 6.5, 6.8, it can be seen that arbitrarily large (but necessarily finite) families of rational knots sharing the same Jones polynomial can be constructed; this was already known [10], where a special case of the construction given here was presented.

7 Pivoting pairs

Out of the 80,317 rational knots up to determinant 899, there were found in [17], 223 Jones rational coincidences (up to mirror image), two of which are amphicheiral pairs and three of which are triplets. Out of the 220 ‘simple’ coin- cidences, 144 can be identified as examples of Template I and 177 as examples of Template II, while 108 are examples of both. Note that since there are an infi- nite number of continued fraction representations of a rational corresponding to

18 a particular rational knot, it is possible that even some of those identified as ex- amples of one template, may be as-yet-unfound examples of the other template. On the other hand, of the three triple coincidences, these can be established by an ‘intersecting pair’ of coincidences as in Example 6.7; in two cases, both come from Template I while in the third case one comes from Template I and the other from Template II. Example 7.1. The first Jones rational coincidence (smallest determinant) is be- ∗ tween K 49 = 1035 and K 49 = 10 (see [2]). It appears as an example of 22 36 22 Template I with n = (2, 4), k = 1, d1 = 1, ǫ = (1, ) which gives the integer sequences (2, 4, 1, 4, 2) and ( 4, 2, 1, 2, 4); this is∗ trivially also an example of Template II with− n−= (2, 4), k−=− 0, m = 1, ǫ = and φ = . 0 0 → 0 ← Example 7.2. Here is an example of a Jones rational equivalence which can be explained both by Template I and Template II, but where the generat- ing continued fraction expansions are quite different in the two cases. Tem- plate I applied with n = (2, 6), k = 2, d1 = 0, d2 = 1, ǫ = (1, 1, 1) gives the integer sequences (2, 6, 0, 2, 6, 1, 2, 6) and (2, 6, 1, 2, 6, 0, 2, 6) whose contin- 275 275 ued fractions evaluate to 147 and 158 respectively. Template II applied with n = (4, 3), k = 2, m = (0, 0, 0), d1 = 0, d2 = 1, ǫ = φ = ( , , ) gives the integer sequences (4, 3, 0, 4, 3, 0, 4, 3, 0,− 4, 3, 1, 4, 3, →0, →4, →3) and ( 3, 4, 0, 3, 4, 0, 3, 4, 0, 3, 4, −1, −3, 4, 0, 3, 4)− whose− − continued fraction− − − − 275 − −275 − − − evaluations are 58 and 117 , respectively. Note that 147 58 1 (275) while 158 117 (275) so that− these templates both give rise· to the≡ same Jones rational≡ − coincidence. This leaves 11 Jones polynomial coincidences amongst rational knots with determinant < 900 which are still unaccounted for by the templates of Theorems 6.5, 6.8. Example 7.3. The first case of a Jones rational coincidence which doesn’t seem to appear as an example of either template from §6 has determinant 377, namely K 377 and K 377 . Integer sequences for these rational knots are 70 278 (5, 2, 4, 4, 0, 5, 3, 6, 4), (5, 3, 6, 4, 0, 5, 2, 4, 4) − − − − − − − − − − − − This first example was what led to the definition of pivoting pairs below. Before this, we make some observations about the form of a matrix which can be expressed as a product of Bn matrices.

Remark 7.4. Suppose that C is a product of matrices Bn. Since det Bn = 1 ← T thus det C = 1. Recall that C = Cu→u−1 and so by Lemma 6.2(v),(vi), 1 1 1 d C = 1 d CT 0 u→u−1 0     (9)
1
1 1 d C = (1 d2) 1 0 CT d − · u→u−1 0 −   

α iβ Matrices C = satisfying (9) have iγ δ   α + idγ = α′ + idβ′, α idβ + idγ d2δ = (1 d2)α′ − − −

19 where the prime denotes the image under u u−1. It follows that such a matrix → ′ α′−α ′ β−β′ is determined by the entries in the first row, γ = β + u−u−1 and δ = α + u−u−1 .

′ ′ α iβ ′ ′ α β αβ ′ ′ C = ′ α −α ′ β−β with αα + ββ + − −1 = 1 (10) iβ + i −1 α + −1 u u  u−u u−u  − Matrices of the form (10) where α, β Z[u,u−1] form a group, , containing as ∈ C a subgroup , the set of those matrices which can be written as a product of Bn B matrices. Such matrices which arise as product of Bn matrices will have either α Z[t,t−1], β uZ[t,t−1] or α uZ[t,t−1], β Z[t,t−1], according as the number∈ of even indices∈ (n even) in the∈ product is even∈ or odd, respectively. This follows from the same fact about individual Bn matrices and its invariance under matrix product. Observe that the condition on α, β in (10) can be reformulated as

(α + (u−1 u)β)(α′ + (u u−1)β′) = (u2 1+ u−2)αα′ (u u−1)2 − − − − − while the second factor on the left hand side is, by Theorem 4.2, the Jones poly- nomial of the associated rational knot, up to a sign and power of u. Recalling that t = u−2, we arrive at the following corollary, which seems to be new. − Corollary 7.5. The Jones polynomial of a rational knot K satisfies

−1 −1 −1 −1 VK (t)VK (t )=(2+ t + t ) (1 + t + t )a(t)a(t ) − where a(t) Z[t,t−1]. ∈ Denote by those elements of for which α = 0. C0 C Lemma 7.6. = B BnB n Z . C0 {± 0 0| ∈ } Proof. In the case of α = 0, the condition in (10) reduces to ββ′ = 1. Since β Z[u,u−1], thus β = un for some n Z. The matrix C is now determined ∈ ± ∈ 0 iβ 0 iun by (10) to be C = ′ β′−β = −n = B0BnB0 iβ −1 ± iu [n] ∓  u−u   − 

Definition 7.7. A pair of matrices C1, C2 are called a pivoting pair if they satisfy ∈B 1 1 1 1 1 d C 1 0 C = 1 d C 1 0 C 1 d · 2 0 2 d · 1 0 −    −   
′
′

Equivalently, by (9), α1α2 = α1α2.

It follows immediately from the definition that, (i) for any C and C , the pair C , C is a pivoting pair; 0 ∈C0 ∈B 0 (ii) for pairs of matrices in 0, being a pivoting pair is an equivalence rela- B\C α1 α2 tion, namely it states that ′ = ′ . We denote the equivalence relation α1 α2 on the corresponding integer sequences by . ∼

20 Example 7.8. The matrices Bn all lie in the same equivalence class since α = ′ ∈B [n]= α . Similarly, any product of Bn matrices corresponding to a palindromic integer sequence will lie in this same equivalence class.

Example 7.9. For any C , n Z, the pair C, CBnC is a pivoting pair. This ∈B ∈ follows from the fact that the (1, 1) matrix entry of C CBnC is 2 ≡ unα u−nα′ α = α − unβ u−nβ′ 2 u u−1 − −  −  where the factor in brackets is invariant under u u−1. Thus all equivalence classes are infinite. → Theorem 7.10 (Pivoting Pairs Template). If two rational knots have inte- ger sequence presentations related by the move

ǫ0 ǫ1 ǫk ǫk ǫk−1 ǫ0 (n , d , n ,...,dk, n ) (n , dk, n ,...,d , n ) 0 1 1 k −→ k k−1 1 1 for some d ,...,dk Z, ǫ ,...,ǫk 1, and integer sequences n n 1 ∈ 0 ∈ { ∗} 0 ∼ 1 ∼ ... nk, then their Jones polynomials coincide. ∼ Proof. The proof is as for Theorem 6.5 using Lemma 6.3(2), now with Uj = ǫj Vj = Cj . The first condition is automatically satisfied since Uj = Vj . The second condition follows from the pivot condition along with Lemma 6.2(ii). Remark 7.11. Theorem 7.10 is a generalization of the first template (Theorem 6.5), the latter being the special case where nj are chosen to be identical j. ∀ Example 7.12. Some examples of pivoting pairs which do not come from the gen- eral processes of Examples 7.8 or 7.9 are (5, 3, 3, 1) (2, 6, 2, 2) (1, 6, 3, 4) while (5, 3, 6, 4) (5, 2, 4, 4). These are− checked∼ by brute− ∼ force, that is, computation− − of−α.∼ For example− − − in the case of the last pair,

α = u−1[3][4](u2 1+u−2)(u8 +3u6+5u4+5u2+5+4u−2+3u−4+2u−6+u−8) 1 − − α = u−1[5](u8 +3u6 +5u4 +5u2 +5+4u−2 +3u−4 +2u−6 + u−8) 2 − in which the differing factors are all invariant under u u−1. This last pair is responsible for the Jones rational coincidence of Example→ 7.3 using the pivoting pairs template with k = 1, ǫ0 = ǫ1 = 1 and d1 = 0. Conjecture 7.13. All Jones rational coincidences can be obtained by using the moves in the templates of Theorems 7.10 and 6.8. This is verified explicitly in [17] for all Jones rational coincidences with deter- minant < 900.

21 8 Jones rational coincidences, determinant< 900

Here is a list of all Jones rational coincidences with determinant < 900, dis- counting mirror images, in order of increasing determinant. A star indicates an amphicheiral knot and a hash indicates pairs whose HOMFLYPT polyno- mials also coincide (suspected mutants). Triple coincidences are listed as such. Since rational knots are alternating, the span of their Jones polynomial gives the crossing number of the knot [14]; the crossing numbers of knots here range 841 841 from 10 in the first coincidence listed, up to 32 for the pair ( 782 , 434 ).

49 49 81 81 121 121 121 121 135 135 147 147 153 153 ( 22 , 36 ), ( 44 , 62 ), ( 56 , 100 ), ( 32 , 76 ), ( 62 , −28 ), ( 64 , 106 ), ( 70 , −32 ), 161 161 169 169 169 169 171 171 189 189 207 207 209 209 ( 60 , 74 ), ( 40 , 118 ), ( 90 , 142 ), ( 74 , −40 ), ( 44 , −82 ), ( 146 , 164 ), ( 74 , 150 ), 225 225 225 225 231 231 243 243 245 245 245 245 253 253 ( 106 , 196 ), ( 122 , 158 ), ( 86 , 170 ), ( 136 , 190 ), ( 106 , 176 ), ( 108 , −88 ), ( 104 , 196 ), 255 255 259 259 261 261 275 275 275 275 279 279 279 279 ( 92 , −112 ), ( 96 , 180 ), ( 148 , 184 ), ( 128 , 228 ), ( 102 , 202 ), ( 58 , −128 ), ( 196 , 214 ), 287 287 289 289 289 289 289 289 297 297 297 297 # 301 301 ( 88 , 130 ), ( 86 , 120 ), ( 50 , 186 ), ( 152 , 254 ), ( 136 , −62 ), ( 116 , 134 ) , ( 80 , 136 ), 315 315 # 319 319 329 329 329 329 333 333 # 343 343 343 343 # ( 68 , 122 ) , ( 196 , 225 ), ( 68 , −214 ), ( 122 , 136 ), ( 76 , 130 ) , ( 148 , 246 ), ( 90 , 104 ) , 351 351 351 351 351 351 # 361 361 361 361 361 361 361 361 ( 190 , 226 ), ( 152 , −82 ), ( 136 , 154 ) , ( 58 , 248 ), ( 172 , 324 ), ( 134 , 210 ), ( 94 , 284 ), 363 363 363 363 369 369 369 369 377 377 ∗ 385 385 399 399 ( 166 , 298 ), ( 98 , 230 ), ( 200 , 254 ), ( 86 , −160 ), ( 70 , 278 ) , ( 212 , 268 ), ( 122 , 164 ), 405 405 423 423 427 427 441 441 441 441 441 441 441 441 ( 224 , 314 ), ( 88 , −194 ), ( 192 , −88 ), ( 106 , 358 ), ( 202 , −92 ), ( 130 , −164 ), ( 250 , 304 ), 441 441 441 441 451 451 451 451 455 455 459 459 459 459 ( 188 , 314 ), ( 230 , 398 ), ( 96 , 260 ), ( 162 , 184 ), ( 186 , 361 ), ( 100 , −206 ), ( 316 , 352 ), 473 473 473 473 477 477 477 477 481 481 493 493 495 495 ( 140 , 305 ), ( 108 , 174 ), ( 214 , −104 ), ( 176 , −142 ), ( 70 , −226 ), ( 194 , 364 ), ( 184 , −146 ), 495 495 495 495 495 505 505 # 505 505 ∗ 507 507 507 507 ( 338 , 392 ), ( −218 , 112 , 292 ), ( 212 , 192 ) , ( 222 , −182 ) , ( 118 , 352 ), ( 272 , 428 ), 513 513 513 513 513 513 517 517 517 517 527 527 529 529 ( 124 , 238 ), ( 226 , −116 ), ( 278 , 350 ), ( 216 , 404 ), ( 190 , 212 ), ( 336 , −98 ), ( 484 , 254 ), 529 529 529 529 529 529 529 529 531 531 531 531 533 533 ( 392 , 488 ), ( 344 , 68 ), ( 116 , 208 ), ( 160 , 298 ), ( 196 , −158 ), ( 124 , −344 ), ( 138 , 216 ), 539 539 539 539 539 539 549 549 551 551 553 553 567 567 ( 246 , −118 ), ( 244 , 398 ), ( 386 , 232 ), ( 128 , 376 ), ( 326 , 402 ), ( 360 , −114 ), ( 260 , −118 ), 567 567 581 581 # 585 585 589 589 595 595 # 603 603 # 605 605 ( −440 , 314 ), ( 172 , 338 ) , ( 268 , −122 ), ( 88 , 274 ), ( 156 , 326 ) , ( 218 , 272 ) , ( 274 , 494 ), 605 605 621 621 # 621 621 621 621 # 623 623 625 625 625 625 ( 164 , 384 ), ( 136 , 343 ) , ( 352 , 424 ), ( 190 , 280 ) , ( 404 , −130 ), ( 426 , 76 ), ( 224 , 274 ), 625 625 625 625 637 637 637 637 639 639 # 639 639 639 639 ( 574 , 324 ), ( 474 , 524 ), ( 288 , 470 ), ( 456 , 272 ), ( 140 , −286 ) , ( 418 , 346 ), ( 362 , 416 ), 649 649 651 651 651 651 651 651 657 657 # 657 657 663 663 ( −240 , 398 ), ( −292 , 452 ), ( 446 , −142 ), ( 460 , 502 ), ( −290 , 148 ) , ( 356 , 446 ), ( 370 , −98 ), 665 665 675 675 675 675 # 675 675 # 679 679 679 679 ( 142 , 492 ), ( −584 , 314 ), ( −154 , 296 ) , ( 152 , −298 ) , ( 180 , −492 ), ( 284 , 402 ), 689 689 693 693 693 693 693 693 703 703 703 703 711 711 ( 314 , −110 ), ( 304 , 502 , 250 ), ( 184 , 382 , 436 ), ( 146 , 516 ), ( 262 , −308 ), ( 245 , 326 ), 711 711 711 711 715 715 715 715 721 721 721 721 ( −166 , 464 ), ( 500 , −392 ), ( 552 , −292 ), ( 412 , 588 ), ( 324 , −500 ), ( −556 , −268 ), 721 721 729 729 729 729 729 729 729 729 729 729 729 729 ( 318 , 332 ), ( 296 , 136 ), ( 676 , 352 ), ( −170 , 496 ), ( 478 , −152 ), ( 458 , 512 ), ( 620 , 188 ), 729 729 735 735 735 735 737 737 737 737 737 737 741 741 ( 406 , 568 ), ( 302 , −412 ), ( 314 , 524 ), ( 302 , 570 ), ( 332 , 600 ), ( 126 , 394 ), ( 548 , 158 ), 745 745 745 745 747 747 747 747 749 749 749 749 ( 288 , −462 ), ( 328 , −442 ), ( 232 , −266 ), ( −526 , 470 ),( 406 , −550 ), ( −326 , 530 ), 755 755 755 755 759 759 765 765 765 765 777 777 # 779 779 ( 312 , −468 ), ( 272 , 448 ), ( 448 , −172 ), ( 362 , 668 ), ( 592 , −428 ), ( 458 , 214 ) , ( 162 , 572 ),

22 783 783 783 783 783 783 783 783 791 791 # 793 793 801 801 ( 350 , −172 ), ( 620 , −424 ), ( 338 , 164 ), ( −568 , 476 ), ( 466 , 234 ) , ( 342 , −146 ), ( −578 , 590 ), 801 801 801 801 803 803 # 817 817 819 819 819 819 819 819 ( 454 , 544 ), ( 634 , −434 ), ( 288 , 310 ) , ( 124 , 382 ), ( 590 , −346 ), ( 184 , −362 ), ( 374 , −562 ), 819 819 833 833 833 833 833 833 # 833 833 837 837 837 837 ( 628 , −464 ), ( −342 , 484 ), ( 540 , −174 ), ( 232 , 218 ) , ( −536 , 596 ), ( 604 , 512 ), ( 344 , 158 ), 837 837 841 841 841 841 841 841 841 841 841 841 841 841 ( 260 , −298 ), ( 726 , 204 ), ( 378 , −260 ), ( 550 , 86 ), ( 608 , 318 ), ( 666 , 144 ), ( 782 , 434 ), 845 845 845 845 847 847 847 847 847 847 855 855 867 867 ( −586 , 194 ), ( 454 , 714 ), ( 692 , −386 ), ( 354 , −614 ), ( 540 , 232 ), ( −178 , 392 ), ( 358 , −254 ), 867 867 867 867 867 867 871 871 873 873 873 873 875 875 ( 188 , −526 ), ( 562 , −154 ), ( 460 , 766 ), ( −720 , 138 ), ( −182 , 400 ), ( 272 , −310 ), ( 256 , 506 ), 889 889 891 891 891 891 893 893 ( 248 , 502 ), ( −208 , 386 ), ( 692 , 494 ), ( 576 , −184 ) It is striking to see the prevalence of squares or at least multiplicities in the prime decomposition of the determinant in Jones rational coincidences. However a p p direct condition on a pair ( q , q′ ) to form a Jones rational coincidence has yet to be found (other than direct computation of the associated Jones polynomials!)

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