TitleRational Witt classes of pretzel kn. Citation Osaka Journal of Mathematics. 47(4)

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1 TitleRational Witt classes of pretzel kn Author(s) Jabuka, Stanislav Citation Osaka Journal of Mathematics 47(4) Issue Date Text Version publisher URL DOI Rights Osaka University

2 Jabuka, S Osaka J Math 47 (2010), RATIONAL WITT CLASSES OF PRETZEL KNOTS STANISLAV JABUKA (Received January 5, 2009, revised June 29, 2009) Abstract In his two pioneering articles [9, 10] Jerry Levine introduced and completely determined the algebraic concordance groups of odd dimensional knots He did so by defining a host of invariants of algebraic concordance which he showed were a complete set of invariants While being very powerful, these invariants are in practice often hard to determine, especially for knots with Alexander polynomials of high degree We thus propose the study of a weaker set of invariants of algebraic concordance the rational Witt classes of knots Though these are rather weaker invariants than those defined by Levine, they have the advantage of lending themselves to quite manageable computability We illustrate this point by computing the rational Witt classes of all pretzel knots We give many examples and provide applications to obstructing sliceness for pretzel knots Also, we obtain explicit formulae for the determinants and signatures of all pretzel knots This article is dedicated to Jerry Levine and his lasting mathematical legacy; on the occasion of the conference Fifty years since Milnor and Fox held at Brandeis University on June 2 5, Introduction 11 Preliminaries In his seminal papers [9, 10] Jerry Levine introduced and determined the algebraic concordance groups C n of concordance classes of embeddings of S n into S n2 These groups had previously been found by Kervaire [6] to be trivial for n even; for n odd, Levine proved that 1 C n ½ ½ 2 ½ 4 Levine achieved this remarkable result by considering a natural homomorphism ³ n Ï C n I(É) from the algebraic concordance group C n into the concordance group of isometric structures I(É) on finite dimensional vector spaces over É (we describe I(É) in detail in Section 23 below) He constructed a complete set of invariants of concordance of isometric structures and used these invariants to show that I(É) ½ ½ 2 ½ 4 Moreover, he showed that the map ³ n Ï C n I(É) is injective and that its image is large 2000 Mathematics Subject Classification 57M27, 11E12 The author was partially supported by NSF grant DMS For brevity, we denote the infinite direct sum Ä ½ i1 p simply by ½ p hoping the reader will not confuse the latter with the product of an infinite number of copies of p Throughout the article, p denotes p

3 978 S JABUKA ½ enough to itself contain a copy of 2 ½ 4 ½, thereby establishing the isomorphism In this article we focus exclusively on the case of n 1 C n ½ ½ 2 ½ 4 To determine the values of Levine s complete set of invariants for a given knot K, one is required to find the irreducible symmetric factors of the Alexander polynomial ½ K (t) of K As the question of whether or not a given polynomial is irreducible is a difficult one in general, the task of determining all the irreducible factors of a given symmetric polynomial can be quite intractable, more so as the degree of the polynomial grows To circumnavigate this issue, we consider another homomorphism ³Ï C 1 W (É) from the algebraic concordance group C 1 into the Witt ring over the rationals (W (É) is described in detail in Section 22, for a brief description see Section 12 below) The isomorphism type of W (É) as an Abelian group is well understood and is given by W (É) ½ 2 4 ½ The maps ³ and ³ 1 fit into the commutative diagram ³ 1 C 1 à I(É) ³ à W (É) From simply knowing the isomorphism types of C 1 and W (É), it is clear that ³Ï C 1 W (É) cannot be injective and a loss of information must occurs in passing from K ¾ C 1 to ³(K ) ¾ W (É) The payoff being that one is no longer required to factor polynomials Indeed, to determine ³(K ) for a given knot K S 3 one only needs to use the Gram Schmidt orthogonalization process along with a simple reduction argument (described in Section 41) The Gram Schmidt process is completely algorithmic and is readily available in many mathematics software packages To goal of this article then is to underscore the computability and usefulness of the rational Witt classes ³(K ) Their determination is almost entirely algorithmic and often straightforward, if tedious, to calculate We illustrate our point by focusing on a concrete family of knots the set of pretzel knots This family is large enough to reflect a number of varied properties of the invariant ³ and yet tractable enough so that a complete determination of the rational Witt classes is possible We proceed by giving a few details about pretzel knots first and then state our main results à 12 Statement of results Given a positive integer n and integers p 1, p 2,, p n, let P(p 1, p 2,, p n ) denote the n-stranded pretzel knot/link It is obtained by taking n pairs of parallel strands, introducing p i half-twists into the i-th strand and capping the strands off by n pairs of bridges The signs of the p i determine the handedness of the corresponding half-twists Our convention is that p i 0 corresponds to righthanded half-twists, see Fig 1 for an example We limit our considerations to knots and moreover require that n 3 and that p i 0 (the purpose of these two limitations is to exclude connected sums of torus knots/links) There are 3 categories of choices of

4 RATIONAL WITT CLASSES OF PRETZEL KNOTS 979 Fig 1 The pretzel knot P( 1, 3, 5, 3, 4) the parameters n, p 1,, p n which lead to knots, namely (1) (i) n is odd and all exept one of the p i are odd, (ii) n is even and all exept one of the p i are odd, (iii) n is odd and all p i are odd As we shall see, these categories exhibit slightly different behavior as far as the formats of their rational Witt classes Pretzel knots are invariant under the action of n by cyclic permutation, ie P(p 1, p 2,, p n 1, p n ) P(p n, p 1, p 2,, p n 1 ) We use this symmetry to fix the convention that if P(p 1,, p n ) comes from either category (i) or (ii) above, we let p n be the unique even integer among p 1,, p n To state our results, we need to give a brief description of the rational Witt ring W (É), a more copious exposition is provided in Section 22 As a set, W (É) consists of equivalence classes of pairs (,, V ) where V is a finite dimensional vector space over É and, Ï V V É is a non-degenerate symmetric bilinear form We say that a pair (,, V ) is metabolic or totally isotropic is there exits a half-dimensional subspace W V such that, W W 0 We will be adding pairs (, 1, V 1 ) and (, 2, V 2 ) by direct summing them, thus (, 1, V 1 ) (, 2, V 2 ) (, 1, 2, V 1 V 2 ) With this understood, the equivalence relation on W (É) is the one by which (, 1, V 1 )

5 980 S JABUKA is equivalent to (, 2, V 2 ) if (, 1, V 1 ) (, 2, V 2 ) is metabolic One proceeds to check that addition is commutative and indeed well defined on W (É), giving W (É) the structure of an Abelian group It is not hard to obtain an explicit presentation of W (É) (see Theorem 21 in Section 22), for now however it will suffice to point out that W (É) is generated by the set a ¾ W (É) a ¾ É 0 Here a stands for (, a, É) where, a is the form on É specified by 1, 1 a a Given a knot K S 3, pick an oriented, genus g Seifert surface S 3 and consider the linking pairing lkï H 1 (Á ) H 1 (Á ) given by lk(«, ) linking number between «and, where is a small push-off of in the preferred normal direction of determined by its orientation Extending lk to H 1 (Á É) linearly and letting, Ï H 1 (Á É) H 1 (Á É) É be «, lk(«, ) lk(, «), defines a non-degenerate symmetric bilinear pairing on the rational vector space H 1 ( g Á É) We use this to define ³(K ) (,, H 1 (Á É)) ¾ W (É), which we refer to as the rational Witt class of K According to [9], ³(K ) is well defined and only depends on K (as an oriented knot) but not on the particular choice of Seifert surface In fact, ³(K ) only depends on the algebraic concordance class of K REMARK 11 The determinant det K of a knot K S 3 is defined as det K det(v V ) where V is a matrix representative of the linking form lkï H 1 (Á ) H 1 (Á ) In order for the statements of Theorems to appear more symmetric, we allow ourselves the freedom to use a signed version of the knot determinant Thus, the determinant for pretzel knots K P(p 1,, p n ) as it appears in the said theorems and throughout the article, agrees up to sign with the usual definition of det K This signed version of the determinant may well change sign when passing from a knot to its mirror With these descriptions and conventions out of the way, we are now ready to state our main results Theorem 12 Consider category (i) from (1), ie let n 3 be an odd integer, let p 1,, p n 1 be odd integers and let p n 0 be an even integer Then the rational Witt class of the pretzel knot P(p 1,, p n ) is given by Å ³(P(p 1,, p n )) n 1 i1 (s i 1 2 s i 2 3 s i (p i 1) p i )) (p 1 p n 1 ) det P(p 1,, p n 1 ) det P(p 1,, p n 1 ) det P(p 1,, p n ),

6 RATIONAL WITT CLASSES OF PRETZEL KNOTS 981 where s i Sign(p i ) The determinants of the pretzel knot P(p 1,, p n ) and the pretzel link P(p 1,, p n 1 ) appearing above, are computed as det P(p 1,, p n ) n i1 det P(p 1,, p n 1 ) n 1 i1 p 1 Çp i p n, p 1 Çp i p n 1 As is customary in the literature, having a hat decorate a variable in a product indicates that the factor should be left out For example p 1 Çp 2 p 3 stands for p 1 p 3 Theorem 13 Consider category (ii) from (1), that is, let n 3 be an even integer, let p 1,, p n 1 be odd integers and let p n 0 be an even integer Then the rational Witt class of the pretzel knot P(p 1,, p n ) is ³(P(p 1,, p n )) n Å i1 (s i 1 2 s i 2 3 s i (p i 1) p i )) (p 1 p n ) det P(p 1,, p n ), where s i Sign(p i ) and the determinant det P(p 1,, p n ) can again be computed by the formula det P(p 1,, p n ) n i1 p 1 Çp i p n To state the next theorem we introduce some auxiliary notation first: Let j (t 1,, t m ) denote the degree j (with 0 j m) symmetric polynomial in the variables t 1,, t m For example, 1 (t 1,, t m ) t 1 t m while m (t 1,, t m ) t 1 t m We adopt the convention that 0 (t 1,, t m ) 1 With this in mind, we have Theorem 14 Consider category (iii) from (1) Thus, let n 3 and p 1,, p n be odd integers and let i stand as an abbreviation for the integer i (p 1,, p i1 ) Then the rational Witt class of the pretzel knot P(p 1,, p n ) is given by ³(P(p 1,, p n )) n 2 n 1 Moreover, the determinant of P(p 1,, p n ) equals det P(p 1,, p n ) n 1

7 982 S JABUKA REMARK 15 To put the results of Theorems into perspective, we would like to point out that at the time of this writing, the algebraic concordance orders aren t known yet even for the 3-stranded pretzel knots P(p 1, p 2, p 3 ) from category (i) in (1) The chief reason for this is that this family contains knots with Alexander polynomials of arbitrarily high degree In contrast, the algebraic concordance orders of P(p 1, p 2, p 3 ) coming from category (iii) in (1) are well understood and follow easily from Levine s article [10], see Remark 113 below All non-trivial knots in this family are of Seifert genus 1 ½ 2 ½ 4 13 Applications and examples While Theorems give ³(K ) in terms of the generators of W (É), in concrete cases one can determine ³(K ) as a specific element in W (É) 2 ½ 4 ½ We give a host of examples of this nature next Such computations rely on an understanding of the isomorphism between W (É) and This isomorphism is completely explicit and easily computed, we explain it in some detail in Section 22 For now we merely present the results of our computations, the full details are deferred to Section 5 After presenting a several concrete examples, we turn to general type corollaries of Theorems The ultimate goal of course is to have a set of numerical conditions on n, p 1,, p n which would pinpoint the order of ³(K ) in W (É) The obstacle to achieving this is number theoretic in nature and we have been unable to overcome it in its full generality However, we are able to give such conditions for the case of n 3 and for some special cases when n 4 As we shall see in Section 22, a necessary condition for ³(K ) to be zero in W (É) is that (K ) 0 and det K m 2 for some odd integer m If only the first of these conditions holds, then ³(K ) is at least of order 2 in W (É) With this in mind the next examples testify that the rational Witt classes carry significantly more information than merely the signature and determinant We start with a useful definition DEFINITION 16 If p is an odd integer, we shall say that the knot P(p 1,, p i 1, p, p i,, p j 1, p, p j,, p n ) is gotten from P(p 1,, p n ) by an upward stabilization (or conversely that P(p 1,, p n ) is obtained from P(p 1,, p i 1, p, p i,, p j 1, p, p j,, p n ) by a downward stabilization) EXAMPLE 17 Let K 1, K 2 and K 3 be the knots K 1 P(21, 13, 17, 15, 12), K 2 P( 3, 3, 7, 5, 2), K 3 P( 3, 5, 7, 9, 6) from category (i) and let K K 1 # K 2 # K 3 The (K ) 0 but ³(K ) has order 4 in

8 RATIONAL WITT CLASSES OF PRETZEL KNOTS 983 W (É) Thus K has concordance order at least 4 The same holds if K i is replaced by a knot gotten from K i by any finite number of upward stabilizations EXAMPLE 18 Let K 1 and K 2 be the knots K 1 P(7, 3, 5, 2), K 2 P( 19, 15, 21, 10) from category (ii) and let K K 1 # K 2 Then (K )0 but ³(K ) has order 4 in W (É) and therefore also in concordance group The same is true if K i is replaced by a knot gotten from K i by any finite number of upward stabilizations Let K be a knot obtained by a finite number of upward stabi- EXAMPLE 19 lization from either P( 3, 9, 15, 5 5) or P( 3, 5, 11, 15, 15) from category (iii) Then the signature of K is zero, the determinant of K is a square but ³(K ) 0 ¾ W (É) Consequently, no such K is slice EXAMPLE 110 Let K 1, K 2 and K 3 be the knots K 1 P(21, 13, 17, 15, 12), K 2 P( 19, 15, 21, 10), K 3 P( 15, 7, 7, 13, 11) from the categories (i), (ii) and (iii) and let K K 1 # K 2 # K 3 Then (K ) 0 but ³(K ) is of order 4 in W (É) The same holds under replacement of K i by upward stabilizations The details of the above computations can be found in Section 5 We now turn to more general corollaries of Theorems Theorem 111 Consider a 3-stranded pretzel knot K P(p, q, r) with p, q, r odd Then the order of ³(K ) in W (É) is as follows: ³(K ) is or order 1 in W (É) if and only if det K m 2 for some odd m ¾ ³(K ) is of order 2 in W (É) if and only if det K 0, det K is not a square and no prime Ð 3 (mod 4) divides det K with an odd power ³(K ) is of order 4 in W (É) if and only if det K 0 and there exists a prime Ð 3 (mod 4) dividing det K with an odd power ³(K ) is of infinite order W (É) if and only if det K 0 Recall that det K pq pr qr

9 984 S JABUKA Theorem 112 Consider again K P(p, q, r) but with p, q odd and with r 0 even Then ³(K ) is of finite order in W (É) if and only if or p q 0 p q 2 and det K 0 The order of ³(K ) in W (É) in these cases is as follows: If p q 0 then ³(K ) has order 1 in W (É) If p q 2 and det K 0 then, ³(K ) is of order 1 in W (É) if det K m 2 for some odd integer m ³(K ) is of order 2 in W (É) if det K is not a square and no prime Ð 3 (mod 4) divides det K with an odd power ³(K ) is of order 4 in W (É) if there is a prime Ð 3 (mod 4) that divides det K with an odd power Here too, det K pq pr qr A slightly more general version of this theorem is given in Theorem 62 REMARK 113 As already mentioned in Remark 15, the algebraic concordance orders of the knots P(p, q, r) with p, q, r odd are known by work of Levine [10] and agree with the orders of ³(P(p, q, r)) in W (É) The analogues of the results of Theorem 112 are not known for the algebraic concordance group However, according to Theorem 116 below, it is clear that when r is even, the order of ³(P(p, q, r)) in W (É) and the order of P(p, q, r) in C 1 are different in general We point the interested reader towards [14] for a discussion of finite order elements in C 1 REMARK 114 The condition on the congruency class mod 4, appearing in both Theorems 111 and 112, is reminiscent of a similar condition appearing in a beautiful (and much stronger) theorem by Livingston and Naik [13]: If K is a knot with det K Ð where Ð is a prime congruent to 3 mod 4 and gcd(ð, ) 1, then K has infinite order in the topological concordance group Theorem 115 Consider a pretzel knot K P(p 1,, p n ) from category (i) in (1), ie assume that n, p 1,, p n 1 are odd, n 3 and p n 0 is even Additionally, suppose that the p 1,, p n 1 are all mutually coprime Then ³(K ) 0 ¾ W (É) if and only if (K ) 0 and det K m 2 for some odd m ¾ Seeing as the torsion subgroups of C 1 and W (É) are isomorphic, one can t help but speculate whether ³ Tor(C1 )Ï Tor(C 1 ) W (É) is injective Unfortunately this is not the case as the next theorem testifies

10 RATIONAL WITT CLASSES OF PRETZEL KNOTS 985 Theorem 116 Consider the knot K P(5, 3, 8) All Tristram Levine signatures (K ) vanish but K is not trivial in C 1 On the other hand, the rational Witt class ³(K ) is zero Thus, K is a nontrivial element of Ker(³)Tor(C 1 ) REMARK 117 We would like to point out that for knots K with 10 or fewer crossings, K is algebraically slice if and only if ³(K ) is zero in W (É) This follows by inspection, using KnotInfo [2] 2, and relying on the fact that if ³(K )0 then (K )0 and det K m 2 As a byproduct of our computations we obtain closed formulae for the signature and determinants of all pretzel knots The formulae for the determinants have already been stated in Theorems 12 14, the signature formulae are the content of the next theorem While these are not directly relevant to our discussion, we list them here in the hopes that they may be useful elsewhere Theorem 118 Let K P(p 1,, p n ) be a pretzel knot from either of the 3 categories (i) (iii) from (1) As usual, we assume that n 3 Then the signature (K ) of K can be computed as follows: 1 If n, p 1,, p n 1 are odd and p n 0 is even, then (K ) n 1 i1 Sign(p i ) (p i 1) Sign(det P(p 1,, p n 1 ) det P(p 1,, p n )) Sign(p 1 p n 1 det P(p 1,, p n 1 )) The determinants det P(p 1,, p n ) and det P(p 1,, p n 1 ) are computed as in Theorem 12 2 If n, p n are even, p 1,, p n 1 are odd and p n 0, then (K ) n i1 Sign(p i ) (p i 1) where det P(p 1,, p n ) is as computed in Theorem 13 3 If n, p 1,, p n are all odd, then (K ) n 1 i1 where i i (p 1,, p i1 ) as in Theorem 14 Sign(p 1 p n det P(p 1,, p n )), Sign( i 1 i ), 2 A web site created by Chuck Livingston and maintained by Chuck Livingston and Jae Choon Cha The site contains a wealth of information about knots with low crossing number It can be found at

11 986 S JABUKA For example, if K P(p 1,, p n ) with n, p 1,, p n odd and p i 0 for all i, then i 0 for all i also and therefore (K ) n 1 As another example consider the case of n, p n even and p 1,, p n 1 odd and again p i 0 for all i Then (K ) n 1 (p 1 p n ) 14 Organization Section 2 provides background on the three flavors of algebraic concordance groups C 1, I(É) and W (É) encountered in the introduction The relationships between these groups are also made more transparent In Section 3 the first steps towards computing ³(P(p 1,, p n )) are taken in that specific Seifert surfaces are picked for the knots along with specific bases for their first homology These choices allow us to determined a linking matrix for the knots Section 4 explains how one can diagonalize the linking matrices found in Section 3, leading to proofs of Theorems 12, 13 and 14 More detailed versions of these theorems are provided in Theorems 48, 411 and 413 respectively Section 5 is devoted to computations of examples and shows how Theorems imply the results from Examples stated above The final section provides proofs for Theorems 111, 112, 115 and Algebraic concordance groups In this section we describe the three algebraic concordance groups mentioned in the introduction, namely C 1 The algebraic concordance group of classical knots in S 3, I() The concordance group of isometric structures over the field, W () The Witt ring of non-degenerate, symmetric, bilinear forms over We provide a generous amount of details of the constructions of these groups but we omit proofs The interested reader may consult [1, 4, 7, 11, 19] for more details and additional background 21 The algebraic concordance group C 1 This section largely follows the exposition from [9] with a slight bias towards a coordinate free description Our explanation of the algebraic concordance group C 1 runs largely in parallel to the description of the Witt ring W (É) from the introduction Thus, we shall consider pairs (,, L) where L is a finitely generated free Abelian group of even rank and, Ï L L is a bilinear pairing with the property that,, is unimodular Following Levine [10], we shall call such pairs admissible pairs Here, denotes the bilinear form x, y y, x Note that, is not required to be symmetric nor non-degenerate We will say that (,, L) is metabolic or totally isotropic if there exists a splitting L L 1 L 2 with rk L 2(rk L 1 ) and, L1 L 1 0 We shall add pairs (, 1, L 1 ) and (, 2, L 2 )

12 RATIONAL WITT CLASSES OF PRETZEL KNOTS 987 by direct summing them, ie (, 1, L 1 ) (, 2, L 2 ) (, 1, 2, L 1 L 2 ) With these definitions understood, we define the algebraic concordance group C 1 to be the set of pairs (,, L) as above, up to the equivalence relation by which if and only if (, 1, L 1 ) (, 2, L 2 ) (, 1, L 1 ) (, 2, L 2 ) is metabolic We shall refer to this equivalence relation as that of algebraic concordance Under the operation of direct summing, C 1 becomes an Abelian group An easy check reveals that the inverse of (,, L) is (,, L) The group C 1 was introduced by Jerry Levine in [9] and its isomorphism type was completely determined by him in [10] The relation of C 1 to knot theory is as follows: Let K be a knot in S 3 and let S 3 be an oriented genus g Seifert surface for K We shall view the orientation on as being given by an normal nowhere vanishing vector field n on Recall from the introduction that the linking pairing lkï H 1 (Á ) H 1 (Á ) is defined by lk(x, y) linking number of x and y, where, by a customary blurring of viewpoints, we interpret x and y as simple closed curves on With this in mind, y is a small push-off of y in the normal direction of determined by n It is well known (see eg [17]) that (lk, H 1 (Á )) is an admissible pair and therefore the assignment (K, ) (lk, H 1 (Á )) ¾ C 1 is well defined As Levine shows in [9], the algebraic concordance class of (lk, H 1 (Á)) is independent of and by abuse of notation, we shall denote it simply by K, hoping that no confusion will arise Levine also shows that if K 1 and K 2 are (geometrically) concordant as knots then their linking forms are algebraically concordant This statement applies to both smooth and topological (geometric) concordance 22 The Witt ring over the field For an excellent introduction to Witt rings we advise the reader to consult [8], but see also [5] and [18] The first half of this section is a re-iteration of the description for the Witt ring W (É) over the rational numbers extended to arbitrary fields Let be a field and consider pairs (,, V ) where V is a finite dimensional -vector space and, Ï V V is a symmetric, non-degenerate bilinear pairing By non-degenerate we mean that the map Ú, Ú provides an isomorphism from V to V We call a pair (,, V ) metabolic or totally isotropic if there exists a subspace W V with dim V 2 dim W and such that, W W 0 As in the

13 988 S JABUKA case of É, we define addition of (, 1, V 1 ) and (, 2, V 2 ) by direct sum (, 1, V 1 ) (, 2, V 2 ) (, 1, 2, V 1 V 2 ), and we proceed to define the equivalence relation (, 1, V 1 ) (, 2, V 2 ) to mean that (, 1, V 1 ) (, 2, V 2 ) is metabolic The set of equivalence classes of pairs (,, V ) is denoted by W () and called the Witt ring of It becomes an Abelian group under the direct sum operation and a commutative ring with the operation of multiplication given by tensor products (, 1, V 1 )Å(, 2, V 2 ) (, 1, 2, V 1 Å V 2 ) The Witt ring W () was introduced by Witt in [20] and has found renewed prominence in the theory of quadratic forms over fields through the work of Pfister (see for example [15, 16]) As is usual in the literature, we will denote 0 by È Let us recall the notation a already used in the introduction: Given a ¾ È we let a denote the non-degenerate symmetric bilinear form (, a, ) specified by 1, 1 a a Note that (2) a a b 2 ¾ W (), b ¾ È and b b 0 ¾ W (), b ¾ È The first of these follows from the fact that fï (a, ) (a b 2, ) given by f (x) x b is an isomorphism of forms The second form is clearly metabolic and thus zero in W () These harmless observations are incredibly useful in computations and we will rely on them substantially in our sample calculations in Section 5 With this notation in mind, the next theorem can be found in [8] Theorem 21 Let, be a non-degenerate symmetric bilinear form on a finite dimensional -vector space V of dimension n Then there exist scalars d 1,, d n ¾ È such that, d 1 d n ¾ W () Said differently, W () is generated by the set a a ¾ È A presentation of W () (as a commutative ring) is obtained from these generators along with the relators (R1) 1 1, (R2) a b a b, a, b ¾ È, (R3) a b (1a b) a b, a, b ¾ È In other words, W () is isomorphic to quotient of the free commutative ring generated by the set a a ¾ È moded out by the ideal generated by elements of the form as in (R1) (R3) In (R1), the symbol 1 denotes the multiplicative unit of W ()

14 RATIONAL WITT CLASSES OF PRETZEL KNOTS 989 REMARK 22 We shall adopt the use of the symbol as the inverse operation of addition in W () For example, the relations (R1) (R3) from the preceding theorem can be rewritten with the sign as (R1) 1 1, (R2) a b a b, a, b ¾ È, (R3) a b ab(a b) a b, a, b ¾ È With this understood, we turn to studying some specific Witt rings We will chiefly be interested in the cases where is either É or Ð where the latter will be our notation for the finite field Ð of characteristic Ð 2 The next result can again be found in [8] and also in [5] Theorem 23 groups Let Ð ¾ be a prime Then there are isomorphisms of Abelian W ( Ð ) 2 Á Ð 2, 2 2 Á Ð 1 (mod 4), 4 Á Ð 3 (mod 4) The generators of 2 W ( 2 ) and of 4 W ( Ð ) with Ð 3 (mod 4) are given by 1 while the two copies of 2 in W ( Ð ) in the case when Ð 1 (mod 4) are generated by 1 and a where a ¾ È È 2 is any non-square element The origins of the proof of the next theorem go back to Gauss work on quadratic reciprocity, it was re-discovered by Milnor and Tate [5] Theorem 24 There is an isomorphism of Abelian groups Ï W (É) ¼ Å Ð¾Æ Ðprime ½ W ( Ð ), where Ï W (É) is the signature function while Ï W (É) ÐW ( Ð ) is the direct sum of homomorphisms Ð Ï W (É) W ( Ð ) (with Ð ranging over all primes) described on generators of W ( Ð ) as follows: Given a rational number 0, write it as Ð l where l is an integer and a rational number whose numerator and denominator are relatively prime to Ð Then (3) Ð (Ð l ) 0Á l is even, Á l is odd Corollary 25 As an Abelian group, W (É) is isomorphic to ½ 2 ½ 4

15 990 S JABUKA 23 The concordance group of isometric structures For more details on this section, see [10] Let be a field, then an isometric structure over is a triple (,, T, V ) consisting of a non-degenerate symmetric bilinear form (,, V ) and a linear operator TÏ V V which is an isometry with respect to,, ie T Ú, T Û Ú, Û for all Ú, Û ¾ V A triple (,, T, V ) shall be called metabolic or totally isotropic if there is a half-dimensioinal T -invariant subspace W V for which, W W 0 Much as in the case of the algebraic concordance group C 1 and the Witt ring W (), isometric structures too are added by direct sum We define two triples (, 1, T 1, V 1 ) and (, 2, T 2, V 2 ) to be equivalent if (, 1, T 1, V 1 ) (, 2, T 2, V 2 ), is metabolic With these definitions understood, we define the concordance group of isometric structures I() as the set of equivalence classes of triples (,, T, V ) as above Not surprisingly, I() becomes an Abelian group under the operation of direct summing 24 Maps between the algebraic concordance groups Having defined C 1, W () and I(), we turn to describing some natural maps between them in the case when É We start by a lemma proved by Levine in [10] Lemma 26 Let (,, L) be an admissible pair (as in Section 21) Then there exists an admissible pair (, ¼, L ¼ ) algebraically concordant to (,, L) and such that, Ï ¼ L ¼ L ¼ is a non-degenerate bilinear form With this in mind, consider an admissible non-degenerate pair (,, L) Given any basis B «1,, «n of L, let A be the matrix representing,, that is, set a i, j «i, «j and let A [a i, j ] We define the maps ³Ï C 1 W (É), ³ 1 Ï C 1 I(É) and Ï I(É) W (É) as in [10] ³(,, L) (,,, L Å É), ³ 1 (,, L) (A A, A 1 A, L Å É), (,, T, V ) (,, V ) It is not hard to verify that the definition of ³ 1 is independent of the choice of the basis B of L It is also easy to verify that, with respect to B, the matrix A 1 A defines an isometry on L Å É Is should be clear that ³ Æ ³ 1, as already pointed out in the introduction We leave it as an (easy) exercise for the reader to check that these maps are well defined This requires one to show that metabolic elements from any one group map to metabolic elements in the other groups

16 RATIONAL WITT CLASSES OF PRETZEL KNOTS 991 We conclude this section by reminding the reader of the isomorphism types of C 1, W (É) and I(É) stated in the introduction: C 1 ½ ½ 2 ½ 4 ³ 1 Ã I(É) ½ ½ 2 ½ 4 ³ Ã W (É) ½ 2 ½ 4 Ã As already mentioned, Levine showed ³ 1 to be injective Clearly injectivity cannot hold for ³ However, given the above diagram, one cannot help but ask: How much loss of information is there if one restricts ³ to the torsion subgroup of C 1? As Theorem 116 shows, the restriction of ³ to the torsion subgroup of C 1 is unfortunately not injective Nevertheless, Examples show that ³ Tor(C1 ) contains significantly more information than just the knot determinant 3 The linking matrices In this section we compute the linking matrix for K P(p 1,, p n ) associated to a choice of oriented Seifert surface for K along with a concrete basis for H 1 (Á ) The details of these computations for the three cases (i) (iii) from (1) proceed in slightly different manners 31 The case of n, p 1,, p n 1 odd and p n even For the remainder of this subsection, we shall assume the conditions from its title with the additional constraints that n 3 and p n 0 We start by recalling Fig 1 in which we chose a particular projection for the pretzel knot P(p 1,, p n ) We choose 1 to be the Seifert surface for K obtained from that projection via Seifert s algorithm (see for example [17]) Specifically, 1 consists of n 1 disks D 1,, D n 1 of which D i and D i1 are connected with p i bands, each carrying a single half-twist whose handedness is determined by the sign of p i (in that the band obtains a right-handed twist if p i 0 and a left-handed twist if p i 0) The disks D n 1 and D 1 are similarly connected with p n 1 bands Finally, there is a band with p n half-twists (right-handed if p n 0 and left-handed if p n 0) both of whose ends are attached to D 1 Note that the genus of 1 is p 1 p 2 p n 1 3 n We label the bands connecting D i to D i1 by B1 i,, Bi p i and we label those connecting D n 1 to D 1 by B n 1 1,, B n 1 p n 1 The unique band with p n twists is labeled B n All of our conventions and labels are illustrated in Fig 2 With these preliminaries in place, we choose our basis (4) B 1 «1 1,, «1 p 1 1, «2 1,, «2 p 2 1,, «n 1 1,, «n 1 p n 1 1,, Æ for H 1 ( 1 Á ) in the following way:

17 992 S JABUKA 1 We let «i j to be the simple closed curve passing through the bands B i 1 and Bi j1 2 We pick to be the simple closed curve passing over the bands B 1 1, B2 1,, Bn The remaining curve Æ passes once through the band B n These curves, along with our orientation conventions, are also depicted in Fig 2 The orientation of 1 is determined by the normal vector field which points outwards from the page (and towards the reader) on all disks D 1, D 3, D 5, and into the page (and away from the reader) on the disks D 2, D 4, D 6, These conventions are indicated by the symbols and respectively in Fig 2 With these definitions in place, we are ready to start computing entries in the linking matrix L [l i, j ] where l i, j lk(x i, x j ) Here x i is the i-th element of the basis B 1 and lk(x i, x j ) is the linking number of x i and x j The latter is a small push-off of x j in the direction of the normal vector field on 1 determined by its orientation, as already previously indicated Seeing as the loops «k i and «m j are disjoint for any choice of i j, we find that ) 0 for any choices of i, j, k, m with i j For the same lk(«k i, «m) j lk(«m, j «k i reason, one also obtains lk(«k i, Æ) lk(æ, «i k ) 0 for any choices of i, k The contribution of the subset «i 1,, «i p i 1 of B 1 to the linking form L, only depends on p i To see how, let us introduce the n n matrices X n and Y n X n X n by the formulae (5) X n ¾ By consulting Fig 2, one finds that (6) and Y n ¾ lk(«k i, «i m ) 0Á k m 1Á k m if p i 0 and i is even, lk(«i k, «i m ) 1Á k m 0 k m if p i 0 and i is even, lk(«k i, «i m ) 1Á k m 0Á k m if p i 0 and i is odd, lk(«k i, «i m ) 0Á k m 1Á k m if p i 0 and i is odd

18 RATIONAL WITT CLASSES OF PRETZEL KNOTS 993 Fig 2 Our choice of Seifert surface 1 for P(p 1,, p n ) for the case when n, p 1,, p n 1 are odd and p n is even This example shows the knot P( 1, 3, 5, 3, 4) The choices of generators for H 1 ( 1 Á ) along with their orientations are indicated

19 994 S JABUKA Fig 3 This figure computes the linking lk(«1 i, «i 2 ) when i is even The two push-offs «i, 1 and «i, 2 of «1 i and «i 2 respectively, are shown in the bottom two pictures The linking of the two is readily computed from these The case of p i 0 and i even is singled out in Fig 3 From this we find that L i, the restriction of the linking form L to the Span(«1 i,, «i p i 1 ), with respect to the basis «1 i,, «i p i 1 takes on one of 4 possible forms: L i X pi 1Á if p i 0 and i is even, X p i 1Á if p i 0 and i is even, X p i 1Á if p i 0 and i is odd, X pi 1Á if p i 0 and i is odd

20 RATIONAL WITT CLASSES OF PRETZEL KNOTS 995 Even so, the matrix representing L i Li be expressed with a single relation as in each of the four cases above, can then (7) L i L i Sign(p i )Y pi 1 Having worked out all of the linking numbers lk(«i k, «j m), we now turn to exploring how and Æ contribute to L Their linking numbers with the various other curves from the basis B 1 are easily read off from Fig 2: lk(, ) 1 2 (Sign(p 1)Sign(p 2 ) Sign(p n 1 )), (8) lk(, Æ) 0, lk(æ, ) 1, lk(æ, Æ) p n 2, while the linking numbers of with the various «i k are (9) lk(, «k i ) lk(«k i, ) 1Á if p i 0 and i is even, 0Á if p i 0 and i is even, 0Á if p i 0 and i is odd, 1Á if p i 0 and i is odd, 0Á if p i 0 and i is even, 1Á if p i 0 and i is even, 1Á if p i 0 and i is odd, 0Á if p i 0 and i is odd As earlier, we see that while lk(«k i, ) and lk(, «i k ) depend on a number of cases, the quantity lk(, «k i ) lk(«i k, ) always equals Sign(p i) We are thus in a position to assemble all the pieces Theorem 31 Let n, p 1,, p n 1 be odd integers with n 3 and let p n 0 be an even integer To keep notation below at bay, let us also introduce the abbreviations s i Sign(p i ), s s 1 s n 1, i p i 1 Then the symmetrized linking form LL of the pretzel knot P(p 1,, p n ) associated to the oriented Seifert surface 1 and the basis B 1 «1 1,, «1 p 1 1, «2 1,, «2 p 2 1,, «n 1 1,, «n 1 p n 1 1,, Æ

21 996 S JABUKA of H 1 ( 1 Á ) as chosen above (see specifically Fig 2), has the form LL ¾ s 1 0 s 1 Y s 1 0 s s 2 Y 2 0 s 2 0 s n s n 1 Y n 1 s n 1 0 s 1 s 1 s 2 s 2 s n 1 s n 1 s p n The matrices Y are as introduced in (5) 32 The case of n even, p 1,, p n 1 odd and p n even We turn to the next case of choice of parities of n, p 1,, p n and pick it for the remainder of this section to be as listed in the title We also keep our additional assumptions of n 3 and p n 0 The Seifert surface 2 that we choose for P(p 1,, p n ) and the preferred basis B 2 for H 1 ( 2 Á ) are very much like in the case considered in Section 31 Specifically, we let 2 be obtained from 1 ( 1 is the Seifert surface from Section 31) by simply deleting its unique band B n with and even number of half-twists and allowing the number of bands which connect the disks D n and D 1 to be an even number, namley p n We then arrive at a surface 2 as in Fig 4 The same figure also indicates our choice of basis B 2 «1 1,, «1 p 1 1, «2 1,, «2 p 2 1,, «n 1,, «n p n 1, for H 1 ( 2 Á ) which is identical to B 1 from (4) safe that we are presently no longer requiring the generator Æ The orientation convention is as in the previous section and is again indicated by a and in Fig 4 The linking numbers between the various «i k and «j m and indeed between the «i k and are identical to those found in Section 31 We thus immediately arrive at the analogue of Theorem 31: Theorem 32 Let n 3 be an even integer and let p 1,, p n 1 be odd integers and p n 0 an even integer Let us re-introduce the abbreviations s i Sign(p i ), s s 1 s n 1, i p i 1

22 RATIONAL WITT CLASSES OF PRETZEL KNOTS 997 Fig 4 Our choice of Seifert surface 2 for P(p 1,, p n ) for the case when n is even, p 1,, p n 1 are odd and p n is even Our example shows the knot P(3, 5, 3, 2) The choices of generators for H 1 ( 2 Á ) with their orientations are indicated

23 998 S JABUKA Then the symmetrized linking form LL of the pretzel knot P(p 1,, p n ) associated to the oriented Seifert surface 2 and the basis B 2 «1 1,, «1 p 1 1, «2 1,, «2 p 2 1,, «n 1,, «n p n 1, of H 1 ( 2 Á ) as chosen above (see specifically Fig 4), takes the form LL ¾ s 1 s 1 Y s 1 s2 0 s 2 Y 2 0 s s n Y n s n s 1 s 1 s 2 s 2 s n s n s The matrices Y are again as defined in (5) 33 The case of n and p 1,, p n odd In this section we consider the remaining case where all of n, p 1,, p n are odd with n 3 We start by picking a Seifert surface 3 for P(p 1,, p n ) which is this time obtained by taking two disks and connecting them by n bands B 1,, B n, each with p i half twists (right-handed twists if p i 0 and left-handed twists if p i 0) The thus obtained surface looks as in Fig 5 We next choose a basis B 3 «1,, «n 1 of H 1 ( 3 Á ) by letting «i be the curve on 3 which runs through the bands B i and B i1 The orientation conventions for the «i and indeed the orientation for 2 itself (indicated again by a and a ) are depicted in Fig 5 The linking form in this basis is rather easy to determine Note first that lk(«i, «j )0 whenever i j 2 On the other hand, by inspection from Fig 5, it follows that lk(«i, «i ) p i p i1 2, lk(«i, «i1 ) p i1 1 2 s n, lk(«i1, «i ) p i1 1 2 With this in place, here is the analogue of Theorems 31 and 32 for the present case Theorem 33 Let n 3 be an odd integer and let p 1,, p n be any odd integers Then the symmetrized linking form LL of the pretzel knot P(p 1,, p n ) associated

24 RATIONAL WITT CLASSES OF PRETZEL KNOTS 999 Fig 5 Our choice of Seifert surface 3 for P(p 1,, p n ) for the case when n and p 1,, p n are odd This example shows the knot P(5, 3, 3, 3, 1) The choices of generators for H 1 ( 3 Á) with their orientations are indicated to the oriented Seifert surface 3 and the basis B 3 «1,, «n 1 of H 1 ( 3 Á ) as chosen above (see Fig 5) takes the form LL ¾ p 1 p 2 p p 2 p 2 p 3 p p 3 p 3 p 4 p p n 2 p n 2 p n 1 p n p n 1 p n 1 p n 4 Diagonalizing the linking matrices In this section we show how one can diagonalize the matrices L L obtained in Theorems 31, 32 and 33 We do this essentially using the Gram Schmidt process on (,, H 1 (Á É)) with x, y lk(x, y)lk(y, x) We need to exercise a bit of care since, while, is non-degenerate, it is by no means definite and square zero vectors do exist

25 1000 S JABUKA Once L L has been diagonalized, it is an easy matter to read off the rational Witt class of LL in terms of the generators of W (É) 41 The Gram Schmidt procedure and reduction We start by reminding the reader of the Gram Schmidt process on an arbitrary finite dimensional inner product space (,, V ) By convention, such an inner product, is assumed to be positive definite We then address the issue of square zero vectors in (,, H 1 (Á É)) Theorem 41 (Gram Schmidt) Let f 1,, f n be a basis for the inner product space (,, V ) and let e 1,, e n be the set of vectors obtained as e 1 f 1, e 2 f f 2, e 1 2 e 1, e 1 e 1, e 3 f f 3, e 2 3 e 2, e 2 e f 3, e 1 2 e 1, e 1 e 1, e n f n f n, e n 1 e n 1, e n 1 e n 1 f n, e 1 e 1, e 1 e 1 Then e 1,, e n is an orthogonal basis for V and Spane 1,, e i Span f 1,, f i for each i n REMARK 42 In order to keep the scalars in our computations integral (rather than rational and non-integral), we will often use the slightly modified Gram Schmidt process by which we set e i d i f i f i, e i 1 e i 1, e i 1 e i 1 f i, e 1 e 1, e 1 e 1 where d i is some common multiple of e 1, e 1,,e i 1, e i 1 Clearly the conclusions of Theorem 41 remain valid for the set e 1,, e n for any choice of d i 0 The next theorem addresses the failure of the Gram Schmidt procedure in the presence of square zero vectors (on non-definite inner product spaces) The result should be viewed as an iterative prescription to be applied as many times in the Gram Schmidt process as is the number of square zero vectors e i encountered Theorem 43 Let (,, V ) be a pair consisting of a finite dimensional -vector space V and a non-degenerate bilinear symmetric form, Let f 1,, f n be a basis for V and let, for some m n, e 1,, e m be obtained from f 1,, f n as in Theorem 41 (or alternatively as in Remark 42) Assume that e i, e i 0 for all,

26 RATIONAL WITT CLASSES OF PRETZEL KNOTS 1001 i m but that e m, e m 0 Additionally, suppose also that e m, f m1 0 (which can always be achieved by a simple reordering, if necessary, of f m1,, f n ) Then (,, V ) is equal to (, ¼, V ¼ ) in the Witt ring W () where V ¼ Span(e 1,, e m 1, f ¼ m2,, f ¼ n ) and, ¼, V ¼ V ¼, with f ¼ m1 m 1 f m1 j1 f ¼¼ mk f mk m 1 j1 f ¼ mk f ¼¼ f ¼¼ mk f m1, e j e j, e j e j, f mk, e j e j, e j e j, mk, e m fm1 ¼, e m f ¼ m1 fmk ¼¼, f m1 f ¼ m1 ¼, e m f ¼¼ mk, e m fm1 ¼, f m1 ¼ fm1 ¼, e m fm1 ¼, e e m m, where the last two equations are valid for k 2 Proof Let A be the symmetric non-degenerate n n matrix representing, with respect to the basis e 1,, e m 1 e m, f m1,, f n Then A is of the form ¾ e 1, e e 1, f m1 e 1, f n A 0 e m 1, e m 1 0 e m 1, f m1 e m 1, f n e m, f m1 e m, f n f m1, e 1 f m1, e m 1 f m1, e m f m1, f m1 f m1, f n For k 1, let f ¼¼ mk f n, e 1 f n, e m 1 f n, e m f n, f m1 f n, f n be given by f ¼¼ mk m 1 f mk j1 f mk, e j e j, e j e j, so that f ¼¼ mk, e i 0 for all k 1 and all i m 1 Thus the matrix A ¼¼ representing

27 1002 S JABUKA LL with respect to the basis e 1,, e m 1 e m, fm1 ¼¼,, f n ¼¼ looks like A ¼¼ ¾ e 1, e 1 0 ¾ 0 e m 1, e m 1 0 e m, f m1 e m, f n f m1, e m fm1 ¼¼, f m1 ¼¼ fm1 ¼¼, f ¼¼ f n, e m fn ¼¼, f ¼¼ m1 fn ¼¼, f ¼¼ Note that e m, fmk ¼¼ e m, f mk for all k 1 To simplify the second summand, we introduce a further change of basis by setting mk, e m f ¼ mk f ¼¼ f ¼¼ mk f ¼¼ fmk ¼¼, f ¼¼ m1, e m f ¼¼ m1 m1 fm1 ¼¼, e m fmk ¼¼, e m f ¼¼ fm1 ¼¼, e m fm1 ¼¼, e m n n m1, f ¼¼ m1 for all k 2 and for convenience, set fm1 ¼ fm1 ¼¼ A quick check reveals that now f ¼ mk, e m 0 and f ¼ mk, f m1 ¼ 0, k 2 Therefore the second summand of A ¼¼ above, when expressed with respect to the basis e m, fm1 ¼ f¼ m2,, f n ¼, takes the form 0 e m, f m1 f m1, e m f ¼ m1, f ¼ m1 ¾ f ¼ m2, f ¼ m2 f ¼ m2, f ¼ n f ¼ n, f ¼ m2 f ¼ n, f ¼ n Since the first summand is metabolic and therefore equals zero in W (), the claim of the theorem follows We shall refer to the passage from (,, V ) to (, ¼, V ¼ ), as described in Theorem 43, as reduction, seeing as the dimension of V gets reduced by 2 in the process 42 The case of n, p 1,, p n 1 odd and p n even, revisited The goal of this subsection is to diagonlize the symmetrized linking matrix L L obtained in Theorem 31 Specifically, we want to find a regular matrix P of the same dimension as LL such that P (LL )P is a diagonal matrix By way of shortcut of notation, we will write x, y to denote lk(x, y)lk(y, x) e m

28 RATIONAL WITT CLASSES OF PRETZEL KNOTS 1003 As the matrix L L from Theorem 31 consists of a number of matrix blocks of the form Y m (see (5) for the definition of Y m ), we first take the time to apply the Gram Schmidt process to the latter We let P m denote the upper triangular m m matrix given by (10) P m ¾ m m Lemma 44 Consider the inner product space (,, m ) where the inner product, with respect to the standard basis «1,, «m of m is given by «i, «j (i, j)-th entry of the matrix Y m from (5) Then defining a 1 «1 and a i i«i «i 1 «i 2 «1 for 2 i m, yields an orthogonal basis for (,, m ) with a i, a i i(i 1) Said differently, 3 P m Y m P m Diag(1 2, 2 3, 3 4,, m (m 1)) Proof This is a straightforward application of the Gram Schmidt process Let a i be as stated in the lemma and assume that a 1,, a i is an orthogonal set for all i k m with the stated squares a i, a i i(i 1) (the case of i 1 being clearly true) We prove that the statement remains true if i is chosen to be k Note that «k, a i «k, i«i «i 1 «1 i , for any choice of i k Using the Gram Schmidt process gives a k ««k, a k 1 k a k 1, a k 1 a k 1 «k, a 1 a 1, a 1 a 1 1 «k (k 1)k ((k 1)«k 1 «k 2 «1 ) «1 1 k (k«k «k 1 «k 2 «1 ) Proceeding as in Remark 42, we let a k be equal to a k k«k «k 1 «k 2 «1 3 Here and in the remainder of the article, we let Diag(x 1, x 2,, x m ) denote the m m square matrix whose off-diagonal entries are zero and whose diagonal entries are given by x 1,, x m

29 1004 S JABUKA which already showes that a 1,, a k is orthogonal To complete the proof of the lemma, we need to compute a k, a k : a k, a k k«k «k 1 «1, k«k «k 1 «1 which is as claimed k 2 «k, «k 2k«k, «k 1 «1 k 1 i1 2k 2 2k(k 1)2(k 1)((k 1) 2 (k 1)) k(k 1), We proceed by defining vectors a i k as a i k k«i k «i k 1 «i 1, «i, «i k 1 i, j1 i j «i, «j for each i 1,, n 1, and where the various «i k are the elements of the basis B 1 defined in (4) from Subsection 31 Lemma 44 then shows that for each such index i, the set a i 1,, ai p i 1 is an orthogonal set with respect to, L L and a i k, ai k Sign(p i )k(k 1) Moreover, since «i k, «j m 0 whenever i j, we see that in fact the set (11) B 1,prelim a 1 1,, a1 p 1 1, a2 1,, a2 p 2 1,, an 1 1,, a n 1 p n 1 1 is also an orthogonal set We then turn to finding two additional vectors (related to and Æ), which we shall label X and Y, needed to complete B1,prelim to an orthogonal basis for B 1 for H 1 ( 1 Á É) We find X using again the Gram Schmidt process Lemma 45 Setting X equal to X p 1 p n 1 n 1 i1 n 1 k1, ki p k p i 1 makes the set B 1,prelim X an orthogonal set Additionally, the square of X is k1 X, X (p 1 p n 1 ) n 1 p 1 Çp i p n 1 i1 «i k,

30 RATIONAL WITT CLASSES OF PRETZEL KNOTS 1005 Proof An easy induction argument on p i shows that p i 1 k1, a i k a i k, ai k ai k 1 p i («i 1 «i 2 «i p i 1 ) Letting X be given by the Gram Schmidt formula applied to the linearly independent set B 1,prelim, ie X n 1 i1 p i 1 k1 leads, in conjunction with the preceding formula, to X n 1 i1, a i k a i k, ai k ai k, p i 1 1 p i To keep coefficients integral (see Remark 42) we multiply the right-hand side of the above by p 1 p n 1 and set X instead equal to X p 1 p n 1 n 1 i1 k1 n 1 k1, ki «i k p k p i 1 as in the statement of the lemma Thus, B 1,prelim X is indeed an orthogonal set We next compute X, X: X, X (p 1 p n 1 ) 2, n 1 2 n 1 i1 (p 1 p n 1 ) 2 p i i1 ki p i 1, k1 p k 2 pi 1 k1 «i k, p i 1 «k i, In the second term of the right-hand side, we relied on the fact that «i k, «j l 0 whenever i j Using the linking form L from Theorem 31, it is easy to see that (for example by induction on p i ) pi 1 k1 p i 1 «k i, k1 p i 1, k1 k1 «i k «i k Sign(pi )p i (p i 1), «i k Sign(pi )(p i 1), k1 «i k

31 1006 S JABUKA which in turn shows that n 1 i1 ki n 1 n 1 i1 i1 p k 2 pi 1 k1 p i 1 «k i, k1 «i k 2 n 1 i1 Sign(p i ) (p 1 p n 1 ) 2 n 1 (p p i i 1)2 Sign(p i ) (p 1 p n 1 ) 2 (p p i i 1) p 1 p n 1 n 1 i1 Sign(p i ) ki (p 1 p n 1 ) 2 p i 1, p i i1 p k (p i 1) k1 «i k Sign(p i ) (p 1 p n 1 ) 2 p i (p i 1) Finally, recalling (see Theorem 31) that, (Sign(p 1 ) Sign(p n 1 )), we are able to assemble all the pieces to compute X, X: 1 p 1 p n 1 X, X p 1 p n 1 (Sign(p 1 ) Sign(p n 1 )) n 1 i1 n 1 i1 Sign(p i ) ki Sign(p i ) ki p k (p i 1) p k, and so X, X (p 1 p n 1 ) 2 1 p 1 1 p n 1, as claimed in the statement of the lemma In the final step, we would like to find a vector Y ¾ H 1 ( 1 Á É) such that B 1,prelim X, Y is an orthogonal basis While a i k, ai k 0 for any choice of i, k, and thus the Gram Schmidt process worked well for finding X, it is possible, and it does happen, that X, X 0 This of course obstructs us from finding Y by means of the Gram Schmidt process, calling instead for an application of Theorem 43 We proceed by treating the two cases X, X 0 and X, X 0 separately Lemma 46 Let X ¾ H 1 ( 1 Á É) be as defined in Lemma 45 and assume that X, X 0 Define Y ¾ H 1 ( 1 Á É) as Y p 1 p n 1 1 p 1 1 Æ p n 1 X

32 RATIONAL WITT CLASSES OF PRETZEL KNOTS 1007 Then B 1 B1,prelim X, Y is an orthogonal basis and Y, Y n 1 n p 1 Çp i p n 1 p 1 Çp i p n i1 i1 Proof to find Y as Our assumption X, X 0 allows us to use the Gram Schmidt process Y Æ Æ, X n 1 X, X X i1 p i 1 k1 Æ, a i k a i k, ai k ai k Since Æ, «k i 0 for all i, k it follows that Æ, ak i 0 also, reducing the above formula to Æ, Y X Æ X, X X With X, X already computed in Lemma 45, the same lemma (using also the result of Theorem 31) implies that showing that Æ, X p 1 p n 1, Y Æ 1 p 1 p n 1 (1p 1 1p n 1 ) X To keep our coefficients integral (see Remark 42) we instead set showing that B1 remains to calculate Y, Y: Y p 1 p n 1 1 p 1 1 Æ p n 1 X, B1,prelim X, Y is an orthogonal basis for (,, H 1 ( 1 Á É)) It Y, Y(p 1 p n 1 ) 2 1 p 1 1 p n 1 2Æ, Æ 2p 1 p n 1 1 p 1 1 p n 1 Æ, XX, X (p 1 p n 1 ) 2 1 p 1 1 p n 1 2 p n 2(p 1 p n 1 ) 2 1 p 1 1 p n 1 (p 1 p n 1 ) 2 1 p 1 1 (p 1 p n 1 ) 2 1 p 1 1 p n 1 1 p 1 1 p n 1 p n 1 p n 1