The rational Witt class and the unknotting number of a knot
TTHE RATIONAL WITT CLASS AND THE UNKNOTTINGNUMBER OF A KNOT
STANISLAV JABUKA
Abstract.
We use the rational Witt class of a knot in S as a tool for addressingquestions about its unknotting number. We apply these tools to several low crossingknots (151 knots with 11 crossing and 100 knots with 12 crossings) and to the familyof n -stranded pretzel knots for various values of n ≥
3. In many cases we obtainnew lower bounds and in some cases explicit values for their unknotting numbers.Our results are mainly concerned with unknotting number one but we also address,somewhat more marginally, the case of higher unknotting numbers. Introduction
Preliminaries and statement of results.
The unknotting number u ( K ) of aknot K in the 3-sphere is the minimum number of crossing changes, in any regularprojection of K , that renders it unknotted. While u ( K ) is easy to define, computingit in practice is often unwieldy. Some of the lower bounds for u ( K ) come from theTristram-Levine signatures σ ω ( K ), ω ∈ S (see Definition 3.4) and bound u ( K ) as (1) | σ ω ( K ) | ≤ u ( K ) ∀ ω ∈ S On the other hand, upper bounds for u ( K ) are most easily found from explicit unknot-tings of K . It is when the upper and lower bounds are disparate, that u ( K ) is difficultto determine.The last three decades have furnished an impressive array of tools for studyingunknotting numbers, tools stemming from varied sources such as gauge theory [3, 21,20], polynomial knot invariants [22], linking forms [16] and 3-manifold theory [5]. In thisarticle we propose to add yet another tool to this list by using the rational Witt class ϕ ( K ) to extract information about u ( K ). The rational Witt class ϕ ( K ) is associatedto an oriented knot K ⊂ S and takes values in the Witt ring W ( Q ) ∼ = Z ⊕ Z ∞ ⊕ Z ∞ ofthe field Q of rational numbers. As a commutative ring, W ( Q ) is obtained by applyingthe Grothendieck group construction to the Abelian semiring of isomorphism classes ofnon-degenerate, symmetric, bilinear forms on finite dimensional rational vector spaces.The operations on the latter are given by direct sums and tensor products of vectorspaces along with summing and multiplying their bilinear forms. The Witt ring W ( Q )is well understood and we describe it in some detail in Section 2. For the time being,we content ourselves with saying that W ( Q ) is generated by 1-dimensional forms (cid:104) a (cid:105) , The author was partially supported by NSF grant DMS 0709625. We indicate a simple proof of this bound at the end of Section 3. The usual knot signature σ ( K )agrees with σ − ( K ). Here and below, we write Z p to mean Z /p Z while we use Z ∞ p as a shorthand for ⊕ ∞ i =1 Z p . a r X i v : . [ m a t h . G T ] J u l STANISLAV JABUKA a ∈ ˙ Q where (cid:104) a (cid:105) : Q × Q → Q is the unique bilinear form that sends (1 ,
1) to a (and,as usual, ˙ Q = Q − { } ). Thus, given a non-degenerate, symmetric, bilinear form q : Q n × Q n → Q , there exist rational numbers a , ..., a n ∈ ˙ Q such that q = (cid:104) a (cid:105) ⊕ ... ⊕ (cid:104) a n (cid:105) ,i.e. such that q (( x , ..., x n ) , ( y , ..., y n )) = a x y + ... + a n x n y n .Given an oriented knot K in S , we shall label crossings in a projection of K as positive or negative according to the usual convention, see Figure 1. Crossing changesthemselves shall be similarly labeled as positive or negative according to whether theychange a negative crossing to a positive one or vice versa, see again Figure 1. Negative crossing changePositive crossing changeA positive crossing. A negative crossing.
Figure 1.
Our convention for positive and negative crossings as well aspositive and negative crossing changes.Our main results, Theorem 1.1 and its various corollaries, are founded on the ob-servation that ϕ ( K ) changes rather predictably when K undergoes a single crossingchange. This phenomenon is described in the next statement. Theorem 1.1.
Let K + be a knot obtained from the knot K − by a positive crossingchange. Then the rational Witt classes of K + and K − are related as follows, dependingon how their signatures σ ( K ± ) compare: ϕ ( K + ) = ϕ ( K − ) ⊕ (cid:68) K + det K − (cid:69) ⊕ (cid:104)− (cid:105) ; σ ( K + ) = σ ( K − ) ϕ ( K − ) ⊕ (cid:68) − K + det K − (cid:69) ⊕ (cid:104)− (cid:105) ; σ ( K + ) = σ ( K − ) − ϕ ( K − ) = ϕ ( K + ) ⊕ (cid:68) − K − det K + (cid:69) ⊕ (cid:104) (cid:105) ; σ ( K − ) = σ ( K + ) ϕ ( K + ) ⊕ (cid:68) K − det K + (cid:69) ⊕ (cid:104) (cid:105) ; σ ( K − ) = σ ( K + ) + 2 The conditions on σ ( K ± ) stated on the right-hand sides above, cover all possible cases. A similar, and rather beautiful formula for how the algebraic concordance class of aknot changes under a crossing switch, was found by S.-G. Kim and C. Livingston in[11].As the rational Witt class of the unknot is trivial, the Theorem 1.1 gives restrictionson what ϕ ( K ) can be if K has a given unknotting number. While such restriction HE RATIONAL WITT CLASS AND THE UNKNOTTING NUMBER OF A KNOT 3 exist regardless of the value of u ( K ), they are easiest to state, and have proven mosteffective, when u ( K ) = 1. Corollary 1.2.
Let K be a knot with unknotting number . Then the rational Wittclass ϕ ( K ) of K must be as follows, depending on whether K can be unknotted by apositive or a negative crossing change. a) If K can be unknotted by a positive crossing change, then ϕ ( K ) = (cid:104) K (cid:105) ⊕ (cid:104) (cid:105) ; σ ( K ) = 2 (cid:104)− K (cid:105) ⊕ (cid:104) (cid:105) ; σ ( K ) = 0b) If K can be unknotted with a negative crossing change, then ϕ ( K ) = (cid:104) K (cid:105) ⊕ (cid:104)− (cid:105) ; σ ( K ) = 0 (cid:104)− K (cid:105) ⊕ (cid:104)− (cid:105) ; σ ( K ) = − As before, σ ( K ) denotes the signature of K . The next corollary provides similar constraints on the rational Witt class ϕ ( K ) of aknot K with u ( K ) = 2. It is inherently weaker than Corollary 1.2 in that it involvesinformation about the knot L obtained from K after only one crossing change, a knotwhich one generally knows little about. Corollary 1.3.
Let K be a knot with unknotting number and let L be the knotobtained from K after only a single crossing change. Then the rational Witt class ϕ ( K ) is determined by det K , det L and σ ( K ) and the type of crossing changes involved, asindicated below. To reduce the number of cases to state, we make the assumption that σ ( K ) ≤ . a) If K can be unknotted by two negative crossing changes, then ϕ ( K ) = (cid:104)− K det L (cid:105) ⊕ (cid:104)− L (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) ; σ ( K ) = − (cid:104)± K det L (cid:105) ⊕ (cid:104)∓ L (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) ; σ ( K ) = − (cid:104) K det L (cid:105) ⊕ (cid:104) L (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) ; σ ( K ) = 0 The signs in the second line have to be chosen consistently either as (+ , − ) (if σ ( L ) = − ) or as ( − , +) (if σ ( L ) = 0 ). b) If K can be unknotted by one positive and one negative crossing change, then ϕ ( K ) = (cid:104)− K det L (cid:105) ⊕ (cid:104)− L (cid:105) ; σ ( K ) = − (cid:104)± K det L (cid:105) ⊕ (cid:104)∓ L (cid:105) ; σ ( K ) = 0 Here too the signs in the σ ( K ) = 0 case have to be chosen consistently. Thechoice of (+ , − ) corresponds to the case where L is either obtained from K by The assumption of σ ( K ) ≤ K by its mirror image ¯ K . Clearly u ( ¯ K ) = u ( K ) while σ ( ¯ K ) = − σ ( K ) and ϕ ( ¯ K ) = − ϕ ( K ). STANISLAV JABUKA a negative crossing change and σ ( L ) = σ ( K ) + 2 or L is obtained from K bya positive crossing change and σ ( L ) = σ ( K ) . The choice ( − , +) represents theother two possibilities. c) If K can be unknotted with two positive crossing changes then σ ( K ) = 0 and ϕ ( K ) = (cid:104)− K det L (cid:105) ⊕ (cid:104)− L (cid:105) ⊕ (cid:104) (cid:105) ⊕ (cid:104) (cid:105) Our techniques apply equally well to knots with higher unknotting numbers. How-ever, the indeterminacy of ϕ ( K ) of a knot K with u ( K ) = n grows with n in that itinvolves the determinants of all the knots that K “goes through on its way to the un-knot”. This phenomenon substantially diminishes the usefulness of our approach, moreso since the number of cases describing ϕ ( K ) grows with n as well. We list the nextcorollary more as an illustration of our methods rather than a tool we deem practicallyuseful. Corollary 1.4.
Let K be a knot with signature − n and with u ( K ) = n ≥ . Let L i be the knot obtained from K by changing i − of the n crossings (e.g. L is just K while L n +1 is the unknot). Then ϕ ( K ) = n (cid:77) i =1 ( (cid:104)− L i +1 det L i (cid:105) ⊕ (cid:104)− (cid:105) )We remark that both the σ ( K ) = − σ ( K ) = − n (cid:77) i =1 (cid:104)− (cid:105) = (cid:104)− (cid:105) ⊕ (cid:0)(cid:76) n − i =1 (cid:104)− (cid:105) (cid:1) ; n is odd (cid:76) ni =1 (cid:104)− (cid:105) ; n is evenholds in W ( Q ) for each n ∈ N .1.2. Applications and examples.
The main utility of Theorem 1.1 and its corollariesis to provide obstructions for a given knot K to satisfy the equation u ( K ) = n . To alarge degree, our emphasis shall be on the case n = 1.We start this section by subjecting 3 different families of knots to Corollary 1.2. Thefirst two of these families are finite and consist of 151 knots with 11 crossings and 100knots with 12 crossings respectively. The third family is that of n -stranded pretzelknots for various n ≥
3. We then apply Corollary 1.3 to the knot K = 10 which, atthe time of this writing, has unknown unknotting number [2] (though it is either 2 or3).As Witt rings live at the interface of number theory, algebra and – to a minordegree – topology, the reader will likely detect a number theoretic flair in many of oursubsequent statements, especially those regarding pretzel knots.1.2.1. Eleven crossing knots.
We consider the family of alternating and non-alternating11 crossing knots 11 a x and 11 n y with x and y ranging through the following parametersets, organized by signature (see Remark 1.6 below for an explanation of the color HE RATIONAL WITT CLASS AND THE UNKNOTTING NUMBER OF A KNOT 5 highlighting).(2) x ∈ ; σ (11 a x ) = 2
4, 5, 16 , 36, 37 , 39, 58, 87, 103, 109, 112, 128 , 135,153, 164, 165, 169, 170 , 201, 214 , 228 , 249, 270, 274 ,278 , 285 , 288 , 303, 313 , 315, 317, 332 , 350 ; σ (11 a x ) = 0
1, 6, 21, 23, 32, 42, 45 , 46, 50, 61, 97, 99 , 107 , 118,125, 133, 134, 148 , 163 , 171, 172, 181, 197, 202 , 239 ,258, 268, 269, 271, 277, 279, 281 , 284, 286, 314 , 327,349, 352, 362 ; σ (11 a x ) = − y ∈ (cid:8)
3, 17, 58 , 91, 92, 102, 113 , 122, 127, 129, 140 , 170 (cid:9) ; σ (11 n y ) = 2 (cid:26)
49, 51, 83, 94 , 115 , 116, 119 , 132, 139, 141 , 142, 157,165 , 172, 179 , 182 (cid:27) ; σ (11 n y ) = 0 (cid:26)
15 , 29 , 54, 60, 79 , 112, 117 , 120, 128, 138, 146, 148,150, 155 , 160, 161, 162, 163 , 166, 167, 168, 177, 178 (cid:27) ; σ (11 n y ) = − a , a , a , a , a and 11 n , n which were listed as having unknotting number either 1, 2 or 3.While Greene’s results [6] show that none of these knots can have unknotting number1, and his results thereby subsume our findings, we nevertheless list here the outcomeof applying Corollary 1.2 to the above knots as an illustration of its efficacy and in thehopes that the reader may appreciate an alternate and independent proof of some ofthe results from [6]. Corollary 1.5.
Consider the knots a x and n y with x and y as in (2) . STANISLAV JABUKA (a)
Each of the knots a x and n y , with x and y , from x ∈ {
7, 33, 137, 219, 296, 297 } ; σ (11 a x ) = 2 {
16, 170, 274, 288 } ; σ (11 a x ) = 0 (cid:26)
45, 99, 107, 148, 163, 202, 239, 281,314 (cid:27) ; σ (11 a x ) = − y ∈ {
58, 113, 140 } ; σ (11 n y ) = 2 { } ; σ (11 n y ) = 0 {
15, 29, 79, 117, 155, 163 } ; σ (11 n y ) = − have unknotting number 2, with the possible exception of a , a and a which have unknotting number at least . (b) None of the signature zero knots a x or n y with x and y from x ∈ { , , , } and y ∈ { } can be unknotted with a single negative crossing change. (c) None of the signature zero knots a x or n y with x and y from x ∈ { , , , } and y ∈ { , , , , } can be unknotted with a single positive crossing change. Remark 1.6.
The knots from part (a) of the preceding corollary have been shaded greenin (2) while those from parts (b) and (c) are represented by a yellow shading. We notethat of the 151 knots from (2) , Corollary 1.5 provides unknotting information for of them. Twelve crossing knots.
At the time of this writing, no data concerning the un-knotting numbers of 12 crossings knots is available on KnotInfo [2]. In order to startcollecting such data, we have applied Corollary 1.2 to the first 50 alternating and thefirst 50 non-alternating 12 crossing knots. Considering that there are 1288 alternatingand 888 non-alternating 12 crossing knots total, this is but a very modest beginningto a somewhat daunting program.Among the knots 12 a x and 12 n y with 1 ≤ x, y ≤
50, the following have signaturegreater than 2 or less than − x ∈ { , , , , , , , , , } y ∈ { , , , } Since such knots have unknotting number at least 2 (by virtue of (1)), we exclude thesefrom the next corollary.
HE RATIONAL WITT CLASS AND THE UNKNOTTING NUMBER OF A KNOT 7
Corollary 1.7.
Consider the knots a x and n y with ≤ x, y ≤ but with theexception of those x and y listed in (3) . (a) The knots a x and n y with x ∈ {
29, 32, 39 } ; σ (12 a x ) = 2 { } ; σ (12 a x ) = 0 { } ; σ (12 a x ) = − y ∈ {
10, 15, 17 } ; σ (12 n y ) = 2 {
27, 33 } ; σ (12 n y ) = − have unknotting number at least . (b) None of the signature zero knots a x and n y with x ∈ { } and y ∈ { , , } can be unknotted with a single negative crossing change. (c) None of the signature zero knots a x and n y with x ∈ { , , , , , , } and y ∈ { , , , } can be unknotted with a single positive crossing change. Corollary 1.7 produces unknotting information for 25 of the 86 examined knots.1.2.3.
Pretzel knots.
For an integer n ≥ p , ..., p n , we let P ( p , ..., p n ) denote the corresponding n -stranded pretzel knot/link. It is obtained bytaking n pairs of parallel strands, introducing | p i | half-twists into the i -th pair (with p i > p i < n pairs of bridges. Figure 2 shows the example P (7 , − , Figure 2.
The pretzel knot P (7 , − , P ( p , ..., p n ) to be a knot, at most one of p , ..., p n can be even. In fact, if n itself is STANISLAV JABUKA even, then precisely one of p , ..., p n needs to be even. We shall assume these parityconditions to be satisfied throughout.The computation of ϕ ( P ( p , ..., p n )) in [8], for any choice of parameters p , ..., p n ,provides a fertile testing ground for Theorem 1.1 and Corollary 1.2. We list here onlya few select applications and examples, leaving a more comprehensive exploration ofunknotting numbers of pretzel knots, for a future occasion. We start with the followingremark. Remark 1.8. In [10] , A. Kawauchi showed that a pretzel knot P ( p , ..., p n ) is a two-bridge knot precisely when at most two of p , ..., p n differ from ± . As T. Kanenobuand H. Murakami [9] determined all two-bridge knots with unknotting number , weomit such knots from our applications below.In [12] , T. Kobayashi showed that the pretzel knots P ( p , p , p ) with p , p , p oddand with u ( P ( p , p , p )) = 1 , are precisely those non-trivial knots for which { a, b } ⊂{ p , p , p } where { a, b } is either {± , ± } or {± , ∓ } . We shall therefore also excludesuch knots from our examples. As Corollary 1.2 is sensitive to signatures, we remark that the signatures of pretzelknots P ( p , ..., p n ) have also been computed in full generality in [8]. Corollary 1.9.
Consider the pretzel knot P ( p , p , p ) with p , p odd and with p even.If p ≥ , p > − p (4 − p )4 and the equality (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104) p − p ( p − (cid:105) = (cid:104)− p − p ( p − (cid:105) fails to hold in W ( Q ) , then u ( P ( p , − p , p )) ≥ . There are many examples meeting the hypothesis of Corollary 1.12. Here are acouple.
Example 1.10.
Consider the pretzel knot P (7 , − , p ) with p ≥ an even integer.If there exists a prime p dividing p − with an odd power and such that − is nota square in Z p , then u ( P (7 , − , p )) ≥ . For example, any of p = 2 k · (cid:96) +1 with k, (cid:96) ∈ N , satisfies these conditions (with p = 7 ). Example 1.11.
The unknotting number of P (17 , − , p ) with p = 15+(2 k +1) · (cid:96) +1 and k, (cid:96) ∈ N , is at least . Corollary 1.12.
Let p > be an odd integer and consider the -stranded pretzel knot P ( p, p, p, − p − . If the equality (cid:104) (cid:105) ⊕ (cid:104) p (cid:105) ⊕ (cid:104) p (cid:105) ⊕ (cid:104) p (cid:105) ⊕ (cid:104)− p − (cid:105) ⊕ (cid:68) − p +3 p (3 p +1) (cid:69) = (cid:104) (cid:105) ⊕ (cid:104) p + 3) (cid:105) fails to hold in W ( Q ) , then u ( P ( p, p, p, − p − ≥ . Many choices of p are possible in Corollary 1.3. Here is an infinite family of suchchoices. Example 1.13.
Taking p = 2 + (2 k + 1) · (cid:96) +1 with k, (cid:96) ∈ N , meets the conditions ofCorollary 1.12. Consequently, each of the corresponding knots P ( p, p, p, − p − hasunknotting number at least . HE RATIONAL WITT CLASS AND THE UNKNOTTING NUMBER OF A KNOT 9
Given a pretzel knot P ( p , ..., p n ) and an odd integer p , we say that the knot P ( p , ..., p m , p, p m +1 , ..., p (cid:96) , − p, p (cid:96) +1 , ..., p n ) was obtained from P ( p , ..., p n ) by upwardstabilization (a term already introduced in [8]). With this in mind, we have: Corollary 1.14. If K = P ( p , ..., p n ) is a pretzel knot for which the equalities for ϕ ( K ) from Corollary 1.2 fail (so that u ( K ) ≥ ), then the same is true for any pretzel knot L obtained from K by a finite number of upward stabilizations. Consequently, for anysuch L one has u ( L ) ≥ . Combining this corollary with previous examples, supplies pretzel knots with anarbitrarily high number of strands and with unknotting number at least 2. For instance, u ( P (7 , − , , p , − p , p , − p , ..., p m , − p m )) ≥ p , ..., p m .1.2.4. Obstructing unknotting number . As already mentioned, Corollary 1.3 is inher-ently weaker than Corollary 1.2 as it involves the unknown quantity det L . Even so,it is still possible to gain some unknotting information from it. To demonstrate this,we consider the knot K = 10 which has signature 4, determinant 41 and, as of thiswriting, has unknotting number 2 ≤ u (10 ) ≤
3, according to KnotInfo [2]. ApplyingCorollary 1.3 to this knot, we find:
Corollary 1.15.
Suppose K = 10 can be unknotted by crossing changes and let L be the knot obtained from K by a single of these crossing changes. If L has or fewercrossings, then L must be contained in the list of knots (out of knots with orfewer crossings, not counting mirror images): ¯3 , ¯5 , ¯6 , ¯7 , ¯7 , ¯8 , ¯8 , ¯9 , ¯9 , , , . A bar on top of a knot indicates its mirror image.
We finish this section by pointing out that all of our applications of Theorem 1.1made the choice of either K + = unknot or K − = unknot. The usefulness of Theorem1.1 certainly stretches beyond this. We leave it as an exercise for the motivated readerto verify, for example, that the knots 8 and 9 cannot be gotten from one another bya single crossing change.1.3. Organization.
The remainder of this article is organized into 6 sections. Section2 provides background material on Witt rings with a special emphasis on the Witt ringof the rationals. Section 3 defines the rational Witt class ϕ ( K ) associated to a knot K and explores how the former changes when K is altered by a single crossing change.Doing so enables us to prove Theorem 1.1. Section 4 supplies the proofs for Corollaries1.2, 1.3 and 1.4 while Section 5 explains how the results from Corollaries 1.5, 1.7 and1.15 were obtained. Section 6 provides proofs of our claims concerning pretzel knotswhile the final Section 7 provides a comparison of our work to that of R. Lickorish from[16]. Acknowledgements
During the preparation of this article, I have enjoyed and ben-efited from conversations with Brendan Owens, whose input I gratefully acknowledge.
I would also like to thank Josh Greene for generously sharing his results from [6] and forproviding helpful comments on an earlier version of this work. Additional thanks aredue to Chuck Livingston and Swatee Naik for many stimulating conversations aboutWitt rings. 2.
Background material on Witt rings
This section reviews some of the basic algebra underlying the definition of Wittrings W ( F ) over arbitrary fields F . We then focus in on the case of F = Q and give acompletely explicit description of the isomorphism W ( Q ) ∼ = Z ⊕ Z ∞ ⊕ Z ∞ which wasalready mentioned in the introduction. For more information we advise the interestedreader to consider the sources [4, 7, 13, 19].To begin with, let us fix a field F and let B F be the set of isomorphism classesof symmetric, bilinear, non-degenerate forms over finite dimensional F -vector spaces.Thus, an element of B F is a pair ( V, B ) where V is a finite dimensional F -vector spaceand B : V × V → F is a symmetric, bilinear and non-degenerate form where by thelatter we mean that if B ( x, y ) = 0 for all y ∈ V , then x = 0. The set B F becomes anAbelian semiring (often, somewhat humorously, referred to as an Abelian rig ) underthe operations ⊕ and ⊗ given by( V , B ) ⊕ ( V , B ) = ( V ⊕ V , B + B ) & ( V , B ) ⊗ ( V , B ) = ( V ⊗ F V , B · B )When the importance of V is minor and the danger of confusion little, we will onlywrite B to mean ( V, B ) and likewise B ⊕ B to mean ( V , B ) ⊕ ( V , B ).Recall that Grothendieck’s group construction turns an Abelian semigroup ( G, +)into an Abelian group by considering the set ( G × G ) / ∼ where ∼ is the equivalencerelation defined by ( x , y ) ∼ ( x , y ) if x + y = x + y (intuitively we should regard ( x, y ) as representing x − y , even though the latter is ofcourse not defined). With respect to the addition ( x , y ) + ( x , y ) = ( x + x , y + y )on ( G × G ) / ∼ , the inverse of ( x, y ) is then given by ( y, x ). The semigroup G injectsnaturally into ( G × G ) / ∼ by sending x to ( x, G has the structure of an Abeliansemiring, ( G × G ) / ∼ itself becomes an Abelian ring.With this understood, here is the definition of the Witt ring. Definition 2.1.
The Witt ring W ( F ) associated to the field F , is the Abelian ringobtained by applying the Grothendieck construction to the Abelian semiring ( B F , ⊕ , ⊗ ) . As is customary, we shall use ˙ F to denote F − { } . Given an element a ∈ ˙ F , let (cid:104) a (cid:105) denote the unique bilinear, symmetric, non-degenerate form on F × F which sends(1 ,
1) to a ∈ F . Note that (cid:104) a (cid:105) = (cid:104) a · d (cid:105) for any choice of d ∈ ˙ F since f : ( F , (cid:104) a · d (cid:105) ) → ( F , (cid:104) a (cid:105) ) given by f ( λ ) = d · λ is anisomorphism of bilinear forms. We will often tacitly rely on the equality (cid:104) a (cid:105) = (cid:104) a · d (cid:105) in the remainder of the article.The next theorem is basic and can be found in each of [4, 7, 13, 19]. HE RATIONAL WITT CLASS AND THE UNKNOTTING NUMBER OF A KNOT 11
Theorem 2.2.
For any field F , the Witt ring W ( F ) is generated by the set {(cid:104) a (cid:105) | a ∈ ˙ F } .A presentation of W ( F ) as a commutative ring is obtained by adding the next relatorsto these generators: ( R (cid:104) (cid:105) ⊕ (cid:104)− (cid:105) ( R (cid:104) a (cid:105) ⊗ (cid:104) b (cid:105) ⊕ (cid:104)− a · b (cid:105) a, b ∈ ˙ F ( R (cid:104) a + b (cid:105) ⊕ (cid:104) ab ( a + b ) (cid:105) ⊕ (cid:104)− a (cid:105) ⊕ (cid:104)− b (cid:105) a, b ∈ ˙ F For the next discussion, we assume that char F (cid:54) = 2. The hyperbolic form over thefield F is the 2-dimensional bilinear form ( F , H ) where H , with respect to the standardbasis { e = (1 , , e = (0 , } of F , is represented by the matrix H = (cid:20) (cid:21) With respect to the basis { f , f } of F , given by f = e + e , f = − e + e , thehyperbolic form H is represented by the matrix H = (cid:20) − (cid:21) = (cid:104) (cid:105) ⊕ (cid:104)− (cid:105) = 0 ∈ W ( F )showing that it equals zero in W ( F ).Hyperbolic forms are rather special, indeed, they arise as summands of all “isotropicforms”. A form ( V, B ) is called isotropic if there exists a non-zero vector v ∈ V with B ( v, v ) = 0, otherwise ( V, B ) is called anisotropic . Thus, if B is isotropic then B = B (cid:48) ⊕ H for some form B (cid:48) (see Proposition 2.25 in [4]) and, consequently, B = B (cid:48) ∈ W ( F ). If B (cid:48) itself is isotropic, there is a further decomposition B (cid:48) = B (cid:48)(cid:48) ⊕ H andagain B (cid:48) = B (cid:48)(cid:48) ∈ W ( F ). This process ends after a finite number of steps giving us adecomposition of the original form B , called the Witt decomposition , as B = B ⊕ H ⊕ ... ⊕ H n where B is anisotropic (but possibly zero) and each of H , ..., H n (with n also possiblyzero) is a hyperbolic form. While this decomposition is not unique, the integer n andthe isomorphism type of B are uniquely determined by B (see Section 2.5 in [4]).With this in mind, we can define W ( F ) (as an Abelian group) alternatively as the setof equivalence classes of B F / ∼ with the operation ⊕ , where ∼ is defined as B ∼ B ⇔ The anisotropic parts of B and B are isomorphic.In this description it is easy to see that the inverse of ( V, B ) in W ( F ) is the form( V, − B ) since ( V ⊕ V, B + ( − B )) is a direct sum of hyperbolic forms.We now turn to examining some concrete Witt rings, including the case of F = Q . Fora prime integer p , let Z p denote the finite field Z p = { , , ..., p − } of characteristic p .The Witt rings W ( Z p ) are well understood as should be evident from the next theorem(which can be found in Section 2.8 in [4]). Theorem 2.3.
Let p be a prime integer. Then there are isomorphisms of Abeliangroups W ( Z p ) ∼ = Z ; p = 2 Z ⊕ Z ; p ≡ mod Z ; p ≡ mod The generators of Z ∼ = W ( Z ) and of Z ∼ = W ( Z p ) with p ≡ mod , are given by (cid:104) (cid:105) , while the two copies of Z in W ( Z p ) when p ≡ mod , are generated by (cid:104) (cid:105) and (cid:104) a (cid:105) for any choice of a ∈ ˙ Z p − ( ˙ Z p ) . The reason for stating a separate theorem about Witt rings of finite fields is thatthey are instrumental in understanding the Witt ring of the rationals. The relationbetween the former to the latter is elucidated in the next key theorem (which can befound on page 88 of [7]).
Theorem 2.4.
There is an isomorphism of Abelian groups σ ⊕ ∂ : W ( Q ) → Z ⊕ ( ⊕ p W ( Z p )) where ⊕ p is a sum over all prime integers p . The homomorphism σ : W ( Q ) → Z is thesignature function while ∂ : W ( Q ) → ⊕ p W ( Z p ) is the direct sum of homomorphisms ∂ p : W ( Q ) → W ( Z p ) described on generators of W ( Q ) as follows: Given a rationalnumber λ (cid:54) = 0 , write it as λ = p (cid:96) · β where (cid:96) is an integer and β a rational numberwhose numerator and denominator are relatively prime to p . Then (4) ∂ p ( (cid:104) p (cid:96) · β (cid:105) ) = (cid:26) (cid:96) is even (cid:104) β (cid:105) ; (cid:96) is odd The preceding theorem makes is possible to determine precisely, and completelyexplicitly, when two forms B and B over Q are equal in W ( Q ). Namely, if B = (cid:104) a (cid:105) ⊕ ... ⊕ (cid:104) a n (cid:105) and B = (cid:104) b (cid:105) ⊕ ... ⊕ (cid:104) b m (cid:105) then B = B ∈ W ( Q ) if and only if (cid:104) a (cid:105) ⊕ ... ⊕ (cid:104) a n (cid:105) ⊕ (cid:104)− b (cid:105) ⊕ ... ⊕ (cid:104)− b m (cid:105) = 0 ∈ W ( Q )This latter equation in turn holds if and only if σ and each ∂ p map its left-hand sideto zero. Here is an example illustrating Theorem 2.4. Example 2.5.
Let B be the bilinear form on Q given by B = (cid:104)− (cid:105)⊕(cid:104) (cid:105)⊕(cid:104)− (cid:105)⊕(cid:104) (cid:105) .Note that the only primes p for which ∂ p B can be nonzero, are p = 3 , , , . For thesechoices of p , we obtain ∂ B = (cid:104)− (cid:105) = (cid:104)− (cid:105) = (cid:104) (cid:105) ∈ W ( Z ) ∂ B = (cid:104)− (cid:105) = (cid:104) (cid:105) ∈ W ( Z ) and ∈ ˙ Z − ( ˙ Z ) ∂ B = (cid:104) (cid:105) ∈ W ( Z ) ∂ B = (cid:104)− (cid:105) = (cid:104)− (cid:105) = (cid:104) (cid:105) ⊕ (cid:104) (cid:105) ⊕ (cid:104) (cid:105) ∈ W ( Z ) Since σ ( B ) = 0 , it follows that B is torsion of order in W ( Q ) . HE RATIONAL WITT CLASS AND THE UNKNOTTING NUMBER OF A KNOT 13 The rational Witt class of a knot under a crossing change
In this section we make precise the definition of ϕ ( K ) – the rational Witt class ofa knot K ⊂ S (see Definition 3.1). We then examine how ϕ ( K ) is altered when K undergoes a crossing change (Theorem 3.2).Let K be an oriented knot in S and let Σ be a Seifert surface of K whose orientationis compatible with that of K . We shall view the orientation of Σ as being given by anormal and nowhere vanishing vector field (cid:126)n on Σ. The linking form or Seifert form on H (Σ; Z ) is the bilinear form (cid:96)k : H (Σ , Z ) × H (Σ , Z ) → Z given by (cid:96)k ( α, β ) = linking number of α with β + Here we view α and β both as curves on Σ and β + is a small push-off of β fromΣ in the direction of (cid:126)n . Thus α and β + are disjoint curves in S and their linkingnumber is (cid:80) p ε ( p ) where p ranges over the double points of any regular projectionof α (cid:116) β + and where ε ( p ) = 1 if p is a positive crossing and ε ( p ) = − p is anegative crossing (see Figure 1 for the definition of positive/negative crossings). Weextend (cid:96)k linearly to a form, of the same name, from H (Σ; Q ) × H (Σ; Q ) to Q , andlet B K : H (Σ; Q ) × H (Σ; Q ) → Q be the bilinear, symmetric and non-degenerateform B K ( α, β ) = (cid:96)k ( α, β ) + (cid:96)k ( β, α ). Definition 3.1.
With the notation as in the preceding paragraph, the rational Wittclass ϕ ( K ) of a knot K ⊂ S is the element of the rational Witt ring W ( Q ) given by ( H (Σ; Q ) , B K ) for any choice of an oriented Seifert surface Σ of K . The fact that ϕ ( K ) is well defined, i.e. independent of the choice of Σ, followsfrom work of Levine [14, 15] but can also be easily verified directly. Namely, anytwo oriented Seifert surfaces Σ and Σ (cid:48) of the same knot K , differ from one anotherby a sequence of 1-handle attachments/detachments. These operations change theassociated bilinear forms by adding/subtracting a hyperbolic summand and thus donot affect their rational Witt classes.We now turn to exploring how ϕ ( K ) changes when K is altered by a single crossingswitch. For concreteness sake, we take the crossing change to be a positive one, cf.Figure 1. Let K − be an oriented knot, let c be a negative crossing in some projection of K − and let K + be the knot obtained from K − by switching the distinguished crossing c as in Figure 3. c ce g − e g + Σ − Σ + K − K + ( a ) ( b ) Figure 3.
Changing the negative crossing c in K − to a positive one in K + . The shaded areas indicate the Seifert surfaces Σ − and Σ + . Pick oriented Seifert surfaces Σ ± for K ± so that Σ − and Σ + are identical safe in aneighborhood of the crossing c where they differ as in Figure 3. For the purpose ofcomparing ϕ ( K − ) to ϕ ( K + ), it will prove advantageous to pick bases { e ± , ..., e g ± } of H (Σ ± ; Z ) with e i − = e i + for i = 1 , ..., g −
1, with e g ± near c as indicated in Figure4 and, additionally, such that none of e ± , ..., e g − ± pass through the crossing c . Suchbases can always be chosen though one may have to revise the initial choice of theSeifert surfaces Σ ± . Figure 4 shows how to do this by a simple stabilization argumentsupported in a neighborhood of the distinguished crossing c . e g − e g + Figure 4.
One can always adjust the initial choices of Seifert surfacesΣ ± by stabilizing them in a neighborhood of the distinguished crossing c , enabling one to find a preferred basis { e ± , ..., e g ± } for H (Σ ± ; Z ) with e i − = e i + for i = 1 , ..., g −
1. If initially a curve e i ± with i ≤ g − c , we simply replace it by e i ± − e g ± (or by e i ± + e g ± depending on the orientations of the curves).Let (cid:96)k ± : H (Σ ± ; Q ) × H (Σ ± ; Q ) → Q be the linking pairings associated to Σ ± .Note that our choice of bases implies (cid:96)k − ( e i − , e j − ) = (cid:96)k + ( e i + , e j + ) ∀ ( i, j ) (cid:54) = (2 g, g ) (cid:96)k − ( e g − , e g − ) = (cid:96)k + ( e g + , e g + ) + 1Let V ± be the (2 g ) × (2 g ) integral matrices representing the linking pairings (cid:96)k ± withrespect to the bases { e ± , ..., e g ± } , so that ϕ ( K ± ) = ( Q g , V ± + V τ ± ) ∈ W ( Q ). We notethat while the determinant of ϕ ( K ) is only well defined as an element of ˙ Q / ˙ Q (ratherthan as a rational number), the determinant of V ± + V τ ± agrees with the determinantof the knots K ± .To be able to compare ϕ ( K − ) to ϕ ( K + ), we shall express each as a sum of 1-dimensional forms by diagonalizing V ± + V τ ± . We accomplish this by changing our pre-ferred bases { e ± , ..., e g ± } to new bases { f ± , ..., f g ± } via, essentially, the Gram-Schmidtalgorithm. For simplicity of notation, we shall write (cid:104) v, w (cid:105) ± or simply (cid:104) v, w (cid:105) for( (cid:96)k ± + (cid:96)k τ ± )( v, w ). HE RATIONAL WITT CLASS AND THE UNKNOTTING NUMBER OF A KNOT 15
With this in mind, we define the vectors f i ± as f ± = e ± and f i ± = e i ± − i − (cid:88) j =1 (cid:104) e i ± , f j ± (cid:105)(cid:104) f j ± , f j ± (cid:105) f j ± for i ≥ (cid:104) f i ± , f i ± (cid:105) could equal zero.To account for this, we divide our discussion into three separate cases. Case of (cid:104) f i ± , f i ± (cid:105) (cid:54) = 0 , i = 1 , ..., g − . For the moment, we assume that noneof (cid:104) f i ± , f i ± (cid:105) vanishes. In this case we find that { f ± , ..., f g ± } are orthogonal bases for H (Σ ± , Q ) where we think of (cid:96)k ± + (cid:96)k τ ± as giving us an inner product (cid:104)· , ·(cid:105) = (cid:104)· , ·(cid:105) ± .Since e i − = e i + for all i = 1 , ..., g −
1, the same is true for the new basis elements: f i − = f i + for i = 1 , ..., g −
1. From this, and since (cid:104) e i − , e j − (cid:105) = (cid:104) e i + , e j + (cid:105) whenever( i, j ) (cid:54) = (2 g, g ), we obtain (cid:104) f i − , f j − (cid:105) = (cid:104) f i + , f j + (cid:105) ∀ ( i, j ) (cid:54) = (2 g, g )On the other hand, for i = j = 2 g , we obtain (cid:104) f g + , f g + (cid:105) = (cid:42) e g + − g − (cid:88) j =1 (cid:104) e i + , f j + (cid:105)(cid:104) f j + , f j + (cid:105) f j + , e g + − g − (cid:88) j =1 (cid:104) e i + , f j + (cid:105)(cid:104) f j + , f j + (cid:105) f j + (cid:43) = (cid:104) e g + , e g + (cid:105) − (cid:42) e g + , g − (cid:88) j =1 (cid:104) e i + , f j + (cid:105)(cid:104) f j + , f j + (cid:105) f j + (cid:43) + (cid:42) g − (cid:88) j =1 (cid:104) e i + , f j + (cid:105)(cid:104) f j + , f j + (cid:105) f j + , g − (cid:88) j =1 (cid:104) e i + , f j + (cid:105)(cid:104) f j + , f j + (cid:105) f j + (cid:43) = (cid:104) e g − , e g − (cid:105) − − (cid:42) e g − , g − (cid:88) j =1 (cid:104) e i − , f j − (cid:105)(cid:104) f j − , f j − (cid:105) f j − (cid:43) + (cid:42) g − (cid:88) j =1 (cid:104) e i − , f j − (cid:105)(cid:104) f j − , f j − (cid:105) f j − , g − (cid:88) j =1 (cid:104) e i − , f j − (cid:105)(cid:104) f j − , f j − (cid:105) f j − (cid:43) = (cid:104) f g − , f g − (cid:105) − (cid:104) f i − , f i − (cid:105) = a i , we see that ϕ ( K − ) and ϕ ( K + ) take the forms ϕ ( K − ) = (cid:104) a (cid:105) ⊕ ... ⊕ (cid:104) a g − (cid:105) ⊕ (cid:104) a g (cid:105) ϕ ( K + ) = (cid:104) a (cid:105) ⊕ ... ⊕ (cid:104) a g − (cid:105) ⊕ (cid:104) a g − (cid:105) Since we have taken care to make our basis change an orthogonal one, and since thedeterminant of ϕ ( K ± ) when expressed in the bases { e ± , ..., e g ± } agreed with det K ± ,we see thatdet K − = | a · ...a g − · a g | and det K + = | a · ... · a g − · ( a g − | Given this, we can summarize our findings for the current special case by stating that,given our notation above, there exists a rational number a ∈ ˙ Q (with a = a g above)such that ϕ ( K + ) = ϕ ( K − ) ⊕ (cid:28) − a (cid:29) ⊕ (cid:104) a − (cid:105) and det K + = det K − · (cid:12)(cid:12)(cid:12)(cid:12) a − a (cid:12)(cid:12)(cid:12)(cid:12) (5)As we shall see, this statement remains true in the two subsequent cases as well. Case of (cid:104) f m ± , f m ± (cid:105) = 0 for some m ≤ g − . This case has been addressed inTheorem 4.3 from [8]. It is shown there that by passing to another basis, one can splitoff a hyperbolic summand from ( (cid:96)k ± + (cid:96)k τ ± , H (Σ ± ; Q )) thereby ridding oneself of an element of square zero. Here are the specifics: Suppose that f i ± , i = 1 , ..., m have beendefined as above and satisfy (cid:104) f i ± , f i ± (cid:105) (cid:54) = 0 for i = 1 , ..., m − (cid:104) f m ± , f m ± (cid:105) = 0. We then define a new basis { f ± , ..., f m − ± , f m ± , e m +1 ± , ..., e g ± } according to e m +1 ± = e m +1 ± − m − (cid:88) j =1 (cid:104) e m +1 ± , f j ± (cid:105)(cid:104) f j ± , f j ± (cid:105) e j ± g m + k ± = e m + k ± − m − (cid:88) j =1 (cid:104) e m + k ± , f j ± (cid:105)(cid:104) f j ± , f j ± (cid:105) e j ± e m + k ± = g m + k ± − (cid:104) g m + k ± , f m ± (cid:105)(cid:104) e m +1 ± , f m ± (cid:105) e m +1 ± −− (cid:104) g m + k ± , e m +1 ± (cid:105) · (cid:104) e m +1 ± , f m ± (cid:105) − (cid:104) g m + k ± , f m ± (cid:105) · (cid:104) e m +1 ± , e m +1 ± (cid:105)(cid:104) e m +1 ± , f m ± (cid:105) · (cid:104) e m +1 ± , f m ± (cid:105) f m ± (6)with the last two equations valid for k ≥
2. An explicit computation (addressed in theproof of Theorem 4.3 in [8]) shows that this basis decomposes as { f ± , ..., f m − ± } ∪ { f m ± , e m ± } ∪ { e m +1 , ..., e g ± } with the spans of each of the three sets perpendicular (with respect to (cid:104)· , ·(cid:105) ± ) to thespans of the other two. Additionally, the restriction of (cid:104)· , ·(cid:105) ± to the span of { f m ± , e m ± } is a hyperbolic form showing that( (cid:96)k ± + (cid:96)k τ ± , H (Σ ± ; Q )) = ( (cid:104)· , ·(cid:105) ± | Span , Span ) ∈ W ( Q )where Span is Span ( f ± , ..., f m − ± , e m +1 , ..., e g ± ). However, in passing from H (Σ ± , Q )to Span ( f ± , ..., f m − ± , e m +1 , ..., e g ± ) we have eliminated the square zero vector f m . Notethat e g ± = e g ± + a ± with a − = a + so that (cid:104) e g + , e g + (cid:105) = (cid:104) e g − , e g − (cid:105) + 2still holds, allowing us to re-derive (5) just as before (after first eliminating additionalvectors of square zero, if any). Case of (cid:104) f g − ± , f g − ± (cid:105) = 0 . If (cid:104) f g − ± , f g − ± (cid:105) = 0 then (cid:104) f g − ± , e g ± (cid:105) (cid:54) = 0 sinceotherwise the from (cid:96)k ± + (cid:96)k τ ± would be degenerate. Thus, we introduce the new basiselement f g ± as f g ± = e g ± − g − (cid:88) i =1 (cid:104) e g ± , f i ± (cid:105)(cid:104) f i ± , f i ± (cid:105) f i ± achieving (cid:104) f g ± , f i ± (cid:105) = 0 for i = 1 , ..., g −
2. This shows that the spans of the twosubsets { f ± , ..., f g − ± } and { f g − ± , f g ± } of our new basis, are orthogonal (again, withrespect to (cid:104)· , ·(cid:105) ± ). Moreover, since (cid:104) f g − ± , f g − ± (cid:105) = 0, the vectors { f g − ± , f g ± } span ahyperbolic space showing that (with Span ± = Span ( f ± , ..., f g − ± )) ϕ ( K − ) = ( (cid:96)k − + (cid:96)k τ − , H (Σ − ; Q )) = ( (cid:104)· , ·(cid:105) − | Span − , Span − ) == ( (cid:104)· , ·(cid:105) + | Span + , Span + ) = ( (cid:96)k + + (cid:96)k τ + , H (Σ + ; Q )) = ϕ ( K + ) HE RATIONAL WITT CLASS AND THE UNKNOTTING NUMBER OF A KNOT 17
Even though we obtain ϕ ( K − ) = ϕ ( K + ) in this case, both ϕ ( K − ) and ϕ ( K + ) stilladhere to the form from (5), using a little trick. Namely, we can add the form (cid:104) (cid:105)⊕(cid:104)− (cid:105) to ϕ ( K − ) (since the former is trivial in W ( Q )) and observe that (cid:104) (cid:105) ⊕ (cid:104)− (cid:105) = (cid:104) (cid:105) ⊕ (cid:104)− − (cid:105) since − − − · and therefore (cid:104)− − (cid:105) = (cid:104)− (cid:105) . Thus, even in this current case, ϕ ( K − ) and ϕ ( K + ) satisfy formula (5).We see that each of the three cases we just discussed, leads to formula (5), thus provingthe next proposition. Proposition 3.2.
Let K − and K + be two oriented knots that have projections that areidentical save near one crossing c which is a negative one for K − and a positive onefor K + . Then there exists a rational numbers a ∈ ˙ Q such that ϕ ( K + ) = ϕ ( K − ) ⊕ (cid:28) − a (cid:29) ⊕ (cid:104) a − (cid:105) and det K + = det K − · (cid:12)(cid:12)(cid:12)(cid:12) a − a (cid:12)(cid:12)(cid:12)(cid:12) Note that the preceding theorem shows that the signatures of K − and K + are relatedas σ ( K + ) = σ ( K − ) − < a < σ ( K − ) ; a < a > ω = − a from that same proposition. Namely, if σ ( K + ) = σ ( K − ) then a − a > σ ( K + ) = σ ( K − ) − a − a <
0. From thisone easily arrives at:(7) If σ ( K + ) = σ ( K − ) then a = K − det K − − det K + and a − K + det K − − det K + . If σ ( K + ) = σ ( K − ) − a = K − det K − +det K + and a − − K + det K − +det K + . Combining Proposition 3.2 with the equations from (7), provides a proof of the nexttheorem.
Theorem 3.3.
Let K − and K + be two oriented knots that have projections that areidentical safe near one crossing c which is a negative one for K − and a positive onefor K + . Then their rational Witt classes are related as follows: ϕ ( K + ) = ϕ ( K − ) ⊕ (cid:68) − det K − − det K + K − (cid:69) ⊕ (cid:68) K + det K − − det K + (cid:69) ; σ ( K + ) = σ ( K − ) ϕ ( K − ) ⊕ (cid:68) − det K − +det K + K − (cid:69) ⊕ (cid:68) − K + det K − +det K + (cid:69) ; σ ( K + ) = σ ( K − ) − Theorem 1.1 follows directly from the preceding theorem after an easy applicationof relation ( R
3) from Theorem 2.2. To see this, let us for brevity of notation writedet K − = b and det K + = c . One then computes as (cid:68) − det K − − det K + K − (cid:69) ⊕ (cid:68) K + det K − − det K + (cid:69) = (cid:10) − b − c b (cid:11) ⊕ (cid:10) cb − c (cid:11) = (cid:104)− b ( b − c ) (cid:105) ⊕ (cid:104) c ( b − c ) (cid:105) ( now use ( R (cid:104)− b − c ) (cid:105) ⊕ (cid:104) bc ( b − c ) (cid:105) = (cid:104)− (cid:105) ⊕ (cid:104) bc (cid:105) = (cid:104)− (cid:105) ⊕ (cid:28) cb (cid:29) = (cid:104)− (cid:105) ⊕ (cid:68) K + det K − (cid:69) and more time as (cid:68) − det K − +det K + K − (cid:69) ⊕ (cid:68) − K + det K − +det K + (cid:69) = (cid:10) − b + c b (cid:11) ⊕ (cid:10) − cb + c (cid:11) = (cid:104)− b ( b + c ) (cid:105) ⊕ (cid:104)− c ( b + c ) (cid:105) ( now use ( R (cid:104)− b + c ) (cid:105) ⊕ (cid:104)− bc ( b + c ) (cid:105) = (cid:104)− (cid:105) ⊕ (cid:104)− bc (cid:105) = (cid:104)− (cid:105) ⊕ (cid:28) − cb (cid:29) = (cid:104)− (cid:105) ⊕ (cid:68) − K + det K − (cid:69) With these in place, Theorem 1.1 follows.To illustrate our discussion thus far, we turn to an example. We shall consider theknot K = ¯7 (the mirror image of the knot 7 ) from Figure 5a and change the negativecrossings c and c indicated in that same figure.To facilitate the crossing change at c , we pick the Seifert surface Σ − and the basis { e − , e − , e − , e − } for H (Σ − ; Q ) as indicated in Figure 5b (where we have dropped thesubscripts “ − ”from the notation). Let V − be the matrix representing the linking form (cid:96)k − with respect to this basis. An explicit computation of linking numbers then showsthat V − + V τ − = − − −
10 1 0 10 − Diagonalizing this matrix using the Gram-Schmidt process yields ϕ (¯7 ) as ϕ (¯7 ) = (cid:104) (cid:105) ⊕ (cid:104) (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104) (cid:105) Since changing the crossing c only affects the linking number of e − with e − , therational Witt class of the knot L obtained after changing c differs from that for 7 by HE RATIONAL WITT CLASS AND THE UNKNOTTING NUMBER OF A KNOT 19 c c c ( a ) ( b ) ΣˆΣ e e e e ( c ) ( d ) ˆ e ˆ e ˆ e ˆ e Figure 5. (a)
The knot 7 with two distinguished crossings c and c . (b) The Seifert surface Σ and the basis { e , e , e , e } of H (Σ; Z ) usedfor analyzing how ϕ (7 ) changes if the crossing c is switched. The curve e is oriented so that (cid:104) e , e (cid:105) = 1. (c) The knot 7 after the crossing c has been switched, has undergone a simply isotopy showing the newknot to be 5 . Its remaining distinguished crossing c is still indicated. (d) The Seifert surface ˆΣ and the basis { ˆ e , ˆ e , ˆ e , ˆ e } of H ( ˆΣ; Z ) usedfor describing how ϕ (5 ) is affected by the change of the crossing c . Thecurve ˆ e is oriented so as to yield (cid:104) ˆ e , ˆ e (cid:105) = 1.subtracting 2 from the –summand: ϕ ( L ) = (cid:104) (cid:105) ⊕ (cid:104) (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104) − (cid:105) = (cid:104) (cid:105) ⊕ (cid:104) (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104) (cid:105) This shows that ϕ (¯7 ) = ϕ ( L ) ⊕ (cid:104)− a (cid:105) ⊕ (cid:104) a − (cid:105) with a = , as claimed in Theorem3.3. It is rather easy to verify that L is the knot ¯5 .Turning now to changing the crossing c , we pick a new Seifert surface ˆΣ − and basis { ˆ e − , ..., ˆ e − } as in Figure 5(d) (where again the subscripts are omitted). Let ˆ V − be thematrix expressing the linking form (cid:98) (cid:96)k − with respect to this basis. A quick computationof linking numbers yields ˆ V − + ˆ V τ − = − − Running the Gram-Schmidt procedure on the latter bilinear form yields the rationalWitt class ϕ (¯5 ): ϕ (¯5 ) = (cid:104) (cid:105) ⊕ (cid:104) (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104) (cid:105) The knot M obtained from ¯5 by changing c must have rational Witt class equal tothe one gotten from ϕ (¯5 ) by subtracting 2 from the last –summand: ϕ ( M ) = (cid:104) (cid:105) ⊕ (cid:104) (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104) − (cid:105) = (cid:104) (cid:105) ⊕ (cid:104) (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) = 0By inspection one finds that M is in fact that unknot and so ϕ (unknot) = 0 ∈ W ( Q ),as already mentioned in the introduction. We note that here too, in accordance withTheorem 3.3, we obtain the equation ϕ (¯5 ) = ϕ ( M ) ⊕ (cid:104)− a (cid:105) ⊕ (cid:104) a − (cid:105) , this time with a = .We finish this section with a brief discussion of Tristram-Levine signatures and indi-cate, with few details, how inequality (1) follows easily from the discussion precedingTheorem 3.2 (we note that the bound (1) is only used with ω = − Definition 3.4.
Let K be an oriented knot and Σ an oriented Seifert surface for K .Given a complex number ω of unit modulus, consider the Hermitian form A ( ω ) on H (Σ; C ) given by A ( ω ) = − ω · (cid:96)k + − ω − · (cid:96)k τ The Tristram-Levine signature σ ω ( K ) is defined as the signature σ ( A ( ω )) of A ( ω ) ,provided the latter is non-sigular. If A ( ω ) is singular, then we set σ ω ( K ) = ( σ ω − ( K )+ σ ω + ( K )) where ω ± are points on S on either side of ω and sufficiently close to it. It is not hard to verify that σ ω ( K ) is independent of the choice of Σ and that A ( ω )is singular if and only if ω is a root of the Alexander polynomial ∆ K ( t ) of K . Notealso that σ − ( K ) agrees with the usual signature σ ( K ).If K − and K + are two knots that only differ in a single crossing c which is a negativeone for K − and a positive one for K + , then one can diagonalize the corresponding A ± ( ω )(this time as Hermitian rather than symmetric forms) much as was already done forthe case of ω = − a , ..., a g ∈ ˙ Q such thatthe diagonalized A − ( ω ) and A + ( ω ) look as (cid:104) a (cid:105) ⊕ ... ⊕ (cid:104) a g (cid:105) and (cid:104) a (cid:105) ⊕ ... ⊕ (cid:104) a g + (1 − Re( ω )) (cid:105) HE RATIONAL WITT CLASS AND THE UNKNOTTING NUMBER OF A KNOT 21 respectively. Since for ω ∈ S one obtains 0 ≤ − Re( ω ) ≤
2, we see that either σ ω ( K + ) = σ ω ( K − ) or σ ω ( K + ) = σ ω ( K − ) + 2. Inequality (1) is an easy consequence ofthis observation. 4. Proofs of Corollaries 1.2, 1.3 and 1.4
Proof of Corollary 1.2.
Suppose that K is an oriented knot in S that can beunknotted with a single crossing change. According to (1) (with ω = −
1) we must have σ ( K ) = 0 or σ ( K ) = ±
2. Let U denote the unknot and recall that ϕ ( U ) = 0 ∈ W ( Q ).If K can be unknotted with a negative crossing change, then according to Theorem1.1 (with K − = U and K + = K ) we obtain ϕ ( K ) = (cid:104) K (cid:105) ⊕ (cid:104)− (cid:105) ; σ ( K ) = 0 (cid:104)− K (cid:105) ⊕ (cid:104)− (cid:105) ; σ ( K ) = − K can be unknotted with a positive crossing change, then The-orem 1.1 (this time with K − = K and K + = U ) implies0 = ϕ ( K ) ⊕ (cid:104) K (cid:105) ⊕ (cid:104)− (cid:105) ; σ ( K ) = 0 ϕ ( K ) ⊕ (cid:104)− K (cid:105) ⊕ (cid:104)− (cid:105) ; σ ( K ) = 2Solving each of these for ϕ ( K ) yields ϕ ( K ) = (cid:104)− K (cid:105) ⊕ (cid:104) (cid:105) ; σ ( K ) = 0 (cid:104) K (cid:105) ⊕ (cid:104) (cid:105) ; σ ( K ) = 2This completes the proof of Corollary 1.2.4.2. Proofs of Corollaries 1.3 and 1.4.
We start by considering Corollary 1.3 first.Let K be a knot with unknotting number 2 and let L be the knot obtained from K aftera single crossing change. We let U again denote the unknot. We split our discussionaccording to the type of crossing changes involved in changing K to U via L .4.2.1. K can be unknotted with two negative crossing changes. If K can be unknottedwith two negative crossing changes, then Theorem 1.1 implies that(8) ϕ ( K ) = ϕ ( L ) ⊕ (cid:10) K det L (cid:11) ⊕ (cid:104)− (cid:105) ; σ ( K ) = σ ( L ) ϕ ( L ) ⊕ (cid:10) − K det L (cid:11) ⊕ (cid:104)− (cid:105) ; σ ( K ) = σ ( L ) − ϕ ( L ) = (cid:104) L (cid:105) ⊕ (cid:104)− (cid:105) ; σ ( L ) = 0 (cid:104)− L (cid:105) ⊕ (cid:104)− (cid:105) ; σ ( L ) = − σ ( K ) = − σ ( L ) = − ϕ ( K ) = (cid:10) − K det L (cid:11) ⊕ (cid:104)− L (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) Note that (cid:10) − K det L (cid:11) = (cid:104)− K · det L (cid:105) ∈ W ( Q ) and that, according to relation ( R (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) = (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) also holds in W ( Q ).If σ ( K ) = − σ ( L ) = − σ ( L ) = 0. These two cases, again inconjunction with (8), lead to ϕ ( K ) = (cid:10) K det L (cid:11) ⊕ (cid:104)− L (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) ; σ ( L ) = − (cid:10) − K det L (cid:11) ⊕ (cid:104) L (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) ; σ ( L ) = 0Finally, if σ ( K ) = 0 then σ ( L ) = 0 also so that (8) provides us with ϕ ( K ) = (cid:10) K det L (cid:11) ⊕ (cid:104) L (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) K can be unknotted by two positive crossing changes. If K is knot with σ ( K ) ≤ σ ( K ) = 0 since a positive crossing change cannot increase the signature.Consequently, we also obtain σ ( L ) = 0. Theorem 1.1 now implies that ϕ ( K ) = ϕ ( L ) ⊕ (cid:104)− K det L (cid:105) ⊕ (cid:104) (cid:105) and ϕ ( L ) = (cid:104)− L (cid:105) ⊕ (cid:104) (cid:105) showing that ϕ ( K ) = (cid:104)− K det L (cid:105) ⊕ (cid:104)− L (cid:105) ⊕ (cid:104) (cid:105) ⊕ (cid:104) (cid:105) (since (cid:104) (cid:105) ⊕ (cid:104) (cid:105) = (cid:104) (cid:105) ⊕ (cid:104) (cid:105) ).4.2.3. K can be unknotted with one positive and one negative crossing change. In thissubsection we distinguish further the cases of σ ( K ) = − σ ( K ) = 0.Starting with σ ( K ) = −
2, we note that by signature considerations, it follows thatthe negative crossing change has to increase the signature of K to 0 while the positivecrossing change cannot alter it further. Regardless of whether L is gotten from K bythe positive or the negative crossing change, Theorem 1.1 implies that ϕ ( K ) = (cid:104)− K det L (cid:105) ⊕ (cid:104)− L (cid:105) If σ ( K ) = 0 then there are two possibilities, namely, either both the positive andthe negative crossing change alter the signature, or else, neither does. In both cases,Theorem 1.1 yields ϕ ( K ) = (cid:104)± K det L (cid:105) ⊕ (cid:104)∓ det L (cid:105) This exhausts all possibilities and completes the proof of Corollary 1.3.The proof of Corollary 1.4 follows along the same lines as the proofs of Corollaries1.2 and 1.3. Thus, suppose that K is a knot with σ ( K ) = − n for some n ∈ N and that u ( K ) = n . Let L i be the knot gotten from K by changing i − n crossings sothat, for instance, L = K while L n +1 is the unknot. Signature considerations dictatethat all of these crossing changes by negative crossing changes (since positive crossingchanges cannot increase the signature) and that therefore σ ( L i ) = − n − i + 1). Usingthis observation, Theorem 1.1 implies that ϕ ( L i ) = ϕ ( L i +1 ) ⊕ (cid:104)− L i +1 det L i (cid:105) ⊕ (cid:104)− (cid:105) i = 1 , ..., n Adding these last n equations immediately yields the result of Corollary 1.4. HE RATIONAL WITT CLASS AND THE UNKNOTTING NUMBER OF A KNOT 23 Low crossing examples
The results of Corollaries 1.5 and 1.7 are a direct consequence of applying Corollary1.2 to certain 11 and 12 crossing knots. Our computations were aided by a
Math-ematica computer code written by the author. The Seifert matrices for the variousknots were taken from KnotInfo[2], indeed, without the latter our calculations wouldhave been substantially more time consuming. In the next two subsections, we listwith full details, two sample computations.5.1.
Knots with crossings. In this section we consider the knot K = 11 a as anexample. This knot has signature zero and determinant 105 and so, in order to haveunknotting number 1, its rational Witt class (according to Corollary 1.2) must equal ϕ (11 a ) = (cid:104)± (cid:105) ⊕ (cid:104)∓ (cid:105) for at least one consistent choice of signs.To compute the actual rational Witt class of 11 a , we start with a Seifert form thelatter. From KnotInfo [2], one finds that the symmetrized linking form (cid:96)k + (cid:96)k τ of11 a is represented by the matrix (cid:96)k + (cid:96)k τ = − − − − − − − − −
10 0 0 − −
11 1 − − This matrix is then diagonalized using the Gram-Schmidt procedure (without havingto split off hyperbolic summands) to give ϕ (11 a ) = (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:28) − (cid:29) ⊕ (cid:28) (cid:29) ⊕ (cid:28) (cid:29) ⊕ (cid:28) (cid:29) To compare the latter to (cid:104)± (cid:105) ⊕ (cid:104)∓ (cid:105) , we apply the homomorphism ∂ : W ( Q ) → W ( Z ) to all three forms: ∂ ( ϕ (11 a )) = (cid:104) · (cid:105) = (cid:104) (cid:105) ∂ ( (cid:104)± (cid:105) ⊕ (cid:104)∓ (cid:105) ) = (cid:104)± (cid:105) = (cid:104) (cid:105) Since 1 ∈ ( ˙ Z ) but 3 ∈ ˙ Z − ( ˙ Z ) , the forms (cid:104) (cid:105) and (cid:104) (cid:105) are distinct forms in W ( Z )(see Theorem 2.3). Accordingly, ϕ (11 a ) cannot equal (cid:104) (cid:105) ⊕ (cid:104)− (cid:105) nor (cid:104)− (cid:105) ⊕ (cid:104) (cid:105) in W ( Q ) and therefore u (11 a ) ≥
2. Since an explicit unknotting of 11 a with twocrossing changes is easily found, we arrive at u (11 a ) = 2 as claimed in Corollary 1.5.5.2. Knots with crossings. As an example among 12 crossings knots, we singleout the non-alternating knot K = 12 n . This knot has signature − u (12 n ) = 1, then according to Corollary 1.2, its rational Witt classwould have to equal (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) . The actual rational Witt class of 12 n is again computed by starting with the matrix representing (cid:96)k + (cid:96)k τ which one finds on KnotInfo[2] to be (cid:96)k + (cid:96)k τ = − − − − − − − − −
10 0 − − − − − − − − − − − −
10 0 0 − − −
10 0 0 − − − − − − − − − − − − − − − This, when diagonalized with the Gram-Schmidt algorithm, yields (after a cancellationof (cid:104) (cid:105) ⊕ (cid:104)− (cid:105) ) ϕ (12 n ) = (cid:28) − (cid:29) ⊕ (cid:28) − (cid:29) ⊕ (cid:28) (cid:29) ⊕ (cid:28) (cid:29) ⊕ (cid:28) − (cid:29) ⊕ (cid:28) − (cid:29) Applying ∂ to these gives ∂ ( (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) ) = (cid:104)− (cid:105) = (cid:104) (cid:105) ∂ ( ϕ (12 n )) = (cid:104)− · (cid:105) = (cid:104) (cid:105) Since 16 is a square in Z while 35 isn’t, the two forms ϕ (12 n ) and (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) cannot be equal in W ( Q ) (cf. Theorem 2.3) and so u (12 n ) ≥ Proof of Corollary 1.15.
In Corollary 1.15 we examined the knot 10 . Forconvenience, we work here with the knot K = 10 (the mirror image of the knote10 ) which has signature − u ( K ) = 2, Corollary 1.3implies that the rational Witt class of K must be equal to(9) ϕ (10 ) = (cid:104)− d (cid:105) ⊕ (cid:104)− d (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) where d = det L is the determinant of the knot L obtained from K after only onecrossing change. The signature of L is clearly −
2. On the other hand, the actualrational Witt class of 10 can be computed to be(10) ϕ (10 ) = (cid:104) (cid:105) ⊕ (cid:104) (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) If one assumes that L is a knot with 9 or fewer crossings, then d takes on odd valuesfrom 1 to 75 (as can be seen by examining the tables at KnotInfo [2]). One then asks,for which values of d in that range, are the rational Witt classes from (9) and (10)equal. Using Mathematica, one finds the answer is for d = 3 , , , , , , , , , , , , L with 9 or fewer crossings, with unknotting number 1, withsignature − d ’s from the previous line, arethe knots L = 3 , , , , , , , , , ¯9 , ¯9 , ¯9 as claimed in Corollary 1.15. HE RATIONAL WITT CLASS AND THE UNKNOTTING NUMBER OF A KNOT 25 Pretzel knots
The rational Witt classes and the signatures of pretzel knots P ( p , ..., p n ) have beencompletely determined in [8]. The pertinent statements are contained in Theorems 1.2– 1.4 and in Theorem 1.18 in [8]. For the reader’s convenience, we provide below thoseresults relevant to Section 1.2.3 of the present article.6.1. Case -stranded pretzel knots. Given a pretzel knot P ( p , p , p ) with p , p odd and p (cid:54) = 0 even, its rational Witt class and signature are given by (courtesy of[8]): ϕ ( P ( p , p , p )) = (cid:77) i =1 (cid:16) ⊕ | p i |− k =1 (cid:104)− ε k k ( k + 1) (cid:105) (cid:17) ⊕ (cid:104)− p + p p p (cid:105) ⊕ (cid:104) det P ( p ,p ,p ) p + p (cid:105) σ ( P ( p , p , p )) = ( ε + ε ) − ( p + p ) − Sign ( p + p p p ) + Sign ( detP ( p ,p ,p ) p + p )In the above, ε k = Sign ( p k ) and det P ( p , p , p ) = p p + p p + p p is a signed versionof the determinant used in [8]. We have implicitely assumed that p + p (cid:54) = 0 for if p + p = 0, then ϕ ( P ( p , − p , p )) = 0 and σ ( P ( p , − p , p )) = 0.The knots considered in Corollary 1.9 make the choices p ≥ p = 4 − p and p > − p (4 − p )4 , the latter condition ensuring that det P ( p , − p , p ) >
0. The rationalWitt class and signature of P ( p , − p , p ) then become ϕ ( P ( p , − p , p )) = (cid:77) i =1 (cid:16) ⊕ | p i |− k =1 (cid:104)− ε k k ( k + 1) (cid:105) (cid:17) ⊕ (cid:104)− p (4 − p ) (cid:105) ⊕ (cid:104) p + p (4 − p ) (cid:105) σ ( P ( p , − p , p )) = − ϕ ( P ( p , − p , p )) by using the next lemma. Lemma 6.1.
For any n ≥ and for ε ∈ {± } , the equality ⊕ n − k =1 (cid:104)− ε · k ( k + 1) (cid:105) = (cid:104) ε · n (cid:105) ⊕ (cid:32) n (cid:77) i =1 (cid:104)− ε (cid:105) (cid:33) holds in W ( Q ) .Proof. The claim of the lemma follows easily from an induction argument. When n = 2,the equality (cid:104)− ε · (cid:105) = (cid:104) ε · (cid:105) ⊕ (cid:104)− ε (cid:105) ⊕ (cid:104)− ε (cid:105) follows from an application of relation ( R
3) from Theorem 2.2, by which (cid:104)− ε (cid:105) ⊕(cid:104)− ε (cid:105) = (cid:104)− ε (cid:105) ⊕ (cid:104)− ε (cid:105) . Proceeding by induction, and using again ( R ⊕ n − k =1 (cid:104)− εk ( k + 1) (cid:105) = (cid:104)− ε ( n − n (cid:105) ⊕ (cid:0) ⊕ n − k =1 (cid:104)− εk ( k + 1) (cid:105) (cid:1) = (cid:104)− ε ( n − n (cid:105) ⊕ (cid:104) ε ( n − (cid:105) ⊕ (cid:32) n − (cid:77) i =1 (cid:104)− ε (cid:105) (cid:33) (now use ( R (cid:104)− ε (cid:105) ⊕ (cid:104) εn (cid:105) ⊕ (cid:32) n − (cid:77) i =1 (cid:104)− ε (cid:105) (cid:33) = (cid:104) εn (cid:105) ⊕ (cid:32) n (cid:77) i =1 (cid:104)− ε (cid:105) (cid:33) This proves the lemma. (cid:3)
Lemma 6.1 allows us to substantially simplify the expression for ϕ ( P ( p , − p , p ))from (11), to obtain ϕ ( P ( p , − p , p )) = (cid:104) p (cid:105) ⊕ (cid:104) − p (cid:105) ⊕ (cid:104)− p (4 − p ) (cid:105) ⊕ (cid:104) p + p (4 − p ) (cid:105) ⊕ (cid:32) (cid:77) i =1 (cid:104)− (cid:105) (cid:33) Applying relation ( R
3) to the first two terms on the right-hand side above (and carryingthe third term), yields an additional simplification: (cid:104) p (cid:105) ⊕ (cid:104) − p (cid:105) ⊕ (cid:104)− p (4 − p ) (cid:105) = (cid:104) (cid:105) ⊕ (cid:104) p (4 − p ) (cid:105) ⊕ (cid:104)− p (4 − p ) (cid:105) = (cid:104) (cid:105) With this last equation, we finally arrive at ϕ ( P ( p , − p , p )) = (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104) p + p (4 − p ) (cid:105) ⊕ (cid:32) (cid:77) i =1 (cid:104)− (cid:105) (cid:33) Corollary 1.9 follows immediately from this and from Corollary 1.2.To explain the conclusions from Example 1.10, consider the knot P (7 , − , r ) with r ≥
6. Then u ( P (7 , − , r )) = 1 forces the equality (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) ⊕ (cid:104) r − (cid:105) = (cid:104)− r − (cid:105) in W ( Q ). If there is a prime p with 4 r −
21 = p m +1 · β and with gcd( β, p ) = 1, the ∂ p applied to the above, leads to the equality (cid:104) β (cid:105) = (cid:104)− β (cid:105) in W ( Z p )This latter equality can only be valid if − Z p . If r = 2 k · (cid:96) +1 with k, (cid:96) ∈ N , then 4 r −
21 = 7(8 k · (cid:96) −
3) so we can choose p = 7 since, indeed, − Z . Example 1.11 is analyzed similarly. HE RATIONAL WITT CLASS AND THE UNKNOTTING NUMBER OF A KNOT 27
Case of -stranded pretzel knots. Consider here a 4-stranded pretzel knot P ( p , p , p , p ) with p , p , p odd and with p (cid:54) = 0 even. In this case, the rational Wittclass and signature are given by (as proved in [8], with the use of Lemma 6.1 for aslight simplification): ϕ ( P ( p , p , p , p )) = (cid:77) i =1 (cid:104) p i (cid:105) ⊕ | p i | (cid:77) k =1 (cid:104)− ε k (cid:105) ⊕ (cid:68) − det P ( p ,p ,p ,p ) p p p p (cid:69) σ ( P ( p , p , p , p )) = (cid:88) i =1 ( ε i − p i ) − Sign ( p p p p det P ( p , p , p , p ))where again ε k = Sign ( p k ) and where this time det P ( p , p , p , p ) = p p p + p p p + p p p + p p p .Turning to the knot P ( p, p, p, − p −
1) considered in Corollary 1.12, we note that itsdeterminant is given by − p (8 p + 3). We remind the reader that p is an odd, positiveinteger. When inserted into the above form of the rational Witt class, this determinantformula provides us with ϕ ( P ( p, p, p, − p − (cid:104) p (cid:105) ⊕ (cid:104) p (cid:105) ⊕ (cid:104) p (cid:105) ⊕ (cid:104)− p − (cid:105) ⊕ (cid:68) − p +3 p (3 p +1) (cid:69) ⊕ (cid:104) (cid:105) σ ( P ( p, p, p, − p − ∂ to ϕ ( P ( p, p, p, − p − p = 2 + (2 k + 1) · (cid:96) +1 ,yields ∂ ( ϕ ( P ( p, p, p, − p − (cid:104) (cid:105) ∈ W ( Z )On the other hand, if we had u ( P ( p, p, p, − p − ϕ ( P ( p, p, p, − p − (cid:104) (cid:105) ⊕ (cid:104) p + 3) (cid:105) . However, ∂ applied to thislast form gives (with p = 2 + (2 k + 1) · (cid:96) +1 ) ∂ ( (cid:104) (cid:105) ⊕ (cid:104) p + 3) (cid:105) ) = (cid:104) (cid:105) ∈ W ( Z )Since 5 is a square in Z but 2 is not, we see that the equality ϕ ( P ( p, p, p, − p − (cid:104) (cid:105) ⊕ (cid:104) p + 3) (cid:105) cannot be satisfied in W ( Q ). The conclusion of Example 1.13 follows.6.3. Upward stabilizations.
The notion of upward stabilization, used in Corollary1.14, was introduced in Definition 1.6 from [8]. As noted in [8], if L is obtained by anupward stabilization from the pretzel knot K , then ϕ ( L ) = ϕ ( L ) while det L = det K · λ for some integer λ . These facts make Corollary 1.14 evident.7. Comparison with work of Lickorish [16]This final section compares our work to that of R. Lickorish from [16]. To set up theframework for comparison, we explore a few preliminaries first.Given a rational homology 3-sphere Y , let λ : H ( Y ; Z ) × H ( Y ; Z ) → Q / Z be its associated linking pairing defined as follows: Given two curves α, β ∈ H ( Y ; Z ),there exists a nonzero integer n such that n · α bounds a 2-chain, say σ . With such a σ in place, we set λ ( α, β ) = σ · βn ∈ Q / Z where the numerator of the right hand side above is the usual intersection pairingbetween homology groups of complementary dimensions. For our intentions, Y will bethe 2-fold branched cover Σ K of S with branching set a knot K .Lickorish proves the following lemma (Lemma 2 in [16]): Lemma 7.1 (Lickorish [16]) . Let Y be the -manifold obtained by p/q -framed Dehnsurgery on a knot L ⊂ S , with p (cid:54) = 0 . Then H ( Y ; Z ) is cyclic of order | p | , generatedby a meridian µ and moreover λ ( µ, µ ) = qp ∈ Q / Z This lemma establishes a bridge towards studying unknotting number 1 knots since,if u ( K ) = 1, then Σ K is obtained as n/ L with n an odd integer.This fact was already known to Montesinos [17, 18] but a complete proof is also given byLickorish in [16]. Thus, according to Lemma 7.1, if u ( K ) = 1, then H (Σ K ; Z ) is cyclicof order | n | (and hence n = ± det K ) and possesses a generator µ with λ ( µ, µ ) = 2 /n .If K happens to be a 2-bridge knot with Σ K = L ( p, q ), then Σ K is also gotten by p/q -framed surgery on the unknot and therefore, again according to Lemma 7.1, therehas to be a generator µ (cid:48) of H (Σ K ; Z ) with λ ( µ (cid:48) , µ (cid:48) ) = q/p . Since H (Σ K , ; Z ) is cyclic,there is an integer t such that µ (cid:48) = t · µ and hence, by applying λ to this, we also findthat qp = t · n in Q / Z This equation is re-captured in an equivalent format (since n, p = ± det K ) by the nextstatement. Theorem 7.2 (Lickorish [16]) . Let K be a -bridge knot with -fold brached cover thelens space L ( p, q ) . If u ( K ) = 1 , then the congruence q ≡ ± t ( mod det K ) must hold for some t ∈ Z . Lickorish in [16] goes on to apply this theorem to the knot K = 7 for which Σ K = L (15 , had unknotting number 1, there would have to be a solution t ∈ Z of the congruence 4 ≡ ± t (mod 15)It is easy to see that there is no such t showing that u (7 ) > u (7 ) = 2(since an unknotting of 7 with two crossing changes is easily found).Our methods equally well apply to the example K = 7 . Namely, its rational Wittclass was computed in Section 3 as ϕ (7 ) = (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) HE RATIONAL WITT CLASS AND THE UNKNOTTING NUMBER OF A KNOT 29
If we had u (7 ) = 1, then Corollary 1.2 would force the equality (cid:104)− (cid:105) ⊕ (cid:104)− (cid:105) = (cid:104)− (cid:105) ⊕ (cid:104)− · (cid:105) in W ( Q ). This equality is easily seen to fail by applying ∂ to it,reducing it to (cid:104) (cid:105) = (cid:104) (cid:105) in W ( Z ) (which fails since 2 is not a square in Z ).Lickorish’s obstruction and our obstruction to the unknotting number being 1, seemsimilar, at least in their origins, as both derive from an intersection pairing representedby the linking form of the knot. Yet, the two obstructions are not equivalent as thenext example shows. Example 7.3.
Consider the knot K = 8 whose twofold branched cover is the lensspace L (25 , . If the unknotting number of were , Theorem 7.2 would guaranteea solution of the congruence ≡ ± t ( mod . But there is no such solution sinceneither nor are squares mod . It follows that u (8 ) ≥ (in fact, u (8 ) = 2 ).On the other hand, the obstruction to u (8 ) = 1 from Corollary 1.2, just yields ϕ (8 ) = 0 , an equality that is in fact valid. Finding possible examples of two-bridge knots for which our obstruction providespositive results where Lickorish’s doesn’t, will have to wait until the rational Wittclasses of two-bridge knots have been computed (a project currently in progress). Thereare, however, no such examples among two-bridge knots with 10 or fewer crossings.
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