Alexander quandle lower bounds for link genera
aa r X i v : . [ m a t h . G T ] A p r ALEXANDER QUANDLE LOWER BOUNDS FOR LINK GENERA
R. BENEDETTI, R. FRIGERIO
Abstract.
Every finite field F q , q = p n , carries several Alexander quandle structures X =( F q , ∗ ). We denote by Q F the family of these quandles, where p and n vary respectivelyamong the odd primes and the positive integers.For every k –components oriented link L , every partition P of L into h := |P| sublinks,and every labelling z ∈ N h of such a partition, the number of X –colorings of any diagram of( L, z ) is a well–defined invariant of ( L, P ), of the form q a X ( L, P ,z )+1 for some natural number a X ( L, P , z ). Letting X and z vary respectively in Q F and among the labellings of P , wedefine the derived invariant A Q ( L, P ) := sup { a X ( L, P , z ) } .If P M is such that |P M | = k , we show that A Q ( L, P M ) ≤ t ( L ), where t ( L ) is thetunnel number of L , generalizing a result by Ishii. If P is a “boundary partition” of L and g ( L, P ) denotes the infimum among the sums of the genera of a system of disjoint Seifertsurfaces for the L j ’s, then we show that A Q ( L, P ) ≤ g ( L, P ) + 2 k − |P| −
1. We pointout further properties of A Q ( L, P ), mostly in the case of A Q ( L ) := A Q ( L, P m ), |P m | = 1.By elaborating on a suitable version of a result by Inoue, we show that when L = K is aknot then A Q ( K ) ≤ A ( K ), where A ( K ) is the breadth of the Alexander polynomial of K .However, for every g ≥ g knots having the same Alexanderpolynomial but different quandle invariants A Q . Moreover, in such examples A Q providessharp lower bounds for the genera of the knots. On the other hand, we show that A Q ( L )can give better lower bounds on the genus than A ( L ), when L has k ≥ A Q ( L ) it is enough to consider only colorings withrespect to the constant labelling z = 1. In the case when L = K is a knot, if either A Q ( K ) = A ( K ) or A Q ( K ) provides a sharp lower bound for the knot genus, or if A Q ( K ) = 1,then A Q ( K ) can be realized by means of the proper subfamily of quandles { X = ( F p , ∗ ) } ,where p varies among the odd primes. Introduction A quandle X = ( X, ∗ ) is a non-empty set X with a binary operation ∗ satisfying the followingaxioms:(Q1) a ∗ a = a for every a ∈ X ;(Q2) ( a ∗ b ) ∗ c = ( a ∗ c ) ∗ ( b ∗ c ) for every a, b, c ∈ X ;(Q3) for every b ∈ X , the map S b : X → X defined by S b ( x ) = x ∗ b is a bijection.Every set X admits the trivial quandle structure X (0) with the operation defined by a ∗ b = a for every a, b ∈ X . Given a quandle X = ( X, ∗ ) := ( X, ∗ ) := X (1), for every integer n > X ( n ) := ( X, ∗ n ), where for every a, b ∈ X one sets a ∗ n b = ( a ∗ n − b ) ∗ b . Every finite quandle X has a well defined type t X ≥
1, such that X ( n ) = X ( m ) if and only if m = n mod ( t X ).1.1. Quandle colorings.
Let K ⊂ S be an oriented (smooth or PL) knot. The fundamentalquandle of K was defined independently by Joyce [11] and Matveev [14]. They also showedthat the fundamental quandle is a classifying invariant of knots. If X is a finite quandle, thenfor every natural number z ≥ c X ( K, z ) ∈ N which counts the Mathematics Subject Classification.
Key words and phrases.
Alexander quandle, quandle colorings, Alexander ideals, genus, tunnel number. representations of the fundamental quandle of K in X ( z ). It turns out that c X ( K, z ) can becomputed as the number of suitably defined X ( z )- colorings of any diagram D of K . In orderto simplify the notation, we denote by ( K, z ) a knot labelled by a natural number z . Anylabel of K obviously defines a label on every diagram of K , and if ( D, z ) is any diagram of(
K, z ), then we define a X -coloring of ( D, z ) to be a X ( z )-coloring of D . Of course, if X hastype t X ≥
1, then we may (and we will) actually consider Z t X -valued (rather than N -valued)labels, where we understand that, for every j ≥
2, we identify Z j = Z /j Z with the set ofcanonical representatives { , . . . , j − } . The definition of c X ( K, z ) easily extends to the caseof oriented labelled links. In fact, let L = K ∪ . . . ∪ K k be an oriented link with k components,and let P = ( L , . . . , L h ) be a partition of L , where the L i ’s are disjoint sublinks of L suchthat L = L ∪ . . . ∪ L h . We denote by |P| = h the number of links in the partition P . Aspecial rˆole is played by the maximal (resp. minimal) partition P M (resp. P m ) of L , which canbe characterized as the unique partition such that |P| = k (resp. |P| = 1), so that L i = K i for i = 1 , . . . , k (resp. L = L ). A ( N –valued) P –cycle for L is a map z : { , . . . , h } → Z thatassigns the non–negative integer z i = z ( i ) to every component of the sublink L i of L . In whatfollows, we often denote such a cycle ( z , . . . , z h ) simply by z , and we denote by 0 (resp. by1) the cycle that assigns the integer 0 (resp. 1) to every component of L .If D is a diagram of L , then any P –cycle z for L descends to a P –cycle ( D, z ) for D . InSection 2 we recall the definition of X –coloring of ( D, z ). The total number of such colorings isdenoted by c X ( D, P , z ), and turns out to be independent of the chosen diagram, thus definingan invariant c X ( L, P , z ) of the partitioned and labelled link ( L, P , z ).1.2. Alexander quandles.
In this paper we deal with a concrete family Q F of finite quan-dles, that we are now going to introduce. Let us fix some notation we will extensively usefrom now on. For every odd prime p ≥
3, we denote by Λ (resp. Λ m ) the ring Z [ t, t − ](resp. Z m [ t, t − ]). Moreover, π m : Λ → Λ m is the ring homomorphism induced by the pro-jection Z → Z m . For every p ( t ) ∈ Λ (resp. p ( t ) ∈ Λ m ) we define the breadth br p ( t ) of p ( t )as the difference between the highest and the lowest exponent of the non–null monomials of p ( t ). In particular, the breadth of any constant polynomial (including the null polynomial)is equal to 0 (the reason why we set br 0 = 0 will be clear soon). If p ( t ) , q ( t ) are elements ofΛ (resp. of Λ m ), we write p ( t ) . = q ( t ) if p ( t ) and q ( t ) generate the same ideal of Λ (resp. Λ m ), i.e. if and only if p ( t ) = ± t k q ( t ), k ∈ Z (resp. p ( t ) = at k q ( t ), a ∈ Z ∗ m , k ∈ Z ).Recall that a finite Alexander quandle is a pair ( M, ∗ ), where M is a finite Λ m -module andthe quandle operation is defined (in terms of the module operations) by a ∗ b := ta + (1 − t ) b . We now define the family Q F of finite Alexander quandles we are interested in. Fix an oddprime p , let h ( t ) be an irreducible element of Λ p with positive breadth br h ( t ) = n ≥
1, andlet us define F ( p, h ( t )) as the quotient ring F ( p, h ( t )) = Λ p / ( h ( t )) . If ˆ h ( t ) ∈ Z p [ t ] ⊆ Λ p is such that ˆ h ( t ) . = h ( t ) and h (0) = 0, then it is readily seen the theinclusion Z p [ t ] ֒ → Λ p induces an isomorphism Z p [ t ] / (ˆ h ( t )) → F ( p, h ( t )). Since deg ˆ h ( t ) =br h ( t ) = n , it follows that F ( p, h ( t )) is a finite field of cardinality q = p n .We may therefore define the Alexander quandle X := ( F ( p, h ( t )) , ∗ ) by setting a ∗ b := ta + (1 − t ) b for every a, b ∈ F ( p, h ( t )) , where t is the class of t in F ( p, h ( t )). Once q = p n is fixed, there exists only one finite field F q up to field isomorphism. However, even in the case when h ( t ) ∈ Λ p and h ( t ) ∈ Λ p have the same breadth, it may happen that the quandles ( F ( p, h ( t )) , ∗ ) and ( F ( p, h ( t )) , ∗ ) arenot isomorphic (see Remark 2.2).We now set Q F ( m ) = { ( F ( p, h ( t )) , ∗ ) | ≤ br h ( t ) ≤ m } and Q F = [ m ≥ Q F ( m ) . The invariant A Q ( L, P ) . Let us fix a quandle X = ( F ( p, h ( t )) , ∗ ). Let D be a diagramof a link L , let P be a partition of L and z be a P –cycle for L . If X = ( F ( p, h ( t )) , ∗ ), then itturns out that the space of the X -colorings of ( D, z ) is a F ( p, h ( t ))-vector space of dimension d X ( L, P , z ) ≥
1. Hence c X ( L, P , z ) = q d X ( L, P ,z ) , so the whole information about c X ( L, P , z )is encoded by the integer a X ( L, P , z ) := d X ( L, P , z ) − ≥
0. For instance, if L = K isa knot, then the A Q - marked spectrum of K , that is the set { a X ( K, n ) | X ∈ Q F , n ∈ Z t X } ,considered as a map defined on a subset of Q F × N , carries the whole information provided bythese quandle coloring invariants. In this paper we concentrate our attention on the derivedinvariant defined by A Q ( L, P ) := sup { a X ( L, P , z ) } , where X varies in Q F and z varies among the P –cycles of L . We show in Lemma 2.5 that A Q ( L, P M ) is an invariant of the unoriented link L . On the contrary, for a generic partition P the invariant A Q ( L, P ) can depend on the orientations of the components of L . For everypartition P we have of course A Q ( L, P m ) ≤ A Q ( L, P ) ≤ A Q ( L, P M ) . When L = K is a knot, of course there is only one partition ( P m = P M ) and we simply write A Q ( K ). Moreover, henceforth the invariant A Q ( L, P m ) will be denoted simply by A Q ( L ).1.4. A lower bound on the tunnel number of links.
Recall that the tunnel number t ( L )of a link L ⊂ S is the minimum number of properly embedded arcs in S \ L to be attachedto L in such a way that the regular neighbourhood of the resulting connected spatial graphis an unknotted handlebody ( i.e. it is the regular neighbourhood also of a graph lying on a2–dimensional sphere S ⊆ S ). Of course, the tunnel number is an invariant of unoriented links.The argument of Proposition 6 in [9] (originally given for quandles of type 2) easily extendsto our situation (see Proposition 2.6) and allows us to prove (in Subsection 2.3) the following: Proposition 1.1.
For every link L we have A Q ( L, P M ) ≤ t ( L ) . In particular, A Q ( L, P ) is always finite.1.5. Lower bounds on genera of links.
We say that P = ( L , . . . , L h ) is a boundarypartition of L = K ∪ . . . ∪ K k if there exists a system (Σ , . . . , Σ h ) of disjoint connectedoriented surfaces such that Σ i is a Seifert surface of L i (i.e. ∂ Σ i = L i as oriented 1–manifolds,where ∂ Σ i inherits the orientation induced by Σ i ), for every i = 1 , . . . , h . If P is a boundarypartition of L , then we define the genus of ( L, P ) by g ( L, P ) := min ( h X i =1 g (Σ i ) ) , R. BENEDETTI, R. FRIGERIO where (Σ , . . . , Σ h ) varies among such systems of Seifert surfaces. If P is not a boundarypartition, we set g ( L, P ) = + ∞ . Every link admits a connected Seifert surface, so P m is always a boundary partition, andthe number g ( L ) := g ( L, P m ) is usually known as the genus of L . On the other hand, P M isa boundary partition if and only if L is a boundary link. It is immediate that g ( L, P M ) is aninvariant of the unoriented link L .The following result provides the fundamental estimate on link genera provided by quandleinvariants, and is proved in Section 5 (note that the statement below is non–trivial only when P is a boundary partition): Theorem 1.2.
Let ( L, P ) be a k –component partitioned link, and let z , z be P –cycles for L . Then we have: | a X ( L, P , z ) − a X ( L, P , z ) | ≤ g ( L, P ) + k − |P| . Recall that 0 is the cycle that assigns the integer 0 to every component of L . In thehypotheses of the previous Theorem, for every partition P we have a X ( L, P ,
0) = k − Corollary 1.3. If ( L, P ) is a k –component partitioned link, then: A Q ( L, P ) ≤ g ( L, P ) + 2 k − |P| − . In particular: • If P M is the maximal partition of L , then A Q ( L, P M ) ≤ g ( L, P M ) + k − . • If P m is the minimal partition of L , then A Q ( L ) = A Q ( L, P m ) ≤ g ( L ) + 2 k − . • For every knot K we have A Q ( K ) ≤ g ( K ) . Remark 1.4.
Let L = K be a knot. Clearly if A Q ( K ) = 2 h is even, then g ( K ) ≥ h ; if A Q ( K ) = 2 h − g ( K ) ≥ h . In particular, if g ( K ) = g , the bound on thegenus provided by A Q ( K ) is sharp if and only if A Q ( K ) ≥ g −
1. The very same remarkalso applies in the general case of partitioned links.1.6.
Alexander ideals and quandle coloring invariants of links.
Once Theorem 1.2and Corollary 1.3 are established, we will discuss a bit the performances of the A Q ( L, P )’sas link invariants as well as lower bounds for the link genera. We will mostly concentrate onthe case P = P m .The statement of Theorem 1.3 reminds the classical lower bound (see e.g. [4, Theorem7.2.1]) A ( L ) ≤ g ( L ) + k − , where A ( L ) is the breadth of the Alexander polynomial∆( L )( t ) := det (cid:0) S ( L ) − tS ( L ) T (cid:1) ,S ( L ) being any Seifert matrix of L (of course, the above estimate holds only if we agree thatthe breadth of the null polynomial is equal to 0).Let us introduce some notations that will prove useful in describing the relations betweenAlexander polynomial invariants and quandle coloring invariants of links. We refer to [6, 7] for the definitions and some basic results about Alexander ideals of links and modules. Asusual, we denote by p an odd prime number. If K, K ′ are disjoint oriented knots in S ,we denote by lk( K, K ′ ) the usual linking number of K and K ′ . For every oriented link L = K ∪ . . . ∪ K k , let e X ( L ) the total linking number covering of the complement C ( L ) of L in S , i.e. the covering associated to the kernel of the homomorphism α : π ( C ( L )) → Z , α ( γ ) = P ki =1 lk( γ, K i ). The covering e X ( L ) → C ( L ) is infinite cyclic, so the homology group A ( L ) = H ( e X ( L ); Z ) (resp. A ( p ) ( L ) = H ( e X ( L ); Z p )) admits a natural structure of Λ–module(resp. Λ p –module) such that t ∈ Λ (resp. t ∈ Λ p ) acts on A ( L ) (resp. A ( p ) ( L )) as the mapinduced by the covering translation corresponding to a loop γ ∈ π ( C ( L )) such that α ( γ ) = 1.Let E i ( L ) ⊆ Λ (resp. E ( p ) i ( L ) ⊆ Λ p ) be the i –th elementary ideal of A ( L ) (resp. A ( p ) ( L )).Since Λ is a U.F.D. (resp. Λ p is a P.I.D.), for every i ≥ i ( L ) ∈ Λ(resp. ∆ ( p ) i ( L ) ∈ Λ) as the generator of the smallest principal ideal containing E i − ( L ) (resp. ofthe ideal E ( p ) i − ( L )). Then, ∆ i ( L )( t ) (resp. ∆ ( p ) i ( L )( t )) is well–defined only up to invertibles inΛ (resp. Λ p ), i.e. up to multiplication by ± t k , k ∈ Z (resp. at k , a ∈ Z ∗ p , k ∈ Z ). Since A ( L )admits the square presentation matrix S ( L ) − tS ( L ) T we have∆ ( L )( t ) = det (cid:0) S ( L ) − tS ( L ) T (cid:1) = ∆( L )( t ) , so ∆ ( L )( t ) coincides with the Alexander polynomial of L . Some relations between ∆ i ( L )( t )and ∆ ( p ) i ( L )( t ) are described in Corollary 6.6 (but see also Remarks 6.8 and 6.9).Recall that E ( p ) i ( L ) ⊆ E ( p ) i +1 ( L ) for every i ∈ N , so either ∆ ( p ) i ( L )( t ) = ∆ ( p ) i +1 ( L )( t ) = 0,or ∆ ( p ) i +1 ( L )( t ) divides ∆ ( p ) i ( L )( t ) in Λ p . Therefore, it makes sense to define the polynomial e ( p ) i ( L )( t ) ∈ Λ p as follows: e ( p ) i ( L )( t ) = 0 if ∆ ( p ) i ( L )( t ) = ∆ ( p ) i +1 ( L )( t ) = 0 ,e ( p ) i ( L )( t ) = ∆ ( p ) i ( L )( t )∆ ( p ) i +1 ( L )( t ) otherwise . Also recall (see Lemma 6.5) that there exists a minimum i ∈ N such that ∆ ( p ) i ( L )( t ) = ± i ≥ i , whence e ( p ) i ( L )( t ) = ± i ≥ i .In the very same way we can define the family of polynomials with integer coefficients { e i ( L )( t ) } in Λ.In Section 7 we prove the following result, which is strongly related with the main resultof [8], although there is a subtlety in the statement that we will point out below. Theorem 1.5.
Suppose L is a link, and take a quandle X = ( F ( p, h ( t )) , ∗ ) ∈ Q F . Then thespace of X –colorings of ( L, P m , z ) is in bijection with the module F ( p, h ( t )) ⊕ ∞ M i =1 Λ p . (cid:16) e ( p ) i ( L )( t z ) , h ( t ) (cid:17)! , where z = z ( P m ) is the value assigned by z to every component of L , and ( e ( p ) i ( L )( t ) , h ( t )) ⊆ Λ p is the ideal generated by e ( p ) i ( t ) and h ( t ) . Let us compare our result with Inoue’s Theorem [8, Theorem 1]. We first observe that in [8,Theorem 1] only the case when L = K is a knot and z = 1 is considered. Moreover, our proofof Theorem 1.5 does not make use of Fox differential calculus, and is therefore quite differentfrom Inoue’s argument. However, maybe the most interesting feature of the statement ofTheorem 1.5 is that R. BENEDETTI, R. FRIGERIO
The polynomial e ( p ) i ( L )( t ) ∈ Λ p is not the reduction mod (p), say π p ( e i ( L )( t )) , of e i ( L )( t )as it could be suggested by the original statement of [8, Theorem1]. In fact, in Remark 6.8we show that the statement of Theorem 1.5 does not hold if the e ( p ) i ( L )( t )’s are replaced bythe π p ( e i ( L )( t ))’s. In other words, in the following statement from the abstract of [8]: “ The number of all quandle homomorphisms of a knot quandle of a knot to an Alexanderquandle is completely determined by Alexander polynomials of the knot” the mentioned Alexander polynomials are not just the ones relative to the usual AlexanderΛ–module A ( L ), but one has to consider the polynomials associated to the whole family ofΛ p –modules A ( p ) ( L ).In the case of knots, building on Theorem 1.5 we deduce (in Section 7) the following: Theorem 1.6.
For every knot K we have A Q ( K ) ≤ A ( K ) . Moreover A Q ( K ) = 0 if and only if A ( K ) = 0 . In particular, as a bound on the genus of knots, the invariant A Q ( K ) is dominated by A ( K ). Moreover, the following example shows that, when L = K is a knot, the differencebetween A ( K ) and A Q ( K ) may become arbitrarily large.For any pair p, q of coprime integers, the torus knot T p,q has tunnel number t ( T p,q ) = 1(and its unknotting tunnels have been classified in [2]). Denoting by ∆ p,q ( t ) the Alexanderpolynomial of T p,q , it is well–known (see e.g [3, page 128]) that:∆ p,q ( t ) = ( t pq − t − t p − t q − , g ( T p,q ) = ( p − q − . In particular, the bound on the genus of T p,q provided by the Alexander polynomial is sharp, i.e. we have A ( T p,q ) = 2 g ( T p,q ). As a consequence, we get the following: Proposition 1.7.
For every n ∈ N there exist an integer n ≥ n and a knot K such that A Q ( K ) ≤ t ( K ) = 1 < n = 2 g ( K ) = A ( K ) . While being dominated by A ( · ) in the case of knots, the quandle invariant A Q ( · ) mayprovide a better lower bound on the genus of k –component links, k ≥
2. Moreover, A Q ( · )can provide a sharp lower bound of the knot genus, and can distinguish knots sharing boththe genus and the Alexander polynomial. More precisely, in Section 9 we prove the followingPropositions: Proposition 1.8.
For every n ∈ N there exists a link L such that A Q ( L ) ≥ n and A ( L ) = 0 . Proposition 1.9.
Let us fix g ≥ . Then, for every r , r such that ≤ r ≤ r ≤ r ≤ g ,there exist knots K and K such that the following conditions hold: g ( K ) = g ( K ) = g, ∆( K ) = ∆( K ) (whence A ( K ) = A ( K )) , while A Q ( K ) = r , A Q ( K ) = r . Moreover, we can require that both A Q ( K ) and A Q ( K ) are realized by means of somedihedral quandle with cycle z = 1 . Further properties of the invariant A Q . Let L = K be a knot, and let us lookfor proper subfamilies of Q F that carry the relevant information for computing A Q ( K ). InLemma 7.1 we show that A Q ( K ) is completely determined by the number of colorings relativeto the cycle z = 1: more precisely, we show that for every knot K there exists X ∈ Q F suchthat A Q ( K ) = a X ( K, δ ( K ) := inf { n ∈ N ∗ | A Q ( K ) = sup { a X ( K, | X ∈ Q F ( n ) }} ∈ N ∗ .θ ( K ) := inf { t X | A Q ( K ) = a X ( K, , X ∈ Q F ( δ ( K )) } ∈ N ∗ . In Subsection 7.4 we prove the following:
Proposition 1.10.
Let K be a knot. (1) If θ ( K ) > , then θ ( K ) ≥ δ ( K ) + 1 . (2) If A Q ( K ) = 1 , then δ ( K ) = 1 . (3) If A ( K ) > , then δ ( K ) ≤ A ( K )max { , A Q ( K ) } . (4) Suppose that A Q ( K ) = A ( K ) or A Q ( K ) = A ( K ) − . Then δ ( K ) = 1 . Moreover,there exist an odd prime p and an element a ∈ Z ∗ p such that ( t − a ) A Q ( K ) divides ∆ ( p )1 ( K )( t ) in Λ p . (5) If A Q ( K ) = A ( K ) , then δ ( K ) = 1 and there exist an odd prime p and an element a ∈ Z ∗ p such that ∆ ( p )1 ( K )( t ) . = ( t − a ) A ( K ) in Λ p . In Section 8, Corollary 8.8, we check directly that if g ( K ) = 1 then either A Q ( K ) = 0 or A Q ( K ) ∈ { , } , and in the last case we have( δ ( K ) , θ ( K )) = (1 , . Question 1.11.
Let n ∈ N be fixed. Does a knot K exist such that δ ( K ) ≥ n ? (SeeRemark 7.10 for a brief discussion about this issue). Question 1.12. Is θ ( K ) bounded from above by an explicit function of g ( K ) (or A ( K ), or δ ( K ))? 2. Quandle invariants
We briefly recall a few details about the definition of quandle invariants of links and aboutour favourite family Q F of finite Alexander quandles.Let X = ( X, ∗ ) be any finite quandle, | X | = m . For every b ∈ X , the permutation of X defined by S b : a a ∗ b has order o ( b ) that divides m ! . If we denote by t X the l.c.m. of theseorders, then for every a, b ∈ X we have S t X b ( a ) = a ∗ t X b = a , that is ∗ t X = ∗ , and it is readilyseen that t X is in fact the type of X , as defined in the Introduction.2.1. Basic properties of finite Alexander quandles.
Let us now turn to our favouriteAlexander quandles X = ( F ( p, h ( t )) , ∗ ) where h ( t ) is an irreducible polynomial of breadth n ≥ p . Hence F ( p, h ( t )) is a finite field with q = p n elements.For every m ≥ p m +1 ( t ) = P mj =0 t j ∈ Z [ t ] and H m ( t ) = 1 − t m ∈ Z [ t ], in such a waythat H m ( t ) = (1 − t ) p m ( t ) for every m ≥ p m ( t ) (resp. H m ( t )) also as elements of Z p [ t ], Λ and Λ p ). Also recall that t denotes the classof t in F ( p, h ( t )). An easy inductive argument shows that for every a, b ∈ F ( p, h ( t )) and every m ≥ a ∗ m b = t m a + H m ( t ) b . R. BENEDETTI, R. FRIGERIO
Lemma 2.1.
Let X = ( F ( p, h ( t )) , ∗ ) be a finite Alexander quandle as above, let n = br h ( t ) and set q = p n . Then: (1) X is trivial if and only if h ( t ) . = t − . (2) If X is non–trivial, then ∗ m = ∗ ( i.e. m is a multiple of t X ) if and only if h ( t ) divides p m ( t ) in Λ p . (3) H m ( t ) = 0 in ( F p , h ( t )) if and only if m is a multiple of t X . (4) Suppose that X is non–trivial. Then t X ≥ n + 1 . Moreover, t X = n + 1 if and only p n +1 ( t ) is irreducible in Z p [ t ] and h ( t ) . = p n +1 ( t ) in Λ p . If this is the case, then p n +1 ( t ) is irreducible in Z [ t ] , and n + 1 is prime.Proof. (1) If h ( t ) . = t −
1, then t = 1 in F ( p, h ( t )), so ta +(1 − t ) b = a for every a, b ∈ F ( p, h ( t )).On the other hand, if X is trivial, then (1 − t )( b − a ) = 0 for every a, b ∈ F ( p, h ( t )), so 1 − t = 0.This implies that t − h ( t ) in Λ p , so h ( t ) . = t − h ( t ).(2) We have a ∗ m b = t m a + (1 − t ) p m ( t ) b = a if and only if ( t − p m ( t )( a − b ) = 0. Bypoint (1) this equality holds for every a, b if and only if p m ( t ) = 0 in ( F p , h ( t )), i.e. if andonly if h ( t ) divides p m ( t ) in Λ p .(3) By point (1), X has type 1 ( i.e. it is trivial) if and only if H ( t ) = 0, so we may supposethat X is non–trivial. In this case, since H m ( t ) = (1 − t ) p m ( t ) and 1 − t = 0 in ( F p , h ( t )),point (3) is an immediate consequence of (2).(4) By point (2) the polynomial h ( t ) divides p t X ( t ) in Λ p , so n = br h ( t ) ≤ br p t X = t X − t X = n + 1 if and only if h ( t ) divides p n +1 ( t ) in Λ p .Since br h ( t ) = br p n +1 ( t ), this condition holds if and only if p n +1 ( t ) . = h ( t ), and this impliesthat p n +1 ( t ) is irreducible in Λ p , whence in Z p [ t ]. Being monic, if p n +1 ( t ) is irreducible in Z p [ t ], then it is irreducible also in Z [ t ], and this implies in turn that n + 1 is prime. (cid:3) The simplest non–trivial quandles in our family Q F are the dihedral quandles D p = ( F ( p, t ) , ∗ ). In this case the quandle operation takes the form a ∗ b = 2 b − a , in terms of the fieldoperations of F ( p, h ( t )) = Z p . Dihedral quandles are involutory , i.e. their type is equal to 2. Remark 2.2. If q = p n , the finite field F q , which is unique up to isomorphism, supportsin general non–isomorphic quandle structures. This phenomenon shows up already when n = 1, i.e. when considering Alexander quandles in Q F (1). For every odd prime p and every a ∈ Z ∗ p , let h a ( t ) = a + t , and let X p,a = ( F ( p, h a ( t )) , ∗ ) be the corresponding Alexanderquandle. We have seen in Lemma 2.1–(1) that X p,a is trivial if and only if a = p −
1. Onthe other hand, if a = 1 then X p,a is a dihedral quandle, and its type is equal to 2. ByLemma 2.1–(4), if a / ∈ { , p − } then t X p,a >
2, so the quandles X p,a , X p, and X p,p − arepairwise non–isomorphic. For example, Lemma 2.1–(2) implies that t X p,a = 3 if and only if a = 0 , − − a is a root of t + t + 1, i.e. if and only p = 3 and the equation a − a + 1 = 0has a root in Z p (such a root is necessarily distinct from 0 , − −
3, so we can conclude that t X p,a = 3 if and only if p = 3, theelement p − c in Z p , and a = (1 ± c )( k + 1).Also observe that, if p > n ( n −
1) + 2, then there exists a ∈ Z p \ { , − } such that − a ∈ Z p is not a root of p i ( t ) ∈ Z p [ t ] for every i = 1 , . . . , n . By Lemma 2.1–(2), this implies that thetype of X p,a exceeds n , and this shows that Q F (1) contains quandles of arbitrarily large type.Here is another construction of non–isomorphic quandles supported by the same finitefield F q . Assume for example that both 1 + t m and p m +1 ( t ) are irreducible in Z p [ t ]. ByLemma 2.1–(4), the type of ( F ( p, p m +1 ( t )) , ∗ ) is equal to m + 1. On the other hand, since(1+ t m ) p m +1 ( t ) = p m ( t ), points (4) and (2) of Lemma 2.1 imply respectively that the type of( F ( p, t m ) , ∗ ) is bigger than m and divides 2 m , and is therefore equal to 2 m . An example of this kind is obtained by taking m = 2 and p = 11, so that we have two non–isomorphicquandle structures (of type 3 and 4 respectively) on F q , where q = 11 .2.2. Quandle colorings of links.
Let L = L ∪ . . . ∪ L h = K ∪ . . . ∪ K k be an orientedpartitioned link with k components, where P = ( L , . . . , L h ) is a partition of L into sublinks,and let D be any diagram of L . A ( Z t X –valued) P –cycle on L is a map z : { , . . . , h } → Z t X ,where z ( i ) labels every component of the sublink L i . Such a cycle naturally descends to D .An arc of D is any embedded open interval in D whose endpoints are undercrossing. An X -coloring of ( D, P , z ) assigns to each arc of D a “color” belonging to X in such a way thatat every crossing we see the local configuration shown in Figure 1. Here a, b ∈ X are colors,and z refers to the value assigned by z to the sublink that contains the overcrossing arc. a bz a*zb Figure 1.
The local configuration of a quandle coloring.
Remark 2.3.
The case when X is a dihedral quandle is particularly simple to handle becausein this case orientations become immaterial from the very beginning, in the sense that therule of Figure 1 is well–defined even if one forgets the orientation of the overcrossing arc.The following Proposition shows that c X ( L, P , z ) := c X ( D, P , z )is a well defined invariant of ( L, P , z ) (up to isotopy of oriented, partitioned and labelledlinks), where c X ( D, P , z ) is the number of X -colorings of ( D, P , z ). Proposition 2.4.
Let ( L, P , z ) be a partitioned link endowed with a fixed Z t X –cycle, and let D, D ′ be diagrams of L . Then we have c X ( D, P , z ) = c X ( D ′ , P , z ) . Proof.
Let us briefly describe how our statement can be deduced from the results provedin [9, 10] (in [9] only the case of involutory quandles is considered, but such a restrictionis overcome in [10]). In order to check that c X ( D, P , z ) is independent of D it is sufficientto prove the statement in the case when D and D ′ are related to each other by a classicalReidemeister move on oriented link diagrams. In the cited papers the authors consider indeeda more general situation, where D and D ′ are trivalent spatial graphs, and D ′ is obtained from D either via a Reidemeister move, or via a Whitehead’s move (by the way, this ensures that D and D ′ have ambient–isotopic regular neighbourhoods in S – see also the discussion inSubsection 2.3 below). In our case we have to deal only with the usual Reidemeister moves.Moreover, every Z t X –cycle on D canonically defines a Z t X –cycle on D ′ , so the argumentsin [9, 10] prove the claimed result. (cid:3) Let X ∈ Q F be a quandle of type k supported by the field F q . It is clear that the X -colorings of a diagram ( D, P , z ) as above correspond to the solutions of a linear system over F q . Therefore, the space of such colorings (which contains all the constant colorings) is a F q -vector space of dimension d X ( D, P , z ) ≥
1, so the whole information about c X ( D, P , z ) isencoded by the natural number a X ( D, P , z ) := d X ( D, P , z ) − . By Proposition 2.4, this number is a well defined isotopy invariant of oriented and Z t X -labelled partitioned links. As a consequence, the following polynomial, that collects all such“monomial” invariants, is an invariant of oriented partitioned links:Φ X ( L, P )( t ) := X z t a X ( L, P ,z ) ∈ N [ t ] . Also observe that by the very definitions we havedeg Φ X ( L, P )( t ) = sup z a X ( L, P , z ) , whence(1) A Q ( L, P ) = sup X ∈Q F deg Φ X ( L, P )( t ) . Lemma 2.5.
Let L be an oriented link, and let P M be its maximal partition. Then thepolynomial Φ X ( L, P M )( t ) is an invariant of L as an unoriented link. As a consequence, A Q ( L, P M ) is an invariant of L as an unoriented link.Proof. Let P M = ( K , . . . , K h ) be the maximal partition of L , and for every ǫ : { , . . . , h } →{± } let us denote by ǫL the link ǫ (1) K ∪ · · · ∪ ǫ ( h ) K h , where as usual the symbols K and − K denote knots having the same support and opposite orientations. We also define thecycle ǫz by setting ( ǫz )( j ) = ǫ ( j ) z ( j ). It is not hard to verify that for every cycle z and every ǫ we have a X ( ǫL, P M , z ) = a X ( L, P M , ǫz ) . We now say that two cycles z and z ′ are equivalent if and only if there exists ǫ such that z ′ = ǫz , and we denote by [ z ] the equivalence class of z . The previous discussion shows thatthe polynomials Φ X ( L, P , [ z ])( t ) := X z ′ ∈ [ z ] t a X ( L, P ,z ′ ) do not depend on the orientation of the components of L . The conclusion now follows fromthe obvious equality Φ X ( L, P M )( t ) := X [ z ] Φ X ( L, P M , [ z ])( t ) . (cid:3) Quandle invariants and tunnel number.
For every finite quandle X , the number c X ( L, P m ,
1) (that is the number of colorings associated to the cycle assigning the value 1 toevery component of L ) is in a sense the most widely considered quandle coloring invariant ofclassical links. The multiset of invariants obtained by varying the Z t X -cycles (when X ∈ Q F ,such a multiset is encoded by the polynomial Φ X ( L, P M )( t )) has been introduced in [9, 10] inorder to extend quandle coloring invariants to spatial graphs and even to spatial handlebodies .It turns out that this approach is useful also in the case of links. An interesting applicationof these extended invariants is given in [9, Proposition 6], where only quandles of type 2 are considered. However, Ishii’s argument applies verbatim to our (more general) case, thusgiving the following: Proposition 2.6.
For every X ∈ Q F and every (unoriented) link L we have t ( L ) ≥ A Q ( L, P M ) . Proof.
We sketch the proof for the sake of completeness. By equality (1), it is sufficient toshow that t ( L ) ≥ deg Φ X ( L, P M )( t ) for every quandle X ∈ Q F . Set m = t ( L ). Then thereis a sequence L := G ⊂ G ⊂ G ⊂ · · · ⊂ G m , where G j is a spatial graph with trivalentvertices obtained by attaching an arc to G j − , and G m is a spine of an unknotted handlebody.According to [9], for every quandle X the X -colorings of any diagram of a trivalent graph like G j verify (in addition to the rule already described in Figure 1) the further vertex conditiondescribed on the left of Figure 2 (here a refers to a color). With such a definition of coloring,the number of X –colorings of the diagram of a spatial graph does depend only on the isotopyclass of a regular neighbourhood of the graph, which is a spatial handlebody (the proof ofTheorem 5 in [9] does not really makes use of condition (K2’) stated there, that is equivalentto asking that the considered quandle has type 2). a aa DD’ Figure 2.
Quandle colorings at vertices of trivalent graphs.We can assume that G j − and G j admit respectively diagrams D and D ′ that differ fromeach other only by the local configurations shown on the right of Figure 2. Every cycle on G j − extends to a cycle on G j that assigns the value 0 to the added arc. Then it is easy toshow that deg Φ X ( G j )( t ) ≥ deg Φ X ( G j − )( t ) − . Moreover, since a regular neighbourhood of G m is an unknotted handlebody, we havedeg Φ X ( G m )( t ) = 0 , hence 0 ≥ deg Φ X ( L )( t ) − t ( L ) . ✷ Proposition 1.1 is now an easy consequence of Proposition 2.6. In [1] we have used Ishii’squandle coloring invariants of graphs (only exploiting the dihedral case) in order to detectdifferent level of knottings of spatial handlebodies. Ribbon tangles
Let us now fix a quandle X ∈ Q F of type t X . The following simple Lemma (it is a straight-forward computation) plays a crucial rˆole in the proof of our main results. Consider the localconfigurations of Figure 3. Here a, b, c, b , b are colors belonging to some X -coloring, wherewe understand that z ∈ { , . . . , t X − } is the same value of the cycle on both the overcrossingstrands. ac zbbb b ca z Figure 3.
The quandle colorings described in Lemma 3.1.
Lemma 3.1.
For the diagram on the left of Figure 3 we have: c = a + t − z H z ( t )( b − b ) . For the diagram on the right we have: c = a + H z ( t )( b − b ) . Let us consider a decorated tangle diagram T as suggested in Figure 4.It is understood that the circular box contains h oriented strings, each of which has an“input” and an “output” endpoint. Moreover, the j –th string is decorated with a sign s j ∈{± } , and its endpoints are endowed with an input color r j and an output color f j . rr f f
1 2 2 r ss
1 2 sfh hhrr f f
1 2 2 r ss
1 2 sfh hh
Figure 4.
A string tangle T . We use such a string tangle to encode an associated ribbon tangle R ( T ) with oriented ribbonboundary tangle D ( T ), by applying the doubling rules suggested in Figure 5, where the left(right) side refers to the string sign s = 1 ( s = − T with oppositeorientations. These boundary components are also ordered by taking first the componentwhich shares the same orientation as the corresponding string of T . TR(T)1 −1
Figure 5.
From a string tangle to a ribbon tangle.If z : { , . . . , h } → Z t X is any cycle defined on the strings of T , we define the associated ribbonboundary cycle ˆ z on D ( T ) by assigning the same value z ( j ) to both boundary componentsof the ribbon associated to the j –th string of T . In this way we have obtained a Z t X -labelledribbon boundary tangle ( D ( T ) , ˆ z ). Arcs of T and of D ( T ) are defined as usual, provided nowthat also the endpoints of the strings of T and D ( T ) have to be considered as endpoints ofarcs of T and D ( T ).The notion of X -coloring extends obviously to any Z t X –labelled ribbon boundary tangle( D ( T ) , ˆ z ). For every such a coloring, along every arc of T we see a couple of ordered arcs of D ( T ) carrying an ordered couple of colors, say ( a, b ). The following result is an immediateconsequence of Lemma 3.1. Lemma 3.2.
For every X -coloring of ( D ( T ) , ˆ z ) , the color difference d = b − a is constantalong every string of T . Then every such X -coloring can be described as follows. At the input point of the j –thstring of T we have an ordered couple of colors ( a j , a j + d j ). Along every arc α of T belongingto the j –th string, we have a couple of colors of the form ( a j + r α , a j + d j + r α ).For obvious reasons, we say that the r α ’s define an X diff - coloring of the arcs of T , whichvanishes at the input points of the strings of T (observe that the definition of differencebetween colors relies on the fact that X is a module, and is not related to the quandleoperation of X ). We now deduce from Lemma 3.1 the rule governing these X diff -coloringsat crossings. We refer to Figure 6. Here r I , r F are X diff -colors, d , z , s are respectively theconstant X -color difference, the value of the cycle and the sign of the string that contains theovercrossing strand, and ǫ = ± r F = r I − ǫH z ( t ) t σ ( s ) z d . As a consequence, every X diff -coloring of ( T, z ) (in particular the corresponding set ofoutput colors { f j } ) is completely determined by the input data { d j } , and every X -coloring of( D ( T ) , ˆ z ) is completely determined by the input data { ( a j , d j ) } . In fact, every X diff -coloringof ( T, z ) can be constructed as follows: we run along every string of T from its input point d,s z r r I F z r r I d,s z r r I F z r r Figure 6.
The behaviour of X diff at crossings. On the top, a positive cross-ing. On the bottom, a negative crossing.to its output point, and at every undercrossing we add to the local input value r I a suitableterm according to equation (2).Given an ordered couple ( i, j ) of string indices, let n + i,j (resp. n − i,j ) be the number of timesthe i –th string passes under the j –th string at a positive (resp. negative) crossing, and let usset M i,j = n + i,j − n − i,j . The following Proposition summarizes the discussion carried out in this Section.
Proposition 3.3.
Let T be a decorated tangle with associated ribbon boundary tangle D ( T ) .Then, every X diff -coloring of ( T, z ) (in particular the corresponding set of output colors { f j } )is completely determined by the input data { d j } , and every X -coloring of ( D ( T ) , ˆ z ) is com-pletely determined by the input data { ( a j , d j ) } . In particular, the f j ’s can be computed interms of the d i ’s by means of the formula f i = − h X j =1 M i,j t σ ( s ( j )) z ( j ) H z ( j ) ( t ) d j . Seifert surfaces and special diagrams
Let us consider a compact oriented surface Σ g,s of genus g having s ≥ g + s ≥
1, and g + s = 1 if and only if g = 0 and s = 1, i.e. if Σ g,s is a disk. Letus assume that g + s >
1. It is well known that Σ g,s is homeomorphic to the model shown inFigure 7, where the case g = 2, s = 3 is considered. The picture stresses also the fact thatΣ g,s is the regular neighbourhood of a 1–dimensional trivalent graph P g,s , which is thereforea spine of Σ g,s .Let now L be an oriented link endowed with a Seifert surface Σ of genus g , and let s be thenumber of components of L . Assume first that g + s > , L ) is the imageof a suitable embedding of the corresponding model (Σ g,s , ∂ Σ g,s ) in S . As a consequence, L admits a special diagram D ( T ) as described in Figure 8: on the top there is a suitabledecorated tangle T with 2 g + s − closing tangle C which closesthe ribbon boundary tangle D ( T ) associated to T . The strings of T correspond to a genericprojection of the image (via the embedding Σ g,s ֒ → S ) of some oriented edges of the spine P g,s of Σ g,s . We say that T is the primary tangle of the special diagram D ( T ). If g + s = 1,then L is a trivial knot and Σ is a spanning disk of L ; in this case we understand that the onlyspecial diagram of L is given by the trivial diagram D of L , and we agree that the closingtangle C coincides with D , while the primary tangle T is empty.Let us now consider an oriented link L endowed with a boundary partition P = ( L , . . . , L h ),and let Σ , . . . , Σ h be a system of disjoint Seifert surfaces such that ∂ Σ i = L i (as oriented Figure 7.
The surface Σ , and the spine P , . r f r f r r r r f f f f TC Figure 8.
A special diagram of a 3–component link endowed with a Seifertsurface of genus 2.1–manifolds). If g i and s i are the genus and the number of boundary components of Σ i ,then the pair (Σ ∪ . . . ∪ Σ h , L ) is the image of a suitable embedding in S of the disjointunion F hi =1 (Σ g i ,s i , ∂ Σ g i ,s i ). It readily follows that L admits a special diagram as described inFigure 9, where the closing tangle C decomposes into the union of h closing tangles C , . . . , C h .Of course, strings of T corresponding to distinct Σ i ’s may be linked to each other.Such a special diagram is adapted to P , in the sense that every arc of the primary tangle T gives rise to a pair of arcs of D ( T ) that belong to the same link of the partition P . Therefore,every P –cycle on L descends to a well–defined cycle on T . Remark 4.1.
Suppose that P is a boundary partition of a k –component link L . The proce-dure described in this Section provides a special diagram of L adapted to P whose primarytangle has exactly 2 g ( P ) + k − |P| strings.5. Lower bounds for link genera
We are now ready to give a
Proof of Theorem 1.2.
Let ( L, P ) be a k –component partitioned link, and let us set α = 2 g ( P ) + k − |P| . f f r r r f r f r f C C T Figure 9.
A special diagram for the link L = L ∪ L , where L is a2–component link bounding a Seifert surface of genus 1, and L is a knotbounding a Seifert surface of genus 1.As pointed out in Remark 4.1, L admits a special diagram D ( T ) adapted to P whose primarytangle T has exactly α strings.Let us take a quandle X ∈ Q F and a P –cycle z : P → Z t X . Such a cycle descends to thediagram D ( T ), whence to the boundary ribbon tangle D ( T ) ⊂ D ( T ). What is more, since D ( T ) is adapted to P , the cycle z induces a cycle on T , which will also be denoted by z . The X -colorings of ( D ( T ) , z ) are the X -colorings of ( D ( T ) , ˆ z ) that extend to the whole ( D ( T ) , z ).In order to study the space of X -colorings of ( D ( T ) , z ) we exploit the results obtained inSection 3. The space of X -colorings of ( D ( T ) , z ) is then obtained by imposing the conditionscorresponding to the fact that colors have to match along the closing tangle C of D ( T ).Let us associate to every string of T four variables ( a i , d i , b i , f i ), i = 1 , . . . , α . As usual, thepair ( a i , a i + d i ) refers to the values of a X -coloring on the arcs of D ( T ) originating at theinput point of the i -th string of T , while ( a i + f i , a i + d i + f i ) refers to the values of sucha coloring on the arcs of D ( T ) ending at the output point. Finally, the auxiliary variable b i encodes the change that an arc of D ( T ) undergoes whenever it undercrosses the bandcorresponding to the i –th string. Therefore, the value of b i depends both on d i and on thevalue assigned by z to the i -th string of T . Henceforth, we denote such a value by z i (so z i = z ( j ( i )) when the i -th string of T corresponds to a band of D ( T ) whose boundary lies on L j ( i ) ).Let us write down the system that computes the space of colorings we are interested in.Proposition 3.3 implies that the space of X -colorings of ( D ( T ) , z ) is identified with the spaceof the solutions of the linear system(3) b i = − t − z i H z i ( t ) d i , i = 1 , . . . , α (4) f i = α X j =1 M i,j b j , i = 1 , . . . , α . In order to obtain the space of X -colorings of ( D ( L ) , z ), we have to add to these equations alsothe conditions arising from the fact that colors must match along the strings of the closingtangle C . These conditions can be translated into a linear system(5) S ( { a i } , { d i } , { f i } ) = 0 , and we stress that such a system does not involve the b i ’s (this system is written down inSubsection 6.1, but this is not relevant to our purposes here). Let now z ′ be another P –cycle, and let us concentrate on the difference | a X ( L, P , z ) − a X ( L, P , z ′ ) | . We have just seen that the linear system that computes the space c X ( L, P , z ) is given by theunion of equations (3), (4), (5). Now, the argument above shows that the system computingthe space c X ( L, P , z ′ ) is given by the union of the systems (4) and (5) with the followingequations:(6) b i = − t − z ′ i H z ′ i ( t ) d i , i = 1 , . . . , α , where z ′ i is the value assigned by z to the i -th string of T . Therefore, the system computing c X ( L, P , z ′ ) is obtained from the system computing c X ( L, P , z ′ ) just by replacing equations (3)with equations (6). Since such equations are in number of α = 2 g ( P ) + k − |P| we finallyobtain | a X ( L, P , z ) − a X ( L, P , z ′ ) | ≤ g ( P ) + k − |P| . This concludes the proof of Theorem 1.2. ✷ Finally we note that the very same argument of the above proof gives the following im-provement of Theorem 1.2 .
Theorem 5.1.
Let P = ( L , . . . , L h ) be a boundary partition of L , where L i is a k i –component link, let z and z ′ be two P –cycles on L , and let I = { i ∈ { , . . . , h } | z ( i ) = z ′ ( i ) } .Let also (Σ , . . . , Σ h ) be a system of disjoint Seifert surfaces for the L i ’s. Then the followinginequality holds: | a X ( L, P , z ) − a X ( L, P , z ′ ) | ≤ X i ∈ I g (Σ i ) + X i ∈ I k i − | I | . An example.
The following example shows that Theorem 5.1 could prove more effectivethan Theorem 1.2 in providing bounds on the genus of links.Let us consider the tangle B showed in Figure 10. Recall that D p is the dihedral quandleof order p , let 1 be the cycle that assigns the value 1 ∈ Z t D p = Z to every arc of B , andlet us denote by C p ( a, b, c, d ) the number of D p –colorings of B (relative to the cycle 1) whichextend the colors a, b, c, d assigned on the “corners” of the diagram. b da cb da c B Figure 10.
The tangle B .The following Lemma is proved in [1]: Lemma 5.2.
We have C p ( a, b, c, d ) = p if a = b, c = d and p = 3 C p ( a, b, c, d ) = 1 if a = b, c = d and p = 3 C p ( a, b, c, d ) = 0 otherwise . For every q ≥
1, let L q be the link described in Figure 11. B B B q Figure 11.
On the top: the link L q ; every B i is a copy of the tangle B . Onthe bottom: the case q = 2.It is obvious from the picture that L q is a boundary link such that g ( L, P M ) ≤ q . Let K q (resp. K ′ q ) be the component of L q on the top half (resp. the bottom half) of the diagramshown on the top of Figure 11. We denote every P M –cycle z : P M → Z simply by the pair( z ( { K q } ) , z ( { K ′ q } )), and the integers a D ( L q , P M , ( z , z )) simply by a D ( L q , ( z , z )). Proposition 5.3.
For every q ≥ we have a D ( L q , (1 , q + 1 , a D ( L q , (1 , a D ( L q , (0 , a D ( L q , (0 , . Proof.
As usual, the only (0 , L q are those which are constant on every com-ponent of L q , so a D ( L q , (0 , , L q . It is immediate to observe that K q and K ′ q are both trivial. Since the cycle (1 ,
0) vanishes on K ′ q , it is immediate to realize thatany such coloring restricts to a coloring of K q (1). Since K q is trivial, this implies that every(1 , L q is constant on K q . The discussion in Section 3 now implies that that the colorings of K ′ q are not affected by the crossings between the bands of K ′ q and the bands of K q . Then, every (1 , L q restricts to a 0-coloring ( i.e. to a constant coloring) of K ′ q . We have proved that the only (1 , L q , so a D ( L, (1 , , a D ( L, (0 , a, b ∈ D . An easy application of Lemma 5.2 shows that thenumber of the colorings of L q which take the value a (resp. b ) on the arc of K q (resp. of K ′ q )joining the tangles B and B q is equal to 3 q . Therefore, the number of (1 , L q is equal to 3 q +2 , whence the conclusion. (cid:3) Corollary 5.4.
For every q ≥ we have g ( L q , P M ) = 2 q . Proof.
Let (Σ q , Σ ′ q ) be a system of disjoint Seifert surfaces for K q , K ′ q . We have to show that g (Σ q ) + g (Σ ′ q ) ≥ q . By Theorem 5.1 we have2 q = | a D ( L q , (1 , − a D ( L q , (0 , | ≤ g (Σ q ) , q = | a D ( L q , (1 , − a D ( L q , (1 , | ≤ g (Σ ′ q ) , so g (Σ q ) ≥ q and g (Σ ′ q ) ≥ q , whence the conclusion. (cid:3) It is maybe worth mentioning that the bound provided by Corollary 1.3 is less effective inorder to compute g ( L, P M ). In fact, Proposition 5.3 implies that A Q ( L, P M ) = 2 q + 1, so theinequality A Q ( L, P M ) ≤ g ( L, P M ) + 1 only implies g ( L, P M ) ≥ q .6. A proof of Theorem 1.5
With notations as in the preceding Section, let us describe more explicitly the systemcomputing the X –colorings of ( L, P m , z ), where X = F ( p, h ( t )) is a quandle in Q F .6.1. More details on the system associated to a special diagram.
Let us now concen-trate on the case P = P m , so that there exists z ∈ N such that z = z ( i ) for every i = 1 , . . . , k (recall that k is the number of components of L ). We also set g = g ( L ) = g ( L, P m ).Then, the linear system described by equations (3), (4) and (5) reduces to the system(7) f i = (1 − t − z ) α X j =1 M i,j d j , S ( { a i } , { d i } , { f i } ) = 0 , where α = 2 g + k − a i , d i , f i have to be considered as variables in F ( p, h ( t )). Moreover,the system S ( { a i } , { d i } , { f i } ) = 0 has integer coefficients.Let us look more closely to the closing conditions S { a i } , { d i } , { f i } ) = 0. By looking at thedefinition of special diagram for L , one can easily show that such closing conditions reduce,after easy simplifications, to the system a i − = a i − d i − , d i = f i − , f i = − d i − , i = 1 , . . . , g,a i = a i +1 + d i +1 , i = 1 , . . . , g − ,a i = a g , f i = 0 , i = 2 g + 1 , . . . , αa g = a + d . An easy inductive argument shows that the condition a g = a + d is a consequence ofequations a i − = a i − d i − , i = 1 , . . . , g , and a i = a i +1 + d i +1 , i = 1 , . . . , g − Therefore, the system (7) is equivalent to the system(8) a i − = a i − d i − , i = 1 , . . . , g,a i = a i +1 + d i +1 , i = 1 , . . . , g − ,a i = a g , i = 2 g + 1 , . . . , α − d i + (1 − t − z ) P αj =1 M i − ,j d j = 0 , i = 1 , . . . , g,d i − + (1 − t − z ) P αj =1 M i,j d j = 0 , i = 1 , . . . , g, (1 − t − z ) P αj =1 M i,j d j = 0 , i = 2 g + 1 , . . . , α , where we have eliminated the f i ’s from the variables.Let us now define two square matrices N ( z ) and J of order α with coefficients in Λ asfollows: J i,j = − i = 2 h − , j = 2 h, h ≤ g i = 2 h, j = 2 h − , h ≤ g , (so J has in fact integer coefficients), and N ( z ) = ( t z − M + t z J .
We also denote by N ( z, p ) the matrix obtained by replacing each coefficient of N ( z ) by itsimage via π p : Λ → Λ p , and by N ( z, p, h ( t )) the matrix obtained by further projecting eachcoefficient of N ( z, p ) onto F ( p, h ( t )).We are now ready to prove the following: Lemma 6.1.
The space of X –colorings of ( L, P m , z ) is in natural bijection with the directsum F ( p, h ( t )) ⊕ ker N ( z, p, h ( t )) , so a X ( L, P m , z ) = dim ker N ( z, p, h ( t )) . Proof.
The previous discussion shows that the space of colorings we are considering is innatural bijection with the solutions of the system (8). It is immediate to realize that, forevery such solution, each a i , i ≥
2, is uniquely determined by a and the d j ’s. Moreover,once a solution of the system (8) is fixed, we can obtain another solution just by adding aconstant term to every a i .Therefore, the space of the solutions of (8) is isomorphic to the direct sum of F ( p, h ( t ))with the space of the solutions of the system − d i + (1 − t − z ) P αj =1 M i − ,j d j = 0 , i = 1 , . . . , g,d i − + (1 − t − z ) P αj =1 M i,j d j = 0 , i = 1 , . . . , g, (1 − t − z ) P αj =1 M i,j d j = 0 , i = 2 g + 1 , . . . , α . The matrix encoding this system is equal to t − z N ( z, p, h ( t )), and t − z is invertible in F ( p, h ( t )),whence the conclusion. (cid:3) Some relations between M and the Seifert matrix of L . Let Σ be the Seifertsurface of L encoded by the fixed special diagram D ( T ) we are considering, and observe thateach (oriented) string of T canonically defines an (oriented) arc lying on Σ. The module H (Σ; Z ) admits a special geometric basis { β , . . . , β α } , where β j is obtained by closing the j –th string of T in the portion of Σ carried by the closing tangle C , in such a way that weintroduce just one intersection point between β i − and β i , i = 1 , . . . , g , while β i is disjointfrom β j for every i = 2 g + 1 , . . . , α , j = 1 , . . . , α . Recall that the Seifert matrix S ( L ) of L is the square matrix with integer coefficients defined by S ( L ) i,j = lk( β i , β + j ), i, j = 1 , . . . , α , where lk( β i , β + j ) is the linking number (in S ) between β i and the knot β + j obtained byslightly pushing β j to the positive side of Σ. ¿From the very definition of linking number wereadily obtain the following: Lemma 6.2.
We have S ( L ) = M + M T + J . Let us point out another interesting property of M that will prove useful later: Lemma 6.3.
We have M − M T = − J .
Proof.
Since both M − M T and J are antisymmetric, it is sufficient to show that for every i < j we have M i,j − M j,i = 1 if j = i + 1, and M i,j − M j,i = 0 otherwise. However, it followsfrom the definition of M that the number M i,j − M j,i is equal to the algebraic intersectionnumber between the projections of the j –th and the i –th string of T (taken in this order)onto the plane containing the special diagram. If j > i + 1 (resp. j = i + 1), such number isequal to the algebraic intersection number between the projections of β j and of β i (resp. isequal to 1 plus the algebraic intersection number between the projections of β j and of β i ).But the algebraic intersection number between the projections of β j and of β i is obviouslynull, whence the conclusion. (cid:3) Putting together Lemmas 6.2 and 6.3 we get the following:
Corollary 6.4.
We have S ( L ) = M + J, t z S ( L ) − S ( L ) T = ( t z − M + t z J = N ( z ) . Proof of Theorem 1.5.
Let e X ( L ) and A ( p ) ( L ) be the cyclic covering and the Λ p –module defined in the Introduction. We have the following: Lemma 6.5.
The module A ( p ) ( L ) admits the square presentation matrix tS ( L ) ( p ) − ( S ( L ) ( p ) ) T = N (1 , p ) . In particular, ∆ ( p ) i ( L )( t ) = e ( p ) i ( L )( t ) = 1 for every i > α .Proof. The usual proof that tS ( L ) − S ( L ) T is a presentation of H ( e X ( L ); Z ) over Λ relies onsome standard Mayer–Vietoris argument and on Alexander–Lefschetz duality, which ensuresthat, if { β , . . . , β g + k − } is any base of the first homology group of a Seifert surface Σfor L , then the first homology group of S \ Σ admits a dual base { γ , . . . , γ α } such thatlk( β i , γ j ) = δ ij (see e.g. [3, Chapter 8]). Both these tools may still be exploited when Z isreplaced by Z p , and this readily implies the conclusion.An alternative proof can be obtained as follows. An easy application of the UniversalCoefficient Theorem for homology shows that A p ( L ) ∼ = A ( L ) ⊗ Z Z p , and this easily impliesthat any presentation 0 / / Λ r / / Λ s / / A ( L ) / / rp / / O O (cid:15) (cid:15) Λ sp / / O O (cid:15) (cid:15) A p ( L ) O O (cid:15) (cid:15) / / r ⊗ Z Z p Λ s ⊗ Z Z p A ( L ) ⊗ Z Z p , whence the conclusion. (cid:3) The following result describes some relations between ∆ i ( L )( t ) and ∆ ( p ) i ( L )( t ), where i ∈ N . Corollary 6.6. (1)
For every i ∈ N we have E ( p ) i ( L ) = π p ( E i ( L )) . (2) For every i ∈ N the polyonomial π p (∆ i ( L )( t )) divides ∆ ( p ) i ( L )( t ) in Λ p . (3) We have ∆ ( p )1 ( L )( t ) = π p (∆( L )( t )) . (4) If f ( t ) ∈ E i ( L ) , then ∆ ( p ) i ( L )( t ) divides π p ( f ( t )) in Λ p .Proof. By Lemma 6.5, π p maps a set of generators (over Λ) of the ideal E i ( L ) onto a set ofgenerators (over Λ p ) of the ideal E ( p ) i ( L ). Since π p is surjective, this readily implies point (1).By point (1), the polynomial π p (∆ i ( L )( t )) divides every element of E ( p ) i − ( L ), whence point (2).Since A ( L ) admits the square presentation matrix S ( L ) − tS ( L ) T , the ideal E ( L ) is prin-cipal. Together with (1), this immediately gives (3).Point (4) is an easy consequence of point (1). (cid:3) Let us now consider the Λ p –linear map ψ z : Λ αp → Λ αp such that ψ z ( x ) = N ( z, p ) · x forevery x ∈ Λ αp . Of course, ψ z induces a quotient map ψ z : F ( p, h ( t )) α → F ( p, h ( t )) α such that ψ z ( x ) = N ( z, p, h ( t )) · x for every x ∈ F ( p, h ( t )) α . Let now z be a P –cycle for L , and denoteby z = z ( P m ) the value assigned by z to every component of L . By Lemma 6.1, the space of X –colorings of ( L, P m , z ) is in bijection with F ( p, h ( t )) ⊕ ker ψ z , whence to F ( p, h ( t )) ⊕ coker ψ z (here we use that F ( p, h ( t )) α is finite).Since Λ p is a P.I.D., there exist square univalent matrices U (1) , V (1) with coefficients in Λ p such that U (1) · N (1 , p ) · V (1) = diag (cid:16) e ( p )1 ( L )( t ) , . . . , e ( p ) α ( L )( t ) (cid:17) , where diag( γ , . . . , γ α ) denotes the diagonal matrix with the γ i ’s on the diagonal. Let U ( z )(resp. V ( z )) be the matrix obtained by applying to every coefficient of U (1) (resp. V (1)) thering endomorphism of Λ p that maps t to t z . Then we obviously have U ( z ) · N ( z, p ) · V ( z ) = diag (cid:16) e ( p )1 ( L )( t z ) , . . . , e ( p ) α ( L )( t z ) (cid:17) . After reducing the coefficients modulo h ( t ), this equality translates into the equality U ( z ) · N ( z, p, h ( t )) · V ( z ) = diag (cid:16) e ( p )1 ( L )( t z ) , . . . , e ( p ) α ( L )( t z ) (cid:17) , where U ( z ) , V ( z ) are invertible over F ( p, h ( t )), and we denote the class of e ( p ) i ( L )( t ) in F ( p, h ( t )) simply by e ( p ) i ( L )( t ). This implies that coker ψ z is isomorphic to α M i =1 F ( p, h ( t )) . (cid:16) e ( p ) i ( L )( t z ) (cid:17) ∼ = α M i =1 Λ p . (cid:16) e ( p ) i ( L )( t z ) , h ( t ) (cid:17) ∼ = ∞ M i =1 Λ p . (cid:16) e ( p ) i ( L )( t z ) , h ( t ) (cid:17) , where the last equality is due to Lemma 6.5. This concludes the proof of Theorem 1.5. Forlater purposes we point out the following easy: Corollary 6.7.
We have a X ( L, P m , z ) > if and only if h ( t ) divides π p (∆( L )( t z )) in Λ p .Proof. Theorem 1.5 implies that a X ( L, P m , z ) > h ( t ) divides e ( p ) i ( L )( t z ) forsome i ≥
1. The conlcusion follows from the fact that π p (∆( L )( t z )) = ∆ ( p )1 ( L )( t z ) = ∞ Y i =1 e ( p ) i ( L )( t z ) . (cid:3) Remark 6.8.
On may wonder if the equality π p ( e i ( L )( t )) . = e ( p ) i ( L )( t ) holds for every i ≥ p / ( e ( p ) i ( L )( t z ) , h ( t ))with the module Λ p / ( π p ( e i ( L )( t z )) , h ( t )) . Such a claim seems also suggested, at least when L is a knot and z = 1, by the originalstatement of [8, Theorem 1]. However, this is not the case, as the following constructionshows.In fact, let k ( t ) = t − t − and k ( t ) = − t + 5 − t − , and observe that k i ( t − ) = k i ( t ), i = 1 ,
2. It is proved in [12, Theorem 2.5] that a knot K exists such that A ( K ) is presentedby the matrix diag ( k ( t ) , k ( t ) , k ( t ) , k ( t )) . This readily implies that E ( K ) = ( k ( t ) k ( t ) ) , E ( K ) = ( k ( t ) k ( t ) , k ( t ) k ( t ) ) ,E ( K ) = ( k ( t ) , k ( t ) , k ( t ) k ( t )) , E ( K ) = ( k ( t ) , k ( t ))and E i ( K ) = Λ for every i ≥
4, whence∆ ( K )( t ) = k ( t ) k ( t ) , ∆ ( K )( t ) = k ( t ) k ( t ) , and ∆ i ( K )( t ) = 1 for every i ≥
3. As a consequence we get e ( K )( t ) = e ( K )( t ) = k ( t ) k ( t ) , e i ( K )( t ) = 1 for every i ≥ . On the other hand, let us fix p = 3, and observe that in this case π ( k ( t )) = π ( k ( t )) = k ( t ) ∈ Λ , where k ( t ) = ( t + 1) . Therefore, from the equality E (3) i ( K ) = π p ( E i ( K )( t )) (seeCorollary 6.6) we easily deduce that∆ (3)1 ( K ) = k ( t ) , ∆ (3)2 ( K ) = k ( t ) , ∆ (3)3 ( K ) = k ( t ) , ∆ (3)4 ( K ) = k ( t ) , and ∆ (3) i ( K ) = 1 for every i ≥
5, so e (3)1 ( K )( t ) = e (3)2 ( K )( t ) = e (3)3 ( K )( t ) = e (4) ( K )( t ) = k ( t ) , e (3) i ( K )( t ) = 1 for every i ≥ . Therefore, if h ( t ) = t + 1 ∈ Λ , then we have ∞ M i =1 Λ / ( e (3) i ( K )( t ) , h ( t )) ∼ = F (3 , h ( t )) , while ∞ M i =1 Λ / ( π ( e i ( K )( t )) , h ( t )) ∼ = F (3 , h ( t )) . Remark 6.9.
Let k ( t ) , k ( t ) ∈ Λ and k ( t ) ∈ Λ be the polynomials introduced in theprevious Remark. It is proved in [13] that a knot K ′ exists whose module A ( K ′ ) is isomorphicto Λ / ( k ( t ) k ( t )) (see also [15, Theorem 7.C.5]). Let K ′′ = K ′ + K ′ . Then we have A ( K ′′ ) = A ( K ′ ) ⊕ A ( K ′ ) (see e.g. [15, Theorem 7.E.1]), and this readily implies that E ( K ′′ ) = ( k ( t ) k ( t ) ) , E ( K ′′ ) = ( k ( t ) k ( t )) , and E i ( K ′′ ) = Λ for every i ≥
2. Therefore,∆ ( K ′′ )( t ) = k ( t ) k ( t ) , ∆ ( K ′′ )( t ) = k ( t ) k ( t ) , and ∆ i ( K ′′ )( t ) = 1 for every i ≥
3. Moreover, since the elementary ideals of K ′′ are principal,we also have∆ (3)1 ( K ′′ )( t ) = π (∆ ( K )( t )) = k ( t ) , ∆ (3)2 ( K ′′ )( t ) = π (∆ ( K )( t )) = k ( t ) , and ∆ (3) i ( K ′′ )( t ) = π (∆ i ( K )( t )) = 1 for every i ≥ K ′′ and the knot K introduced in the previous Remark satisfy thecondition ∆ i ( K )( t ) = ∆ i ( K ′′ )( t ) for every i ≥
1, but have a different number of D –coloringswith respect to the cycle z = 1. What is more, since for every p the Λ p –module A ( p ) ( K )(resp. A ( p ) ( K ′′ )) admits a square presentation matrix of order 4 (resp. of order 2), Theorem 1.5readily implies that A Q ( K ) ≤ A Q ( K ′′ ) ≤ A Q ( K ) = 4 and A Q ( K ′′ ) = 2. Therefore, even if they share every Alexander polynomial∆ i ( K )( t ) = ∆ i ( K ′′ )( t ), i ≥
1, the knots K , K ′′ are distinguished from each other by theinvariant A Q . 7. Comparing A Q with A Let us keep notation from the preceding Section. Of course, since F ( p, h ( t )) is a field, thequotient Λ p / ( e ( p ) i ( L )( t z ) , h ( t )) of F ( p, h ( t )) is null (resp. isomorphic to F ( p, h ( t ))) if and onlyif h ( t ) divides (resp. does not divide) e ( p ) i ( L )( t z ) in Λ p . Therefore, if we set I ( z, p, h ( t ) , L ) = { i ∈ N ∗ | h ( t ) divides e ( p ) i ( L )( t z ) } , | I ( z, L ) | = sup p,h ( t ) | I ( z, p, h ( t ) , L ) | , | I ( L ) | = sup z | I ( z, L ) | , then we easily obtain that A Q ( L ) = | I ( L ) | . Therefore, in order to prove Theorem 1.6 it is sufficient to show that, if L = K is a knot,then: • I ( K ) ≤ A ( K ), • I ( K ) = 0 if and only if A ( K ) = 0.7.1. Reduction to the cycle z = 1 . We first prove that, in order to compute A Q ( K ), it issufficient to restrict our attention to colorings relative to the cycle z = 1. Lemma 7.1.
We have I ( L ) = I (1 , L ) . In the proof we use the following elementary
Lemma 7.2.
Let p ( t ) , . . . , p n ( t ) be polynomials in Λ p , and let d ( t ) be their G.C.D. in Λ p .For every integer z ≥ , the polynomial d ( t z ) ∈ Λ p is the G.C.D. of p ( t z ) , . . . , p n ( t z ) in Λ p .Proof. For every i = 1 , . . . , n , the fact that d ( t ) divides p i ( t ) readily implies that d ( t z ) di-vides p i ( t z ). On the other hand, Λ p is P.I.D., so Bezout’s Identity implies that there exist λ ( t ) , . . . , λ n ( t ) ∈ Λ p such that d ( t ) = λ ( t ) p ( t ) + . . . + λ n ( t ) p n ( t ) , whence d ( t z ) = λ ( t z ) p ( t z ) + . . . + λ n ( t z ) p n ( t z ) . Therefore, if d ′ ( t ) divides every p i ( t z ), then d ′ ( t ) also divides d ( t z ), whence the conclusion. (cid:3) Proof of Lemma 7.1.
It is sufficient to show that, for every odd prime p , every positiveinteger z and every irreducible polynomial h ( t ) ∈ Λ p , there exists an irreducible polynomial h ′ ( t ) ∈ Λ p such that | I ( z, p, h ( t ) , L ) | ≤ | I (1 , p, h ′ ( t ) , L ) | . Let d ( t ) ∈ Λ p be the G.C.D. of the polynomials { e ( p ) i ( t ) , i ∈ I ( z, p, h ( t ) , L ) } . By thevery definitions, h ( t ) divides e ( p ) i ( t z ) for every i ∈ I ( z, p, h ( t ) , L ), so by Lemma 7.2 we havethat h ( t ) divides d ( t z ). This implies that the breadth of d ( t ) is positive, so d ( t ) admits anirreducible factor h ′ ( t ) of positive breadth. By construction we have that h ′ ( t ) divides e ( p ) i ( t )for every i ∈ I ( z, p, h ( t ) , L ), so I ( z, p, h ( t ) , L ) ⊆ I (1 , p, h ′ ( t ) , L ), whence the conclusion. ✷ More details on Alexander ideals of links.
Recall that e X ( L ) is the total linkingnumber covering of the complement of L , and that k denotes the number of components of L .If x ∈ C ( L ) is any basepoint and e X is the preimage of x in e X ( L ), then the relative homologymodule A ′ ( L ) = H ( e X ( L ) , e X ; Z ) also admits a natural structure of Λ–module. Moreover, itis not difficult to show that A ′ ( L ) ∼ = A ( L ) ⊕ Λ (as Λ–modules), so E i ( A ( L )) = E i +1 ( A ′ ( L ))for every i ∈ N , and ∆ i ( L )( t ) ∈ Λ is the generator of the smallest principal ideal containing E i ( A ′ ( L )). If L = K is a knot, this immediately implies that ∆ i ( L )( t ) ∈ Λ concides with theso called i –th Alexander polynomial of K . Lemma 7.3.
We have ∆ ( p ) i ( L )( t ) = 0 for every i ≥ k . Proof.
Let b X ( L ) be the maximal abelian covering of C ( L ) and let b X be the preimage of x in b X ( L ). Then the homology group b A ( L ) = H ( b X ( L ) , b X ; Z ) admits a natural structure of Z [ t , t − , . . . , t k , t − k ]–module (see e.g. [6, 7]). Just as in the case of the total linking numbercovering, one may define the i –th elementary ideal b E i ( L ) ⊆ Z [ t , t − , . . . , t k , t − k ] of thismodule. If τ : Z [ t , t − , . . . , t k , t − k ] → Λ is the ring homomorphism that sends each t ± i into t ± , it is not difficult to show that E i ( L ) = E i +1 ( A ′ ( L )) = τ ( E i +1 ( b A ( L )))(see e.g. [7, page 106]).Let now ε : Z [ t , t − , . . . , t k , t − k ] → Z be the augmentation homomorphism defined by ε ( f ( t , . . . , t k )) = f (1 , . . . , e.g. [7,Lemma 4.1]) ensures that ε ( E k ( b A ( L ))) = Z . In particular, there exists g ( t ) ∈ E k ( b A ( L ))such that g (1 , . . . ,
1) = 1. Let us set f ( t ) = τ ( g ( t )) ∈ E k − ( L ) and f ( p ) ( t ) = π p ( f ( t )). Ourchoices readily imply that f ( p ) (1) = 1 in Z p , so f ( p ) ( t ) = 0 in Λ p . By Corollary 6.6–(4), thepolynomial ∆ ( p ) k ( L )( t ) divides f ( p ) ( t ) in Λ p , so ∆ ( p ) i ( t ) is not null for every i ≥ k . (cid:3) Corollary 7.4.
We have e ( p ) i ( L )( t ) = 0 for every i ≥ k . Proof of Theorem 1.6.
The key step for proving Theorem 1.6 is the following:
Proposition 7.5. If L is a k –component link, then I (1 , p, h ( t ) , L ) ≤ br ∆ ( p ) k ( L )( t )br h ( t ) + k − . Proof.
By the very definitions we have∆ ( p ) k ( L )( t ) = Y i ≥ k e ( p ) i ( L )( t ) , so (since ∆ ( p ) k ( L )( t ) = 0 by Lemma 7.3)br ∆ ( p ) k ( t ) = X i ≥ k br e ( p ) i ( L )( t ) . Let us now set I ′ (1 , p, h ( t ) , L ) = I (1 , p, h ( t ) , L ) ∩ { i ∈ N | i ≥ k } . In order to conclude it issufficient to show that | I ′ (1 , p, h ( t ) , L ) | ≤ br ∆ ( p ) k ( t )br h ( t ) . By Corollary 7.4, if i ∈ I ′ (1 , p, h ( t ) , L ) then e ( p ) i ( L )( t ) is not null and divisible by h ( t ), sobr e ( p ) i ( L )( t ) ≥ br h ( t ). This readily implies thatbr ∆ ( p ) k ( t ) ≥ X i ∈ I ′ (1 ,p,h ( t ) ,L ) br e ( p ) i ( L )( t ) ≥ | I ′ (1 , p, h ( t ) , L ) | · br h ( t ) , whence the conclusion. (cid:3) Let us now point out the following:
Lemma 7.6. If L is any link, then A Q ( L ) = 0 = ⇒ A ( L ) = 0 . Proof.
Recall from Corollary 6.6 that ∆ ( p )1 ( L )( t ) = π p (∆( L )( t )). As a consequence, if A ( L ) >
0, then br ∆ ( p ) ( L )( t ) > p (just choose p to be larger than the absolutevalue of all the coefficients of ∆( L )( t )). This implies that br e ( p ) i ( t ) > i ∈ N . If h ( t )is any irreducible factor of e ( p ) i ( t ) in Λ p , then i ∈ I (1 , p, h ( t ) , L ), so A Q ( L ) ≥ I (1 , p, h ( t ) , L ) =1. (cid:3) The following Corollary readily implies Theorem 1.6.
Corollary 7.7. If L = K is a knot, then I (1 , p, h ( t ) , K ) ≤ A ( K )br h ( t ) ≤ A ( K ) , A Q ( K ) = I ( K ) = I (1 , K ) ≤ A ( K ) , A Q ( K ) = 0 ⇐⇒ A ( K ) = 0 . Proof.
By Corollary 6.6–(3) we have br ∆ ( p )1 ( K )( t ) ≤ br ∆( K )( t ) = A ( K ), so the first inequal-ity follows immediately from Proposition 7.5. As a consequence, we have I (1 , K ) ≤ A ( K ), sothe second inequality is a consequence of Lemma 7.1. The fact that A Q ( K ) = 0 if and onlyif A ( K ) = 0 easily follows from the second inequality and Lemma 7.6. (cid:3) Computing A Q via proper subfamilies of Q F . This Subsection is devoted to de-termine proper subfamilies of Q F that carry the whole information about the invariant A Q .We will be mainly interested in the case when L = K is a knot (some considerations belowhold more generally for ( L, P m )).Let K be a knot, and recall that δ ( K ) has been defined in Subsection 1.7. We begin withthe following: Lemma 7.8.
Let f ( t ) ∈ Z [ t ] be a polynomial and suppose that there exist prime numbers p , . . . , p k such that f ( n ) = ± p α ( n )1 · . . . · p α k ( n ) k for every n ≥ n , where n ∈ N is fixed. Then f ( t ) is constant.Proof. Let d = deg f ( t ), and take h > | f ( n ) | ≤ hn d . Then α ( n ) ln p + . . . + α k ( n ) ln p k ≤ d ln n + ln h for every n ≥ n . In particular, there exist a constant w ≥ α i ( n ) ≤ w ln n for every n ≥ n . Therefore, if n is such that ( n − n ) > d +1)( w ln n ) k , then the interval [ n , n ] contains (at least) 2( d +1) integers m , . . . , m d +1) suchthat α i ( m j ) = α i ( m j ′ ) for every i = 1 , . . . , k , j, j ′ = 1 , . . . , d + 1), whence f ( m i ) = ± f ( m j ′ )for every j, j ′ = 1 , . . . , d + 1). It follows that f takes the same value on at least d + 1 distinctintegers. Since deg f = d , this implies in turn that f is constant. (cid:3) We now prove Proposition 1.10, which we recall here for the convenience of the reader.
Proposition 7.9.
Let K be a knot. (1) If θ ( K ) > , then θ ( K ) ≥ δ ( K ) + 1 . (2) If A Q ( K ) = 1 , then δ ( K ) = 1 . (3) If A ( K ) > , then δ ( K ) ≤ A ( K )max { , A Q ( K ) } . (4) Suppose that A Q ( K ) = A ( K ) or A Q ( K ) = A ( K ) − . Then δ ( K ) = 1 . Moreover,there exist an odd prime p and an element a ∈ Z ∗ p such that ( t − a ) A Q ( K ) divides ∆ ( p )1 ( K )( t ) in Λ p . (5) If A Q ( K ) = A ( K ) , then δ ( K ) = 1 and there exist an odd prime p and an element a ∈ Z ∗ p such that ∆ ( p )1 ( K )( t ) . = ( t − a ) A ( K ) in Λ p .Proof. (1) Suppose that A Q ( K ) = | I (1 , p, h ( t ) , K ) | , where br h ( t ) = δ ( K ). Since θ ( K ) > X = ( F ( p, h ( t )) , ∗ ) cannot be trivial, so Lemma 2.1 implies that t X ≥ δ ( K ) + 1.(2) By Theorem 1.6 we have A ( K ) >
0, whence br ∆( K )( t ) >
0. Let f ( t ) ∈ Z [ t ] ⊆ Λbe such that f ( t ) . = ∆( K )( t ) and f (0) = 0, so that deg f ( t ) = br ∆( K )( t ) = d >
0. ByLemma 7.8, there exists n > | f (0) | such that f ( n ) is divided by a prime number p > | f (0) | .Let a be the class of n in Z p , and let us set h ( t ) = t − a ∈ Z p [ t ] ⊂ Λ p . Since p divides f ( n ), we have that h ( t ) divides ∆ ( p )1 ( L )( t ) = π p (∆( L )( t )) in Λ p . Also observe that p doesnot divide f (0), so p does not divide f ( n ) − f (0), and this readily implies that a = 0 in Z p .It follows that h ( t ) is irreducible of positive breadth in Λ p , so we may set X = ( F ( p, h ( t )) , ∗ ).By construction we have a X ( K, ≥ A Q ( K ), so δ ( K ) = 1.(3) It is well–known that A ( K ) = br ∆( K )( t ) is even, so A ( K ) > A ( K ) ≥ A Q ( K ) ≤ A ( K ), this implies that A ( K ) / max { , A Q ( K ) } ≥ δ ( K ) >
1, whence A Q ( K ) ≥ A Q ( K ) = I (1 , p, h ( t ) , K ), where δ ( K ) = br h ( t ). Corollary 7.7 impliesthat A Q ( K ) = I (1 , p, h ( t ) , K ) ≤ A ( K ) /δ ( K ), whence the conclusion. (4) The case A ( K ) = 0 is trivial, so the first statement is an immediate consequence of(3). Then, we may choose h ( t ) = ( t − a ) ∈ Λ p , a ∈ Z ∗ p , in such a way that A Q ( K ) = | I (1 , p, h ( t ) , K ) | . Now h ( t ) divides e ( p ) i ( K )( t ) for every i ∈ I (1 , p, h ( t ) , K ), so h ( t ) A Q ( K ) divides Q i ∈ I (1 ,p,h ( t ) ,K ) e ( p ) i ( K )( t ), which divides in turn ∆ ( p )1 ( K )( t ).(5) is an immediate consequence of (4). (cid:3) Remark 7.10.
As mentioned in Question 1.11, we are not able to prove that δ ( K ) may bearbitrarily large. Let us point out some difficulties that one has to face in order to prove(or disprove) such a statement. If one tries to construct a knot K with δ ( K ) ≥ n , onehas to find K and X = ( F ( p, h ( t ) , ∗ ) such that A Q ( K ) = a X ( K,
1) = | I (1 , p, h ( t ) , K ) | andbr h ( t ) = n . Once this has been established, the inequality δ ( K ) ≤ n is proved. In order toshow that δ ( K ) = n we are left to prove that for every odd prime q the number of polynomials e ( q ) i ( K )( t ), i ∈ N , admitting a common factor of breadth at most n − A Q ( K ). One may probably start with a knot K whose Alexander polynomial ∆( K )( t )decomposes as the product of irreducible factors of large breadth. This would ensure thatalso the e i ( K )( t )’s have large breadth. However, it is not clear how to control the breadth(and the existence of common divisors) of the e ( q ) i ( K )( t )’s, when q is a generic prime, evenunder the hypothesis that I (1 , p, h ( t ) , K ) collects a maximal subset of indices such that thecorresponding e ( p ) i ( K )( t )’s have a non–trivial common divisor (and such a divisor has breadth n ). 8. Genus–1 knots
In this Section we fully describe the case of knots that admit a Seifert surface of genus 1(that is, knots of genus 1 or the unknot).Let us fix a quandle X = ( F ( p, h ( t )) , ∗ ), take z ∈ N and consider a special diagram ( D ( T ) , z )of ( K, z ) (see Section 3).By Lemma 6.1, the integer a X ( K, z ) is equal to the dimension of ker N ( z, p, h ( t )), where N ( z, p, h ( t )) = (cid:18) ( t z − M , t z ( M , − − M , t z ( M , + 1) − M , ( t z − M , (cid:19) (recall that t denotes the class of t in F ( p, h ( t ))).Recall that the determinant det K of K is defined as det K = | ∆( K )( − | = | det( S ( L ) + S ( L ) T ) | , where S ( L ) is a Seifert matrix for K . Lemma 8.1.
We have M , − M , = 1 , det K = | M − | . Proof.
The first equality is an immediate consequence of Lemma 6.3. Putting together Lem-mas 6.2 and 6.3 we also obtain S ( L ) + S ( L ) T = 2 M + J , whence the conclusion. (cid:3) Proposition 8.2.
Let K be a knot such that g ( K ) = 1 , let us take z ∈ N and a quandle X = ( F ( p, h ( t )) , ∗ ) ∈ Q F . Then a X ( K, z ) = 2 if and only if for (one, and hence for) every special diagram ( D ( T ) , z ) of ( K, z ) the followingconditions hold: M , = M , = 0 , M , = p + 12 , M , = p −
12 in Z p and h ( t ) (cid:12)(cid:12) (1 + t z ) in Λ p . Proof. If X is trivial, then for every z we have a X ( K, z ) = 0. Moreover, h ( t ) = t − t z , so we may suppose that X is non–trivial. Suppose now that z = 0 in Z t X . Thenwe have a X ( K, z ) = 0, and h ( t ) has to divide 1 − t z by Lemma 2.1–(2). As a consequence, h ( t ) cannot divide 1 + t z , so the conclusion holds also in this case. We may therefore assumethat z = 0 in Z t X .Observe that a X ( K, z ) = 2 if and only if N ( z, p, h ( t )) = 0, i.e. if and only if( t z − M , = 0 ( t z − M , = 0 ,t z ( M , − − M , = 0 , t z ( M , + 1) − M , = 0 . Since z = 0 in Z t X , we have t z − = 0, so the first and the second relations give M , = M , =0. By Lemma 8.1, the third equation can be rewritten as M , t z = M , + 1. Together withthe fourth equation, this immediately implies that 2 M , = −
1, whence M , = ( p − / M , = ( p + 1) / t z + 1 = 0, i.e. to the fact that h ( t ) divides t z + 1 inΛ p . (cid:3) Proposition 8.2 readily implies the following:
Corollary 8.3.
With notations as in Proposition 8.2, we have A Q ( K ) = 2 if and only if there exist a special diagram D ( T ) of K and a prime number p ≥ such thatthe following equalities hold in Z p : M , = M , = 0 , M , = p + 12 , M , = p − . Moreover, in this case there exists a dihedral quandle X ∈ Q F (1) such that a X ( K,
1) = 2 , sothat δ ( K ) = 1 , θ ( K ) = 2 . Remark 8.4.
Notice that if a special diagram ( D ( T ) , z ) verifies the conditions of Corol-lary 8.3 that involve M , and M , , then we can easily realize also the conditions on M , and M , via suitable Reidemeister moves of the first type ( i.e. by “adding kinks”) on thetwo strings of T .Let us now discuss the conditions under which a X ( K, z ) = 0 for every X ∈ Q F , z ∈ Z . Webegin with the following: Lemma 8.5.
We have ∆( K )( t ) . = det M + (1 − M ) t + (det M ) t . Proof.
Corollary 6.4 implies that∆( K )( t ) = det (( t − M + tJ ) = ( t − M , M , − (( t − M , − t )(( t − M , + t ) . Since M , − M , = 1, the conclusion follows. (cid:3) Corollary 8.6.
The integer W ( K ) = det M is a well–defined invariant of K , i.e. it does not depend on the special diagram of K defining M . Moreover, ∆( K )( t ) . = ∆( K ′ )( t ) if and only if W ( K ) = W ( K ′ ) . Proof.
Suppose that D ′ ( T ) is a special diagram of K of genus 1, and let M ′ be the matrixencoding the linking numbers of the strings of T . By Lemma 8.5 we havedet M + (1 − M ) t + (det M ) t . = det M ′ + (1 − M ′ ) t + (det M ′ ) t , so det M = ± det M ′ , 1 − M = ± (1 − M ′ ), and det M = det M ′ . (cid:3) Putting together Lemma 8.5 and Lemma 6.7 we readily get the following:
Proposition 8.7.
Let K be a knot such that g ( K ) = 1 , let us take z ∈ N and a quandle X = ( F ( p, h ( t )) , ∗ ) ∈ Q F . Then a X ( K, z ) ≥ if and only if h ( t ) (cid:12)(cid:12)(cid:12) W ( K ) + (1 − W ( K )) t z + W ( K ) t z in Λ p . In particular, if W ( K ) = 0 then a X ( K, z ) = 0 . Corollary 8.8.
Let K be a knot, such that g ( K ) ≤ . Then A Q ( K ) = 0 if and only if W ( K ) = 0 (due to Lemma 8.1, this condition is equivalent to det K = 1 ). In all the othercases there exists a dihedral quandle X ∈ Q F (1) such that A Q ( K ) = a X ( K, ≥ . We havein particular δ ( K ) = 1 , θ ( K ) = 2 . Proof.
By Proposition 8.7, it is sufficient to show that, if W ( K ) = 0, then there exists adihedral quandle X ∈ Q F (1) such that A Q ( K ) = a X ( K, ≥ W ( K ) = 0 we may choose an odd prime p be dividing 1 − W ( K ). Then thepolynomial 1 + t divides W ( K ) t + (1 − W ( K )) t + W ( K ) in Λ p . By Proposition 8.7, thisimplies that a X ( K, ≥
1, where X is the dihedral quandle X = ( F ( p, t ) , ∗ ). ✷ Remark 8.9.
By Corollary 8.8, every genus–1 knot such that A Q ( K ) ≥ δ ( K ) = 1. Since for every such knot we obviously have A ( K ) ≤ g ( K ) = 2, this fact is alsoa consequence of Proposition 1.10.8.1. A few manipulations on special diagrams.
Here below we describe a few simplemanipulations on (genus–1) special diagrams, which are useful to construct large families ofexamples.
Lemma 8.10.
Let K be a genus–1 knot with a X ( K, z ) = 2 for some quandle X ∈ Q F , andlet ( D ( T ) , z ) be a special diagram of ( K, z ) . Then by adding kinks to only one of the twostrings of T we can arbitrarily modify either M , or M , , so that the resulting ( D ( T ′ ) , z ) isa special diagram of some ( K ′ , z ) such that a X ( K ′ , z ) = 1 . Lemma 8.11.
Let K be a genus–1 knot with a X ( K, z ) ≥ for some quandle X = ( F ( p, h ( t )) , ∗ ) ,and let ( D ( T ) , z ) be a special diagram of ( K, z ) . Let us modify T by means of any sequenceof usual second and third Reidemeister moves, of first Reidemeister moves provided that M , and M , are kept constant mod ( p ) , and of positive (negative) linking moves between thetwo strings, (see Figure 12; here the actual sign of the move depends also on the omittedorientations of the strings), provided that their number is equal to mod ( p ) . Then we geta special diagram ( D ( T ′ ) , z ) of some ( K ′ , z ) such that a X ( K, z ) = a X ( K ′ , z ) ≥ . Figure 12.
Linking moves.8.2.
The dihedral case.
Let us specialize the results above to the simplest case of dihedralquandles, i.e. to the case when X = D p = ( F ( p, t ) , ∗ ) and z = 1. In such a case we simplywrite a p ( K ) = a D p ( K, . The following result is an immediate consequence of Propositions 8.2 and 8.7.
Lemma 8.12.
Let K be a knot such that g ( K ) = 1 , represented by a special diagram D ( T ) . (1) a p ( K ) = 2 if and only the following system of relations is satisfied in Z p : M , = p + 12 , M , = p − , M , = 0 , M , = 0 . (2) a p ( K ) ≥ if and only if − W ( K ) = 0 in Z p . By Lemma 8.1, this condition holds if and only if p divides det K . Now we want to show that, for every p >
2, the first set of conditions in the last Lemma canbe actually realized by a special diagram D ( T p ) of some knot K p . Let us consider the tangleof Figure 13. Here p = 2 k + 1 and there are k (resp. k + 1) overcrossing (resp. undercrossing)vertical strands. So it is immediate to verify that (in Z ): M , = 0 , M , = p, M , = − p − , M , = − p + 12 , whence 1 − W ( K ) = 1 − M = p . Hence:
Lemma 8.13.
The family { K p } of genus–1 knots constructed above is such that a p ( K p ) = 2 while a p ′ ( K p ) = 0 for every p ′ = p . In particular, K p is not isotopic to K p ′ if p = p ′ , and A Q ( K p ) = 2 for every p . Let us now modify T p into a tangle T ′ p by adding one positive kink to the second string of T , and let us denote by K ′ p the knot described by the special diagram D ( T ′ p ). Lemma 8.14.
The family { K ′ p } of genus–1 knots just constructed is such that a p ( K ′ p ) = 1 while a p ′ ( K ′ p ) = 0 for every p ′ = p . In particular, K ′ p is not isotopic to K ′ p ′ if p = p ′ , and A Q ( K ′ p ) = 1 for every p . Moreover, ∆( K p )( t ) . = ∆( K ′ p )( t ) for every odd prime p .Proof. It is readily seen that W ( K ′ p ) = W ( K p ) = ( p − /
4, so ∆( K ′ p )( t ) = ∆( K p )( t ) and a p ( K ′ p ) ≥
1, while a p ′ ( K ′ p ) = 0 for every p ′ = p . Moreover, a p ( K ′ p ) = 2 since for the tangle T ′ p the value of M , is equal to p + 1 which is not null in Z p . (cid:3) T p Figure 13.
The tangle T ( p ). h 2k T(h,k)h 2k Figure 14.
The tangle T ( h, k ). The rectangular boxes refer to the tanglesdescribed in Figure 12.Lemmas 8.13 and 8.14 imply that the quandle invariant A Q can be more effective than theAlexander polynomial in distinguishing knots, and this phenomenon shows up already in thecase of genus–1 knots: Corollary 8.15.
There exist genus–1 knots K , K ′ such that ∆( K )( t ) . = ∆( K ′ )( t ) , while A Q ( K ) = 2 and A Q ( K ′ ) = 1 . Moreover, A Q ( K ) and A Q ( K ′ ) may be realized by the samedihedral quandle and the same cocycle z = 1 . An example involving a quandle of order p . All the previous explicit examples areobtained by using some Alexander quandle structure on F p . By Corollary 8.8, such quandlestructures encode the relevant information about the invariant A Q of genus–1 knots.Let us show anyway also an example based on a quandle of order p . Consider the tangle T ( h, k ) of Figure 14, encoding a knot K ( h, k ). One can verify that a X ( K (5 , ,
2) = 2, when X = ( F (11 , t ) , ∗ ), which is of type t X > q = 11 ).8.4. The general picture of genus–1 knots.
The following Proposition summarizes thediscussion carried out in the preceding Subsections:
Proposition 8.16.
Let K and K ′ be knots of genus g ≤ . Then: (1) Let M be the matrix associated to a special diagram of K . Then the integer W ( K ) =det M is a well–defined invariant of K ( i.e. it does not depend on the chosen diagram). (2) ∆( K )( t ) . = ∆( K ′ )( t ) if and only if W ( K ) = W ( K ′ ) . (3) A ( K ) = 0 if and only if A Q ( K ) = 0 if and only if W ( K ) = 0 . (4) A ( K ) ∈ { , } , while A Q ( K ) ∈ { , , } . More precisely, for every η ∈ { , , } thereexists a genus–1 knot K such that A Q ( K ) = η . (5) There exist K and K ′ such that ∆( K )( t ) = ∆( K ′ )( t ) , while A Q ( K ) = 2 and A Q ( K ′ ) =1 .Proof. By Lemmas 8.5, Corollaries 8.6, 8.8, 8.15, we are only left to prove that there existsa genus–1 knot K such that A Q ( K ) = 0. As an example of such a knot, one may takeany genus–1 knot with trivial Alexander polynomial, such as the Whitehead double of thefigure–eight knot. (cid:3) On genus–1 knots with minimal Seifert rank.
Recall that a Seifert surface Σ ofa knot K is said to have minimal Seifert rank if the rank of its Seifert form S equals thegenus g = g (Σ). Moreover, a knot has minimal Seifert rank if it admits a Seifert surface (ofarbitrary genus) having minimal Seifert rank. It is a well–known fact that every knot K withminimal Seifert rank has trivial Alexander polynomial ∆( K )( t ) . = 1 (i.e. A ( K ) = 0). Weclaim that: If K is a knot such that g ( K ) ≤ and A ( K ) = 0 , then every genus– Seifert surface for K has minimal Seifert rank. It follows that a genus– knot has trivial Alexander polynomialif and only if it has minimal Seifert rank. In fact, let D be a special diagram for K associated to a given genus–1 Seifert surface Σ,let M be the matrix associated to D , and let S be the matrix representing the Seifert formon Σ with respect to the geometric basis carried by D . Corollary 6.4 implies that S = (cid:18) M , M , − M , + 1 M , (cid:19) , so S has rank equal to 1 if and only if 0 = det S = det M − M , + M , + 1 = det M = W ( K ).By Proposition 8.16–(3), if A ( K ) = 0 then W ( K ) = 0, so S has minimal rank.It is a non–trivial fact proved in [5] that the last statement of the claim does not hold ingeneral for knots of genus ≥
2. 9.
Sums of genus–1 knots
We can use genus–1 knots as buiding blocks for the construction of examples of arbitrarygenus. Let us first observe that, if K and K ′ are (oriented) knots endowed respectively withspecial diagrams D ( T ) and D ( T ′ ), then the knot K + K ′ admits an obvious special diagram D ( T + T ′ ) (see Figures 15 and 16).Let N ( z ), N ′ ( z ), N ′′ ( z ) be the matrices associated to the special diagrams D ( T ), D ( T ′ ), D ( T + T ′ ) as in Section 5, where z is a natural number. It is immediate to realize that N ′′ ( z ) = (cid:18) N ( z ) 00 N ′ ( z ) (cid:19) . Since N (1) (resp. N ′ (1), N ′′ (1)) is a Seifert matrix for K (resp. K ′ , K + K ′ ), this readilyimplies the well–known: Lemma 9.1.
We have ∆( K + K + · · · + K h )( t ) = h Y j =1 ∆( K j )( t ) , Figure 15.
On the top: two special diagrams D , D ′ of the unknot K . Onthe bottom: the special diagram for K + K = K obtained by “summing”the special diagrams on the top. T T’T+T’
Figure 16. If T and T ′ are the primary tangles of special diagrams of K and K ′ , then T + T ′ is the primary tangle of a special diagram of K + K ′ . whence A ( K + K + · · · + K h ) = h X j =1 A ( K j ) . Let us now fix an odd prime p and an irreducible element h ( t ) ∈ Λ p of positive breadth.Since N ( z, p, h ( t )) (resp. N ′ ( z, p, h ( t )), N ′′ ( z, p, h ( t ))) is obtained from N ( z ) (resp. N ′ ( z ), N ′′ ( z )) just by projecting the coefficients onto F ( p, h ( t )), from Lemma 6.1 we deduce thefollowing: Lemma 9.2.
For every X ∈ Q F , z ∈ Z t X , we have a X ( K + K + . . . + K h , z ) = h X i =1 a X ( K j , z ) . Therefore, A Q ( K + K + · · · + K h , z ) ≤ h X j =1 A Q ( K j ) . We observe that the equality A Q ( K + K + · · · + K h , z ) = P hj =1 A Q ( K j ) does not hold ingeneral. The equality holds if a single quandle X ∈ Q F exists which realizes all the A Q ( K i )’swith respect to the same cycle.We are now ready to prove Proposition 1.9, which we recall here for the convenience of thereader. Proposition 9.3.
Let us fix g ≥ . Then, for every r , r such that ≤ r ≤ r ≤ r ≤ g ,there exist knots K and K such that the following conditions hold: g ( K ) = g ( K ) = g, ∆( K ) = ∆( K ) (whence A ( K ) = A ( K )) , while A Q ( K ) = r , A Q ( K ) = r . Moreover, we can require that both A Q ( K ) and A Q ( K ) are realized by means of somedihedral quandle with cycle z = 1 .Proof. Let
K, K ′ be the genus–1 knots provided by Corollary 8.15, and let K ′′ be a genus–1knot with trivial Alexander polynomial (see Proposition 8.16. Then we may define K as thesum of r copies of K ′ and g − r copies of K ′′ , and K as the sum of 2 r − r copies of K ′ , r − r copies of K and g − r copies of K ′′ . The additivity of the genus gives that g ( K ) = g ( K ) = g ,and Lemma 9.1 readily implies that ∆( K )( t ) = ∆( K )( t ). Moreover, by Lemma 9.2 we havethat A Q ( K ) ≤ r and A Q ( K ) ≤ r . However, Corollary 8.15 ensures that there exists adihedral quandle X such that A Q ( K ) = a X ( K,
1) = 2 and A Q ( K ′ ) = a X ( K ′ ,
1) = 1, so byLemma 9.2 a X ( K ,
1) = r and a X ( K ,
1) = r , whence the conclusion. (cid:3) The case of links.
Let L = K ∪ . . . ∪ K h be a split link, where K i is a knot for every i = 1 , . . . , h . Let also P M be the maximal partition of L , let z : P M → N be a P M –cycleand set z i = z ( K i ). The following Lemma is an immediate consequence of the definition ofquandle coloring: Lemma 9.4.
For every quandle X ∈ Q F we have c X ( L, P M , z ) = h Y i =1 c X ( K i , z i ) , so a X ( L, P M , z ) = h X i =1 a X ( K i , z i ) ! + h − , and A Q ( L, P M ) ≤ h X i =1 A Q ( K i ) ! + h − . Moreover, if h ≥ then ∆( L )( t ) = 0 , so A ( L ) = 0 . Just as in Lemma 9.2, the equality A Q ( L, P M ) ≤ (cid:16)P hi =1 A Q ( K i ) (cid:17) + h − K be a genus–1 knot such that A Q ( K ) = 2 (see Section 8 for examples of suchknots), and let L h be the split link having h components, each isotopic to K . The followingresult implies Proposition 1.8 Proposition 9.5.
We have A Q ( L h ) = 3 h − .Proof. We have a X ( K ,
1) = 2 for some quandle X ∈ Q F , so Lemma 9.4 implies that A Q ( L h ) ≥ a X ( L, P m ,
1) = a X ( L, P M ,
1) = 3 h − . (cid:3) Remark 9.6.
Strictly speaking, the equality A Q ( L h ) = 3 h − g ( L h ), since Corollary 1.3 states that A Q ( L ) ≤ g ( L ) + 2 k − k –componentlink. This inequality provides the bound 2 g ( L h ) ≥ h + 1, which is not sharp since of course g ( L h ) = h . However, it is immediate to see that if L is a split link, then g ( L ) = g ( L, P M ).The inequalities3 h − A Q ( L h ) ≤ A Q ( L h , P M ) ≤ g ( L h , P M ) + h − g ( L h ) + h − g ( L h ) ≥ h . References [1] R. Benedetti, R. Frigerio,
Levels of knotting of spatial handlebodies , arXiv:1101.2151.[2] M. Boileau, M. Rost, H. Zieschang,
On Heegaard decompositions of torus exteriors and related Seifertfibred spaces , Math. Ann. (1988), 553–581.[3] G. Burde, H. Zieschang,
Knots , de Gruyter Studies in Mathematics, 5. Walter de Gruyter & Co., Berlin,1985.[4] P. Cromwell,
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