LLink colorings and the Goeritz matrix
Lorenzo TraldiLafayette CollegeEaston, Pennsylvania 18042
Abstract
We discuss the connection between colorings of a link diagram and theGoeritz matrix.Keywords. Coloring, Goeritz matrix, linkMathematics Subject Classification. 57M25, 05C10, 05C22, 05C50
This paper is inspired by two papers that have appeared previously, the firstwritten by Nanyes [20] and the second written by Lamey, Silver and Williams[15]. These papers involve the connection between two ideas from classical knottheory, the Goeritz matrix and colorings of link diagrams. Goeritz introducedhis matrix in 1933 [7], and it was also discussed in Reidemeister’s classic treatise[22]. The Goeritz matrix has attracted the attention of many researchers overthe decades; see [3, 4, 8, 9, 10, 11, 14, 16, 17, 18, 21, 23, 25] for instance. Linkcolorings were mentioned in the textbook of Crowell and Fox [5, Exercises VI.6 and VI. 7]. Link colorings can be defined easily and they provide very simplenontriviality proofs for some knots and links, so it is natural that they arementioned in many introductory discussions of knot theory, like [1, 2, 13, 19].Crowell and Fox used link colorings for the purpose of providing combina-torial descriptions of certain kinds of representations of link groups (the funda-mental groups of link complements in S ). At first glance, this purpose doesnot suggest a connection with the Goeritz matrix; link groups are nonabelian ingeneral, and we would expect a matrix of integers to be associated with abeliangroups instead. Nanyes [20] provided an indirect connection: link colorings withvalues in an abelian group A are connected with representations of link groupsin a semidirect product of A and Z / Z , and these representations in turn areconnected with the Goeritz matrix. Nanyes’s discussion is quite general; it ap-plies to any link diagram, and any abelian group. His presentation requires thetheory connecting group representations to covering spaces, and the fact thatthe Goeritz matrix is associated with 2-fold coverings of S branched over links[14, 16, 23].Extending a theme established earlier by Kauffman [12] and Carter, Silverand Williams [2], Lamey, Silver and Williams [15] showed that the colorings1 a r X i v : . [ m a t h . G T ] J un ntroduced by Crowell and Fox are related to other types of link colorings, andthey observed that one of these other types of link colorings is directly related tothe Goeritz matrix. Unlike Nanyes, they restricted their attention to coloringswith values in a field, and to alternating link diagrams for which one of theassociated checkerboard graphs is connected. This portion of their paper doesnot require results from algebraic topology; the arguments involve only relativelyelementary properties of matrices and plane graphs.Taken together, the papers of Lamey, Silver and Williams [15] and Nanyes[20] suggest a problem of exposition: to provide an explanation of the connectionbetween link colorings and the Goeritz matrix that is as direct and elementaryas the discussion in [15], and as general as the discussion in [20]. The purposeof the present paper is to provide such an explanation.Before starting, we should thank an anonymous reviewer whose good adviceimproved the exposition in several regards. We use link diagrams to represent links in the usual way. A tame, classicallink diagram begins with a finite number of piecewise smooth, simple closedcurves γ , . . . , γ µ in the plane. The only (self-)intersections of these curves aretransverse double points, called crossings ; and there are only finitely many ofthese. At each crossing a very short segment of one of the incident curves isremoved. The resulting piecewise smooth 1-dimensional subset of R is a linkdiagram. The set of arc components of a link diagram D is denoted A ( D ). The faces of D are the arc components of R − ∪ γ i , and the set of faces of D isdenoted F ( D ). A link diagram D represents a link L ( D ) in R , which consistsof piecewise smooth, simple closed curves K , . . . , K µ such that K i projects to γ i for each i . K , . . . , K µ are the components of L ( D ). The removal of segmentsin D is used to distinguish the underpassing arc at each crossing.A link diagram is split if the union ∪ γ i is not connected. In keeping with thegoal of generality mentioned in the introduction, our discussion includes splitdiagrams as well as non-split diagrams.We use A to denote an arbitrary abelian group. To avoid notational confu-sion with the arcs of a link diagram, we usually use α to represent an elementof A . Definition 1 If D is a link diagram then a Fox coloring of D with values inan abelian group A is a mapping f : A ( D ) → A with the following property. • If there is a crossing of D at which the underpassing arcs are a and a and the overpassing arc is a , then f ( a ) + f ( a ) = 2 · f ( a ) .The set of Fox colorings of D with values in A is denoted F A ( D ) . Notice that we do not require a Fox coloring of D to be 0 on any arc of D . This generality gives F A ( D ) a couple of pleasant (and obvious) naturality2roperties, which do not hold in [2, 15, 20]. Suppose D is the split union ofsubdiagrams D and D , i.e., D = D ∪ D and no crossing of D involves both D and D . Then the union of functions defines a bijective map F A ( D ) × F A ( D ) → F A ( D )for every abelian group A , and an injective map F A ( D ) × F A ( D ) → F A ⊕ A ( D )for every pair of abelian groups A , A . Definition 2
Let D be a link diagram. Then a Dehn coloring of D with valuesin an abelian group A is a mapping d : F ( D ) → A with the following property. • Suppose F and F (cid:48) are two faces of D whose boundaries share a segment ofpositive length, contained in an arc a ∈ A ( D ) . Then the sum d ( F ) + d ( F (cid:48) ) depends only on a .The set of Dehn colorings of D with values in A is denoted D A ( D ) . Definition 2 gives rise to one equation for each crossing of D . If the facesincident at a crossing are indexed as in Figure 1 then the boundaries of F and F share a segment of positive length contained in a , and so do the boundariesof F and F . Consequently Definition 2 requires d ( F ) + d ( F ) = d ( F ) + d ( F ). a a a F F F F Figure 1: The arcs and faces incident at a crossing.We should remark that the term “Dehn coloring” indicates a courteous re-gard for one of the important early contributors to combinatorial group the-ory and geometric topology, but it does not indicate that these colorings wereactually introduced by Dehn. Definition 2 was mentioned by Kauffman [12]and developed further by Carter, Silver and Williams [2], who chose the name“Dehn coloring” because these colorings are connected with a way to presentlink groups that was introduced by Dehn. We should also remark that like Def-inition 1, Definition 2 is generalized from [2, 12] – we allow A to be an arbitraryabelian group, and we do not require any value of a Dehn coloring to be 0.As discussed in [2] and [12], Dehn colorings and Fox colorings are closelyrelated to each other. 3 efinition 3 Let D be a link diagram, and A an abelian group. Then D A ( D ) and F A ( D ) are both abelian groups under pointwise addition of functions. Thereis a homomorphism ϕ : D A ( D ) → F A ( D ) , defined by: if d ∈ D A ( D ) then ϕ ( d )( a ) = d ( F ) + d ( F ) whenever F , F are two faces of F whose boundaries share a segment of positivelength on a . Definition 2 implies that ϕ ( d ) is well defined, and also that ϕ ( d ) satisfiesDefinition 1: with arcs and faces indexed as in Figure 1, ϕ ( d )( a ) + ϕ ( d )( a ) = d ( F ) + d ( F ) + d ( F ) + d ( F )= ( d ( F ) + d ( F )) + ( d ( F ) + d ( F )) = 2 · ϕ ( d )( a ) . As is well known, the faces of a link diagram can be colored in a checkerboardfashion, so that whenever the boundaries of two faces share a segment of positivelength, one face is shaded and the other is not shaded. It is traditional toprefer one of the two possible checkerboard shadings, by specifying whether theunbounded face should be shaded or unshaded. In keeping with our theme ofgenerality, however, we do not follow this tradition. The gain in generality isvacuous at this point, but later it will mean that the two different shadings of alink diagram give rise to two different Goeritz matrices. The theory we developwill apply equally well to both matrices.Arbitrarily choose one of the two checkerboard shadings of a link diagram D , and let σ : F ( D ) → { , } be the map defined as follows. σ ( F ) = (cid:40)
0, if F is unshaded1, if F is shadedIf α, β ∈ A , then D has a Dehn coloring d α,β given by d α,β ( F ) = (1 − σ ( F )) · α + σ ( F ) · β .This mapping satisfies Definition 2 because in Figure 1 each of the pairs { F , F } , { F , F } includes one shaded face and one unshaded face, so that d α,β ( F ) + d α,β ( F ) = α + β = d α,β ( F ) + d α,β ( F ) . The next result is our version of [2, Theorem 2.2]. We have an epimorphismrather than an isomorphism because our Dehn colorings are not required to be0 anywhere. The proof is essentially the same as in [2], but we provide detailsfor the reader’s convenience.
Theorem 4
The homomorphism ϕ : D A ( D ) → F A ( D ) is surjective, and ker ϕ = { d α, − α | α ∈ A } . roof. Suppose f ∈ F A ( D ), and F is some face of D . Choose an arbitraryelement α ∈ A , and define d ( F ) = α . For any other face F of D , choose asmooth path P from a point in F to a point in F . We may presume that P doesnot come close enough to any crossing to intersect any of the short segmentsremoved to indicate undercrossings, and we may also presume that there areonly finitely many intersections between P and D . Suppose that as we follow P from F to F , we encounter faces and arcs in the order F , a (cid:48) , F (cid:48) , . . . , a (cid:48) k , F (cid:48) k = F .Then we define d ( F ) = ( − k α + k (cid:88) i =1 ( − k − i f ( a (cid:48) i ) . (1)It turns out that d ( F ) is independent of the path P . To see why, suppose P (cid:48) is some other smooth path from F to F . Then P can be smoothly deformedinto P (cid:48) . When a smooth deformation does not involve any crossing of D , thereis no effect on d ( F ). When the deformation passes through a crossing, the effectis to replace one passage of an arc of D with three passages. For instance, inFigure 1 we might replace a passage across a from F to F with a sequenceof three passages; the first from F to F across a , the second from F to F across a , and the third from F to F across a . Suppose the original passagefrom F to F was indexed with F = F (cid:48) j , a = a (cid:48) j +1 and F = F (cid:48) j +1 . The valuegiven by (1) is not changed because Definition 1 tells us that( − k − j f ( a ) + ( − k − j − f ( a ) + ( − k − j − f ( a )= ( − k − j · ( − f ( a ) + 2 · f ( a )) = ( − k − j f ( a ) . The same kind of argument holds if any other one of the passages in Figure 1is replaced with the remaining three.By the way, “replacing one passage with three” might seem to indicate thatthe path is getting longer. This is not true in general because some of the threenew passages may be canceled if P includes the opposite passages through thesame crossings. Also, if i < j and F (cid:48) i = F (cid:48) j then the argument above shows thatthe value of (1) is not changed if P is shortened to a path corresponding to thelist F , a (cid:48) , F (cid:48) , . . . , a (cid:48) i , F (cid:48) i , a (cid:48) j +1 , . . . , F (cid:48) k = F .To verify that d ∈ D A ( D ), suppose the d values of the four faces indicatedin Figure 1 are defined using a path P from F that enters the figure in thelower left hand corner. Then equation (1) implies that d ( F ) = f ( a ) − d ( F ), d ( F ) = f ( a ) − d ( F ) and d ( F ) = f ( a ) − d ( F ). It follows that d ( F ) + d ( F ) = f ( a ) = d ( F ) + d ( F ) . As ϕ ( d ) = f , we have verified that ϕ is surjective. The description of ker ϕ in the statement is obvious.It is easy to see that the epimorphism ϕ splits. Theorem 5 If D is a link diagram and A is an abelian group then D A ( D ) isthe internal direct sum of ker ϕ and a subgroup isomorphic to F A ( D ) . roof. Let F be a fixed but arbitrary face of D . Let δ : F A ( D ) → D A ( D )be the function defined by the construction in the proof of Theorem 4, alwaysusing α = 0. Then formula (1) tells us that δ is a homomorphism.As ϕδ ( f ) = f ∀ f ∈ F A ( D ), the theorem follows. Suppose D is a link diagram, and s is one of the two checkerboard coloringsof its faces. Then each crossing of D is assigned a Goeritz index η ∈ {− , } as indicated in Figure 2. That is, η = 1 if the overpassing arc appears on theright-hand sides of the unshaded face(s) incident at the crossing, and η = − η = ( − σ ( F ) . η = ̶ η = 1 Figure 2: The Goeritz index of a crossing.
Definition 6
Let D be a link diagram, and let s be either of the two checker-board shadings of the faces of D . Let F , . . . , F n be the unshaded faces of D .For i, j ∈ { , . . . , n } let C ij be the set of crossings of D incident on F i and F j .Then the unreduced Goeritz matrix of D with respect to s is the n × n matrix G ( D, s ) with entries defined as follows. G ( D, s ) ij = − (cid:80) c ∈ C ij η ( c ) , if i (cid:54) = j − (cid:80) k (cid:54) = i G ( D, s ) ik , if i = j Before proceeding we make five remarks about Definition 6. (i) G ( D, s ) is asymmetric integer matrix, whose rows and columns sum to 0. It is traditionalto remove one row and the corresponding column, to obtain a matrix whosedeterminant might not be 0. However we do not follow this tradition here;that is why we call our matrix “unreduced.” (ii) G ( D, s ) ignores any crossingthat is incident on only one unshaded face. (iii) There are two checkerboardgraphs or Tait graphs associated with D . One graph has vertices correspondingto the shaded faces, and the other graph has vertices corresponding to the6nshaded faces. Both have edges corresponding to the crossings of D . Thematrix G ( D, s ) is the
Laplacian matrix of the unshaded checkerboard graph,with the Goeritz indices interpreted as edge weights. There is a well developedtheory of Laplacian matrices of weighted graphs; the interested reader mightconsult [6, Chapter 13] for an introduction. (iv) If s is the other checkerboardshading of D then the matrices G ( D, s ) and G ( D, s ) may be quite different;for instance, one may be much larger than the other. Nevertheless there is anintimate relationship between the two matrices. See [17] for a discussion. (v)Despite the connection between Goeritz matrices of link diagrams and Laplacianmatrices of graphs, we have chosen to use relatively little terminology fromgraph theory in this paper. One reason for this choice is that the definition ofthe dual of a plane graph always results in a connected graph. In contrast, thecheckerboard graphs of a split link diagram may be disconnected.We use A n to denote the direct sum of n copies of the abelian group A . If F , . . . , F n are the unshaded faces of D then G ( D, s ) defines a homomorphism A n → A n of abelian groups. We are interested in the properties of the kernel ofthis homomorphism. Definition 7
Let D be a link diagram with a shading s whose unshaded facesare F , . . . , F n . If A is an abelian group then ker A G ( D, s ) = { v ∈ A n | G ( D, s ) · v = 0 } . That is, ker A G ( D, s ) is the subset of A n consisting of elements that areorthogonal to the rows of G ( D, s ).The next proposition is concerned with a special property of some split linkdiagrams: they have unshaded faces whose boundaries are not connected.
Proposition 8
Let D be a link diagram with a shading s whose unshaded facesare F , . . . , F n . Let γ be a simple closed curve, which forms part of the boundaryof F i . Let ρ ( γ ) ∈ Z n be the vector defined as follows. ρ ( γ ) j = − (cid:80) c ∈ C ij ∩ γ η ( c ) , if i (cid:54) = j − (cid:80) k (cid:54) = i ρ ( γ ) k , if i = j Then for any abelian group A , ρ ( γ ) · v = 0 ∀ v ∈ ker A G ( D, s ) . Proof. If γ is the entire boundary of F i then ρ ( γ ) is the i th row of G ( D, s ).If γ is not incident on any crossing that involves an unshaded face other than F i , then ρ ( γ ) = 0. In either of these cases it is obvious that ρ ( γ ) · v = 0 ∀ v ∈ ker A G ( D, s ).Suppose γ is not the entire boundary of F i , and γ is incident on some crossingthat involves an unshaded face other than F i . According to the Jordan curvetheorem, R − γ has two components, one inside γ and the other outside γ .Suppose the other unshaded face that shares a crossing on γ with F i lies inside γ . (See Figure 3 for an example of this sort. In the figure, γ is indicated with7 F i γ Figure 3: The boundary of F i consists of three closed curves.dashes; it is displaced a little bit for clarity.) Let D (cid:48) be the subdiagram of D that includes γ and all the arcs of D contained inside γ , and let s (cid:48) be the shadingof the faces of D (cid:48) defined by s . Then F i corresponds to an unshaded face of D (cid:48) ,whose boundary in D (cid:48) is γ . The other unshaded faces of D (cid:48) are the unshadedfaces of D contained inside γ , and if F j is an unshaded face of D containedinside γ then the only difference between the row of G ( D, s ) corresponding to F j and the row of G ( D (cid:48) , s (cid:48) ) corresponding to F j is that the former has extra 0entries in columns corresponding to unshaded faces of D not contained inside γ . That is, if G (cid:48) is the submatrix of G ( D (cid:48) , s (cid:48) ) obtained by removing the rowcorresponding to the face of D (cid:48) that contains F i , then G ( D, s ) = (cid:18) G (cid:48) G (cid:48)(cid:48) G (cid:48)(cid:48)(cid:48) (cid:19) for some submatrices G (cid:48)(cid:48) and G (cid:48)(cid:48)(cid:48) . Definition 6 makes it clear that the sumof the rows of a Goeritz matrix is 0; hence we can obtain the row of G ( D (cid:48) , s (cid:48) )corresponding to the face of D (cid:48) that contains F i by summing the other rows of G ( D (cid:48) , s (cid:48) ), and multiplying by −
1. It follows that − ρ ( γ ) is the sum of the rowsof (cid:0) G (cid:48) (cid:1) , so ρ ( γ ) is an element of the row space of G ( D, s ). We conclude that ρ ( γ ) · v = 0 for every v ∈ ker A G ( D, s ).If the other unshaded face that shares a crossing on γ with F i lies outside γ then the same argument applies, with “inside γ ” changed to “outside γ ”throughout. Proposition 9
Let D be a link diagram with a shading s whose unshaded facesare F , . . . , F n . Suppose A is an abelian group, v = ( v , . . . , v n ) ∈ ker A G ( D, s ) ,and F i and F j are incident at a crossing where only one shaded face is incident.Then v i = v j . F i F j F i F j S λ S λ Figure 4: Examples of D and D (cid:48) in Proposition 9. Proof.
Suppose F i and F j are incident at a crossing c , and S is the only shadedface of D incident at c . Then there is a piecewise smooth closed curve λ thatis contained in the interior of S except for the fact that it passes through c .Interchanging i and j if necessary, we may presume that F i is contained in theregion inside λ and F j is contained in the region outside λ . Let D (cid:48) be the linkdiagram obtained from D by smoothing c in such a way that F i is detachedfrom F j . (See Figure 4 for an example.)Let D (cid:48)(cid:48) be the subdiagram of D (cid:48) contained inside λ , and D (cid:48)(cid:48)(cid:48) the subdiagramoutside λ . If s (cid:48) , s (cid:48)(cid:48) and s (cid:48)(cid:48)(cid:48) are the shadings of D (cid:48) , D (cid:48)(cid:48) and D (cid:48)(cid:48)(cid:48) defined by s thenthe Goeritz matrix of D (cid:48) is G ( D (cid:48) , s (cid:48) ) = (cid:18) G ( D (cid:48)(cid:48) , s (cid:48)(cid:48) ) 00 G ( D (cid:48)(cid:48)(cid:48) , s (cid:48)(cid:48)(cid:48) ) (cid:19) . Let G ( D, s ) i , G ( D (cid:48) , s (cid:48) ) i and G ( D (cid:48)(cid:48) , s (cid:48)(cid:48) ) i be the rows of G ( D, s ) , G ( D (cid:48) , s (cid:48) ) and G ( D (cid:48)(cid:48) , s (cid:48)(cid:48) ) corresponding to F i , respectively. Also, let G (cid:48)(cid:48) be the submatrix of G ( D (cid:48)(cid:48) , s (cid:48)(cid:48) ) obtained by removing G ( D (cid:48)(cid:48) , s (cid:48)(cid:48) ) i . Then H = (cid:0) G (cid:48)(cid:48) (cid:1) is a submatrixof G ( D, s ), and the sum of the rows of H is the negative of G ( D (cid:48) , s (cid:48) ) i . Let w = ( w , . . . , w n ) be the vector whose only nonzero entries are w i = η ( c ) and w j = − η ( c ). Then w is the difference between G ( D, s ) i and G ( D (cid:48) , s (cid:48) ) i . Itfollows that if we add G ( D, s ) i to the sum of the rows of H , we get w .We conclude that w is included in the row space of G ( D, s ), so w · v = 0 ∀ v ∈ ker A G ( D, s ). D A ( D ) and ker A G ( D, s ) In this section we generalize ideas of Lamey, Silver and Williams [15] to theGoeritz matrix. The foundation for this generalization has already been laid:Definitions 1 and 2 allow for colorings with values in arbitrary abelian groups,Definition 6 allows for link diagrams with arbitrary crossing signs, and Proposi-tions 8 and 9 provide useful special properties of link diagrams with disconnectedcheckerboard graphs. Extending the arguments of [15] to the general setting re-quires only a little attention to special cases, and one additional idea given inDefinition 13. 9 roposition 10
Let s be a shading of a link diagram D , whose unshaded facesare F , . . . , F n . If d ∈ D A ( D ) , then the vector v ( d ) = ( d ( F ) , . . . , d ( F n )) is anelement of ker A G ( D, s ) . Proof.
Suppose 1 ≤ i ≤ n , and let G ( D, s ) i be the i th row of G ( D, s ). The dotproduct G ( D, s ) i · v ( d ) is a sum of contributions from the crossings of D incidenton F i . To analyze these contributions, consider a crossing of D incident on F i and F j (cid:54) = F i , pictured in Figure 5. η = ̶ η = 1 F i F j S'' S' F i F j S'' S'
Figure 5: A crossing incident on F i and F j .If η = −
1, then the contribution of this crossing to G ( D, s ) i · v ( d ) includestwo terms: − η · d ( F j ) = d ( F j ) from G ( D, s ) ij · d ( F j ) and η · d ( F i ) = − d ( F i )from G ( D, s ) ii · d ( F i ). Consulting Definition 2, we see that the contribution ofthis crossing is d ( F j ) − d ( F i ) = d ( S (cid:48) ) − d ( S (cid:48)(cid:48) ) . If η = 1, the contribution of this crossing to G ( D, s ) i · v ( d ) includes − η · d ( F j ) = − d ( F j ) from G ( D, s ) ij · d ( F j ) and η · d ( F i ) = d ( F i ) from G ( D, s ) ii · d ( F i ).Consulting Definition 2, we see that the contribution of this crossing is − d ( F j ) + d ( F i ) = d ( S (cid:48) ) − d ( S (cid:48)(cid:48) ) . Either way, the contribution is d ( S (cid:48) ) − d ( S (cid:48)(cid:48) ).Now, consider a crossing as pictured in Figure 5, but with i = j . Thiscrossing is ignored by G ( D, s ), so its contribution to G ( D, s ) i · v ( d ) is 0. Onthe other hand, Definition 2 tells us that F i = F j = ⇒ d ( S (cid:48) ) = d ( S (cid:48)(cid:48) ), so d ( S (cid:48) ) − d ( S (cid:48)(cid:48) ) is 0 too.In every case, then, the contribution of the crossing pictured in Figure 5 to G ( D, s ) i · v ( d ) is d ( S (cid:48) ) − d ( S (cid:48)(cid:48) ). If we follow the boundary component of F i thatcontains this crossing in the clockwise direction, we see that the total of thecontributions of all the crossings is a telescoping sum:( d ( S (cid:48) ) − d ( S (cid:48)(cid:48) )) + ( d ( S (cid:48)(cid:48) ) − d ( S (cid:48)(cid:48)(cid:48) )) + · · · + ( d ( S ( k ) ) − d ( S (cid:48) )) = 0,where k is the number of crossings incident on this boundary component of F i .As every boundary component of F i contributes 0, G ( D, s ) i · v ( d ) = 0.Proposition 10 tells us that d (cid:55)→ v ( d ) defines a map v : D A ( F ) → ker A G ( D, s ).It is easy to see that v is a homomorphism of abelian groups, and that ker v D . Itis more difficult to see another important property of v : it is surjective. Proposition 11
Let D be a link diagram, with a shading s whose unshadedfaces are F , . . . , F n . Let A be an abelian group, and suppose v ∈ ker A G ( D, s ) .Then there is a d ∈ D A ( D ) with v = v ( d ) . Proof.
For each unshaded face F i of D , define d ( F i ) to be the i th coordinateof the vector v . Our job is to define d values for the shaded faces, in such a waythat the resulting function d : F ( D ) → A satisfies Definition 2.Choose any shaded face S of D , choose any element α ∈ A , and define d ( S ) = α . Repeat the following recursive step as many times as possible. If S (cid:48)(cid:48) is a shaded face such that d ( S (cid:48)(cid:48) ) has not yet been defined, and the boundaryof S (cid:48)(cid:48) shares a crossing with the boundary of a shaded face S (cid:48) such that d ( S (cid:48) )has been defined, then define d ( S (cid:48)(cid:48) ) to be the unique element of A that satisfiesDefinition 2 at this crossing.After the process of the preceding paragraph is completed, there may stillbe shaded faces whose d values have not been defined; these faces do not shareany crossing with shaded faces whose d values have been defined. Go back tothe preceding paragraph, and change the first sentence to read “Choose anyshaded face S of D whose d value has not yet been defined, choose any element α ∈ A , and define d ( S ) = α .” Then repeat the entire process of the precedingparagraph as many times as possible.We claim that this recursion yields a well defined function d ∈ D A ( D ). Theclaim certainly holds in case no crossing appears in D as G ( D, s ) is the 0 matrix,all vectors with entries in A have G ( D, s ) · v = 0, and all functions F ( D ) → A liein D A ( D ). We proceed with the assumption that there is at least one crossingin D .Let S be a shaded face of D . If S does not share a crossing of D with anyother shaded face then the value of d ( S ) is handled by the first sentence of thesecond paragraph (as modified in the third paragraph). In this case it is obviousthat d ( S ) is well defined. Proposition 9 tells us that v i = v j whenever F i and F j share a crossing of D with S , so d satisfies Definition 2 at all crossings involving S . If all shaded faces of D are of this type, we are done.The rest of the proof resembles the proof of Theorem 4. Suppose S is ashaded face such that d ( S ) is defined through the process of the second para-graph, by applying Definition 2 at a crossing where S and another shaded faceare incident. Let S , c , S , . . . , c k , S k = S be the sequence of shaded faces andcrossings that was used to determine the value of d ( S ), with S handled by thefirst sentence of the second paragraph. This sequence corresponds to a piece-wise smooth path P from a point in the interior of S near c to a point inthe interior of S near c k , which stays inside shaded faces except when it passesthrough crossings. Any other possible recursive definition of d ( S ) correspondsto a similar path P (cid:48) from S to S , which can be deformed into P by some se-quence of two types of moves: the type illustrated in Figure 6, and the trivialtype in which two consecutive passages in opposite directions through the samecrossing are canceled. 11 F i Figure 6: Deforming P .To verify that d ( S ) is well defined, we show that the deformation illustratedin Figure 6 does not change the value of d ( S ). Let S p , c p +1 , S p +1 , . . . , c q , S q bethe portion of the sequence S , c , S , . . . , c k , S k that is a sequence of shadedneighbors of F i . Then P (cid:48) is obtained by replacing this portion with S p = S (cid:48) p , c (cid:48) p +1 , S (cid:48) p +1 , . . . , c (cid:48) q (cid:48) , S (cid:48) q (cid:48) = S q , where the crossings c p +1 , . . . , c q , c (cid:48) q +1 , . . . , c (cid:48) p +1 appear in this order on a closed curve γ contained in the boundary of F i . Inter-changing the names of P and P (cid:48) if necessary, we may presume that c p +1 , . . . , c q , c (cid:48) q +1 , . . . , c (cid:48) p +1 appear in this order clockwise around γ . For 1 ≤ j ≤ q − p let U j be the unshaded face that shares the crossing c p + j with F i , and for 1 ≤ j ≤ q (cid:48) − p let U (cid:48) j be the unshaded face that shares the crossing c (cid:48) p + j with F i . Then whenthe second paragraph states that at each step the unique value of d ( U j ) or d ( U (cid:48) j )that satisfies Definition 2 is used, it means that for each j ≥ d ( S p + j ) = d ( S p + j − ) + η ( c p + j ) · ( d ( U j ) − d ( F i ))= d ( S p + j − ) + η ( c p + j ) · ( v ( U j ) − v ( F i ))and d ( S (cid:48) p + j ) = d ( S (cid:48) p + j − ) − η ( c (cid:48) p + j ) · ( d ( U (cid:48) j ) − d ( F i ))= d ( S (cid:48) p + j − ) − η ( c (cid:48) p + j ) · ( v ( U (cid:48) j ) − v ( F i )) . When we follow P we conclude that d ( S q ) − d ( S p ) = q − p (cid:88) j =1 ( d ( S p + j ) − d ( S p + j − )) = q − p (cid:88) j =1 η ( c p + j ) · ( v ( U j ) − v ( F i ))and when we follow P (cid:48) , we conclude that d ( S (cid:48) q (cid:48) ) − d ( S p ) = q (cid:48) − p (cid:88) j =1 ( d ( S (cid:48) p + j ) − d ( S (cid:48) p + j − )) = − q (cid:48) − p (cid:88) j =1 η ( c (cid:48) p + j ) · ( v ( U (cid:48) j ) − v ( F i )) . − q − p (cid:88) j =1 η ( c p + j ) · ( v ( U j ) − v ( F i )) − q (cid:48) − p (cid:88) j =1 η ( c (cid:48) p + j ) · ( v ( U (cid:48) j ) − v ( F i ))= q − p (cid:88) j =1 η ( c p + j ) + q (cid:48) − p (cid:88) j =1 η ( c (cid:48) p + j ) · v ( F i ) − q − p (cid:88) j =1 η ( c p + j ) · v ( U j ) − q (cid:48) − p (cid:88) j =1 η ( c (cid:48) p + j ) · v ( U (cid:48) j ) = ρ ( γ ) · v ,where ρ ( γ ) is the vector discussed in Proposition 8. As ρ ( γ ) · v = 0, P and P (cid:48) lead to the same value for d ( S ).It remains to verify that d ∈ D A ( D ). Suppose c is a crossing of D , aspictured in Figure 1. If a single shaded face S appears twice in the figure thenDefinition 2 requires that the two unshaded faces F i , F j in the figure have thesame d value. Proposition 9 assures us that this is the case. If two differentshaded faces appear, then the well-definedness of d allows us to assume that the d value of one of the two shaded faces is calculated directly from the d value ofthe other one, as in the third sentence of the second paragraph of the proof. Butthis calculation is performed precisely to guarantee that d satisfies Definition 2at the crossing c .Combining Propositions 10 and 11, we obtain the following. Theorem 12
Let D be a link diagram with a shading s , and A an abelian group.Then v : D A ( D ) → ker A G ( D, s ) is a surjective homomorphism, whose kernelconsists of the Dehn colorings that are identically on unshaded faces of D . The next definition allows us to give a precise description of ker v . Definition 13
Let D be a link diagram with a shading s . Let Γ s ( D ) denotethe shaded checkerboard graph of D , i.e., Γ s ( D ) has a vertex for each shadedface of D and an edge for each crossing of D , with the edge corresponding toa crossing c incident on the vertex (or vertices) corresponding to the shadedface(s) incident at c . Then β s ( D ) denotes the number of connected componentsof Γ s ( D ) . If d ∈ ker v then d is identically 0 on unshaded faces of D , so Definition 2 isequivalent to the requirement that d ( S ) = d ( S (cid:48) ) whenever S and S (cid:48) are shadedfaces of D incident at the same crossing. It follows that d is constant on eachconnected component of Γ s ( D ), and these constant values are arbitrary. Wededuce that ker v ∼ = A β s ( D ) . The theorem of Nanyes
In this section we discuss two versions of the theorem of Nanyes [20], one in-volving Dehn colorings and the other involving Fox colorings. The first ver-sion asserts that like ϕ : D A ( D ) → F A ( D ), the epimorphism v : D A ( D ) → ker A G ( D, s ) splits.
Theorem 14
Let D be a link diagram, and A an abelian group. Then D A ( D ) is the internal direct sum of ker v and a subgroup isomorphic to ker A G ( D, s ) . Proof.
Choose shaded faces S , . . . , S β s ( D ) of D , one in each connected com-ponent of the graph Γ s ( D ). Let u : ker A G ( D, s ) → D A ( D ) be the map de-fined by the construction in the proof of Proposition 11, always using one of S , . . . , S β s ( D ) when implementing the first sentence of the second paragraph,and always using 0 for the value of d ( S i ). The uniqueness of the d values calcu-lated in the other steps of the recursion guarantees that u is a homomorphism.As vu is the identity map of ker A G ( D, s ), the theorem follows.Theorem 14 implies that D A ( D ) ∼ = ker A G ( D, s ) ⊕ A β s ( D ) . The second version of the theorem of Nanyes [20] is the corresponding descrip-tion of F A ( D ) up to isomorphism. (N.b. If A is an abelian group then A denotes { } , the abelian group with only one element.) Theorem 15
Let D be a link diagram, and A an abelian group. Then F A ( D ) ∼ = ker A G ( D, s ) ⊕ A β s ( D ) − . Proof.
Choose a shaded face S of D , and let D = { d ∈ D A ( D ) | d ( S ) = 0 } .If we apply the construction in the proof of Theorem 4 with S always playingthe role of F and 0 always playing the role of α , we conclude that the restrictedmapping ( ϕ | D ) : D → F A ( D ) is surjective. As ker ϕ = { d α, − α | α ∈ A } ,ker( ϕ | D ) = { } . Hence F A ( D ) ∼ = D .If we apply the construction in the proof of Proposition 11 with S alwaysused in the first implementation of the first sentence of the second paragraphand 0 always used as the arbitrarily chosen value of d ( S ), then we concludethat the restricted mapping ( v | D ) : D → ker A G ( D, s ) is surjective. Thedescription of ker v at the end of the preceding section applies to ker( v | D ),with the exception that for d ∈ ker( v | D ) the value of d on the connectedcomponent of Γ s ( D ) containing S is not arbitrary; it is 0. We deduce thatker( v | D ) ∼ = A β s ( D ) − .Applying the proof of Theorem 14 to D , we conclude that D is the internaldirect sum of ker( v | D ) and a subgroup isomorphic to ker A G ( D, s ).14otice that checkerboard colorings are not mentioned in Definitions 1 and 2,but Theorems 14 and 15 tell us that D A ( D ) and F A ( D ) are determined upto isomorphism by the checkerboard graphs of D : the unshaded checkerboardgraph provides G ( D, s ), and the shaded checkerboard graph provides β s ( D ).Although the two checkerboard graphs play different roles, the theorems applyequally well if the shading s is reversed.We should explain that we refer to Theorems 14 and 15 as “versions” of thetheorem of Nanyes [20] because there are several differences between our setupand Nanyes’s: he required the image of a Fox coloring to generate A , he requireda Fox coloring to be 0 somewhere, and he removed a row and column from theGoeritz matrix. The first difference has the effect of shifting attention from thekernel to the cokernel of a homomorphism represented by G ( D, s ). The seconddifference has the effect of removing a direct summand isomorphic to A from F A ( D ), and the third difference has the effect of removing another such directsummand from ker A G ( D, s ). We leave further articulation of the details of therelationship between our discussion and that of [20] to the reader.We should also point out that Nanyes’s descriptions of β s ( D ) and G ( D, s )in [20] are inaccurate. He described G ( D, s ) as a matrix obtained using all thefaces of D , not just the unshaded faces; and he described β s ( D ) as the number ofconnected components of the graph of unshaded faces, not the graph of shadedfaces. It is easy to see that either of these mistakes can lead to an erroneousdescription of F A ( D ). For instance, let D be a crossing-free diagram of a µ -component unlink. Then 1 ≤ β s ( D ) ≤ µ and D has n = µ + 1 − β s ( D ) unshadedfaces. The corresponding Goeritz matrix has all entries 0, so if we mistakenlyreplace G ( D, s ) with a matrix that has a row and column for every face, wewill conclude that ker A G ( D, s ) ⊕ A β s ( D ) − is isomorphic to A n +2 β s ( D ) − = A µ + β s ( D ) . If we use the correct definition of G ( D, s ) then we obtain the 0 matrixof order n , so ker A G ( D, s ) ∼ = A n ; if we then mistakenly calculate β s ( D ) usingunshaded faces we will conclude that ker A G ( D, s ) ⊕ A β s ( D ) − is isomorphic to A n − . The correct calculation yields ker A G ( D, s ) ⊕ A β s ( D ) − ∼ = A n + β s ( D ) − = A µ . This is isomorphic to F A ( D ) because a Fox coloring of D is simply afunction that is constant on each component of L ( D ). Let T be the (2 ,
8) torus link diagram pictured on the left in Figure 7, and W the Whitehead link diagram pictured on the right. It is easy to see that L ( T )and L ( W ) are inequivalent links: the linking number of the two components of L ( T ) is ±
4, and the linking number of the two components of L ( W ) is 0.Nevertheless, Theorems 14 and 15 imply that every abelian group A has D A ( T ) ∼ = D A ( W ) and F A ( T ) ∼ = F A ( W ). To see why, notice that accordingto Definition 6 the Goeritz matrices of T and W associated with the shadings15 Figure 7: Diagrams of a (2 ,
8) torus link and a Whitehead link.of Figure 7 are G ( T, s ) = (cid:18) − − (cid:19) and G ( W, s ) = − − − .Consequently, if A is an abelian group then ker A G ( T, s ) is the set of orderedpairs ( x, y ) ∈ A such that 8 y − x = 0, and ker A G ( W, s ) is the set of orderedtriples ( a, b, c ) ∈ A such that − a + b + 2 c = 0 and a − b + 2 c = 0.We claim that for every abelian group A , the formula π ( a, b, c ) = ( a, c )defines an isomorphism π : ker A G ( W, s ) → ker A G ( T, s ). The claim is verifiedin four steps. First, notice that if ( a, b, c ) ∈ ker A G ( W, s ) then8 c − a = 3 · ( − a + b + 2 c ) + a − b + 2 c = 3 · π ( a, b, c ) = ( a, c ) is an element of ker A G ( T, s ). Second, notice that π isa homomorphism because it is a restriction of the projection homomorphism A → A defined by ( a, b, c ) (cid:55)→ ( a, c ). Third, notice that if ( x, y ) ∈ ker A G ( T, s )then ( x, x − y, y ) ∈ ker A G ( W, s ), because − x + (3 x − y ) + 2 y = 0 and x − · (3 x − y ) + 2 y = − x + 8 y = 0.As π ( x, x − y, y ) = ( x, y ), we deduce that π is surjective. Fourth, notice thatif ( a, b, c ) , ( a (cid:48) , b (cid:48) , c (cid:48) ) ∈ ker A G ( W, s ) and π ( a, b, c ) = π ( a (cid:48) , b (cid:48) , c (cid:48) ) then a = a (cid:48) and c = c (cid:48) , so b = 3 a − c = 3 a (cid:48) − c (cid:48) = b (cid:48) . We deduce that π is injective.If s denotes the shadings of T and W opposite to those indicated in Figure7, then β s ( T ) = β s ( W ) = β s ( T ) = β s ( W ) = 1. According to Theorem 15, itfollows that ker A G ( T, s ) ∼ = ker A G ( T, s ) and ker A G ( W, s ) ∼ = ker A G ( W, s ). Weleave it as an exercise for the reader to verify these isomorphisms directly.
Several authors have observed that when A = Z / Z , there is an especially simplerelationship between ker A G ( D, s ) and the link L ( D ) [3, 9, 15, 20, 24, 25]. In16act this simple relationship holds more generally when the exponent of A is 2 , i.e., 2 · α = 0 ∀ α ∈ A . We close with a summary of this simple relationship.Let D be a link diagram, with L ( D ) = K ∪ · · · ∪ K µ . For each F ∈ F ( D )there is a piecewise smooth path P F from a point in the interior of F to apoint in the interior of the unbounded face of D , which does not come near anycrossing of D and has only a finite number of intersections with the arcs of D ,all of which are transverse intersections. In general there are many such paths,with different patterns of intersections with D ; but every such path will havethe same number of intersections (mod 2) with the image of each K i . Definition 16 If F ∈ F ( D ) then for ≤ i ≤ µ the index of F with respect to K i is the parity (mod ) of the number of intersection points of a path P F withthe image of K i in the plane. We denote the index i D ( F, K i ) . Proposition 17
Suppose A is of exponent . Then f : A ( D ) → A is a Foxcoloring of D if and only if there is an element ( α , . . . , α µ ) ∈ A µ such that f ( a ) = α i whenever a is an arc of D that belongs to the image of K i . Proof. As A is of exponent 2, Definition 1 simply requires f ( a ) = f ( a ) inFigure 1. Applying this equation at every crossing, we conclude that F A ( D ) isthe set of maps A ( D ) → A that are constant on the image of every K i .That is, F A ( D ) may be identified naturally with A µ . It follows from Theorem5 that D A ( D ) may be identified naturally with A µ +1 . We spell out the details: Proposition 18
Suppose A is of exponent . Then d : F ( D ) → A is a Dehncoloring of D if and only if there is an element ( α , . . . , α µ ) ∈ A µ +1 such thatevery F ∈ F ( D ) has d ( F ) = α + µ (cid:88) i =1 α i · i D ( F, K i ) . Proof.
The formula in the statement is the appropriate version of formula (1)from the proof of Theorem 4.
Corollary 19
Suppose A is of exponent , and F , . . . , F n are the unshadedfaces of a shading s of D . Then an element v = ( v , . . . , v n ) ∈ A n is containedin ker A G ( D, s ) if and only if there is an element ( α , . . . , α µ ) ∈ A µ +1 such that v j = α + µ (cid:88) i =1 α i · i D ( F j , K i ) for every j ∈ { , . . . , n } . Proof.
According to Proposition 11, the elements of ker A G ( D, s ) are the el-ements of A n obtained by evaluating Dehn colorings on the unshaded faces of D . 17he same formula appears in Proposition 18 and Corollary 19, but the twomaps defined by the formula are quite different. In Proposition 18, the formuladefines an isomorphism A µ +1 → D A ( D ). In Corollary 19, instead, the formuladefines a split epimorphism A µ +1 → ker A G ( D, s ), whose kernel is isomorphicto A β s ( D ) . References [1] Colin C. Adams.
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