The shadow nature of positive and twisted quandle cocycle invariants of knots
aa r X i v : . [ m a t h . G T ] O c t The shadow nature of positive and twistedquandle cocycle invariants of knots
Seiichi Kamada [email protected]
Osaka City University Victoria Lebed [email protected]
OCAMI, Osaka City UniversityKokoro Tanaka [email protected]
Tokyo Gakugei UniversitySeptember 25, 2018
Abstract
Quandle cocycle invariants form a powerful and well developed tool in knot theory.This paper treats their variations — namely, positive and twisted quandle cocycleinvariants, and shadow invariants. We interpret the former as particular cases of thelatter. As an application, several constructions from the shadow world are extendedto the positive and twisted cases. Another application is a sharpening of twistedquandle cocycle invariants for multi-component links.
Keywords: quandles; multi-term distributive (co)homology; quandle cocycle invariants;positive quandle cocycle invariants; twisted quandle cocycle invariants; shadow colorings. A quandle is a set Q endowed with binary operations ⊳ and e ⊳ satisfying( a ⊳ b ) ⊳ c = ( a ⊳ c ) ⊳ ( b ⊳ c ) , (1)( a ⊳ b ) e ⊳ b = ( a e ⊳ b ) ⊳ b = a, (2) a ⊳ a = a. (3)A group with the conjugation operation a ⊳ b = b − ab is an important example.Quandles were introduced in [Joy82, Mat82] as an efficient tool for constructingknot invariants. Concretely, for a quandle Q and a knot diagram D , considerthe set C Q ( D ) of Q -colorings of D — that is, assignments C : Arcs ( D ) → Q ofelements of Q to every arc of D , satisfying the condition on Figure 1 A around Operation e ⊳ is deduced from ⊳ using (2); we thus often simply speak about a quandle ( Q, ⊳ ). In this paper by knots we always mean oriented knots and links in R . Axioms (1)-(3) force the number | C Q ( D ) | of such colorings to bestable under Reidemeister moves, and thus to give a knot invariant. a bb a ⊳ b A am m ⊳ a BFigure 1: Quandle and shadow coloringsThese quandle invariants are very powerful, but they do not detect, for instance,the chirality of knots. This flaw was fixed in [CJK +
03] by enriching colorings with weights . Concretely, given an abelian group A , a map ω : Q × Q → A satisfying ω ( a, b ) + ω ( a ⊳ b, c ) = ω ( a ⊳ c, b ⊳ c ) + ω ( a, c ) , (4) ω ( a, a ) = 0 (5)is called a quandle -cocycle of Q . (The word “cocycle” will be justified below.) The ω -weight of a Q -colored knot diagram ( D, C ) is defined as a sum W ω ( D, C ) = X a b ε ( ) ω ( a, b ) (6)over all crossings of D , where ε ( ) is the sign of (cf. Figure 2). Axioms (4)-(5)were chosen so that the the multiset of ω -weights { W ω ( D, C ) | C ∈ C Q ( D ) } yields aknot invariant, called a quandle cocycle invariant . ε ( ) a bb a ⊳ b + ω ( a, b ) b a ⊳ ba b − ω ( a, b )Figure 2: Quandle cocycle invariantsThis construction admits several variations. Instead of quandle 2-cocycles, theyuse some other cohomological data, which we now recall. Together with a quandle Q ,consider a Q -module — that is, a set M endowed with two maps ⊳ , e ⊳ : M × Q → M satisfying Axioms (1)-(2) for all a ∈ M , b, c ∈ Q . The simplest examples are Q itself,and a one-element set (called a trivial Q -module), with evident module operations.Take also an abelian group A . Denote by C k ( M, Q, A ) the abelian group of mapsfrom M × Q × k to A , and put ( d kl φ )( m, a , . . . , a k +1 ) = k +1 X i =1 ( − i − φ ( m ⊳ a i , . . . , a i − ⊳ a i , a i +1 , . . . , a k +1 ) , (7)( d kr φ )( m, a , . . . , a k +1 ) = k +1 X i =1 ( − i − φ ( m, a , . . . , a i − , a i +1 , . . . , a k +1 ) . (8) Here and afterwards diagrams with unoriented arcs mean that all coherently oriented versionsof these diagrams should be considered. Q -modules appeared under the name Q -(quandle-)sets ([FRS95]), and are also known as Q -shadows ([CN11]). The notation is slightly abusive but convenient. Subscripts l & r refer to an interpretation of these maps as left & right differentials, cf. [Leb13]. d kl and d kr turn out to be anti-commuting differentials. Hence for all α l , α r lying in Z (or, more generally, in a commutative ring R if A is an R -module),( C k ( M, Q, A ) , α l d kl − α r d kr ) is a cochain complex. Denote by C k Q ( M, Q, A ) the sub-group of maps vanishing on all tuples ( m, a , . . . , a k ) with a i = a i +1 for some i . Both d kl and d kr restrict to C k Q ( M, Q, A ), and thus so do their linear combinations. The cocycles / coboundaries / cohomology groups of ( C k Q ( M, Q, A ) , α l d kl − α r d kr ) receivethe adjective • quandle if α l = α r = 1 ([FRS95, CJK + • positive quandle if α l = − α r = 1 ([CG14]); • ( t -)twisted quandle if α l = 1, α r = t , and A is a Z [ t ± ]-module ([CES02]); • two-term distributive in the general case ([PS14, Prz11]).Note that a trivial M can be safely excluded from consideration (and from ournotations), since M × Q × k ∼ = Q × k ; we will often talk about ( M -)shadow cocycles /coboundaries / cohomology groups if M is non-trivial. Positive ([CG14]) and twisted ([CES02]) variations of quandle cocycle invari-ants using positive and, respectively, twisted quandle 2-cocycles with trivial M arerecalled in Section 2. Shadow invariants based on shadow quandle 2-cocycles arealso reviewed; together with arc colorings, these use region colorings by elementsof M , respecting the rule on Figure 1 B . In literature, mostly the case M = Q was considered formally and thoroughly ([RS00, CKS01, Kam02]); the general caseseems to be folklore. In this paper, carefully chosen Q -modules allow us to presenttwisted (Section 3) and positive (Section 5) quandle cocycle invariants as shadowinvariants. This instantly gives a proof of their invariance, avoiding technical verifi-cations. Moreover, all results established for shadow invariants can now be appliedin the positive and twisted cases. Such results, discussed in Section 4, include theequality of invariants obtained from cohomologous cocycles; a certain symmetry ofthe invariants; some restrictions on the values of weights one can encounter; higher-dimensional generalizations; and shadow versions. In Section 6, the shadow ideaslead to a sharpening of twisted quandle cocycle invariants for multi-component links. In this section we recall three variations of the quandle cocycle invariant construc-tion. From now on, fix a quandle ( Q, ⊳ ) and an abelian group A .First, together with the arcs of a knot diagram D , one can color its regions (= theconnected components of R \ D ) by elements of a Q -module M . Concretely, define a (shadow) ( Q, M ) -coloring of D as a Q -coloring C of the arcs of D and an assignment C sh : Regions ( D ) → M , compatible in the sense of Figure 1 B . The set of suchcolorings is denoted by C shQ,M ( D ). The color of the exterior region ∆ ex ( D ) of D uniquely determines all the other region colors, hence the ( Q, M )-coloring number | C shQ,M ( D ) | equals | C Q ( D ) | · | M | and says nothing new about the knot. The situation Note the “double-shadow” situation here. In our diagrams, region colors are underlined for a better readability. M -shadow quandle -cocycle of Q — that is, a map ω : M × Q × Q → A satisfying ω ( m, a, b ) + ω ( m ⊳ b, a ⊳ b, c ) + ω ( m, b, c ) = ω ( m ⊳ c, a ⊳ c, b ⊳ c ) + ω ( m, a, c ) + ω ( m ⊳ a, b, c ) , (9) ω ( m, a, a ) = 0 . (10) Definition 2.1.
The (shadow) ω -weight of a ( Q, M )-colored knot diagram ( D, C , C sh )is the sum W shω ( D, C , C sh ) = X a bm ω ( m, a, b ) − X a bm ω ( m, a, b ) (11)taken over all crossings of D . Theorem 1.
For any Q -module M , m ∈ M , and M -shadow quandle -cocycle ω of Q , the multiset { W shω ( D, C , C sh ) | ( C , C sh ) ∈ C shQ,M ( D ) , C sh (∆ ex ( D )) = m } depends only on the underlying knot, and not on the choice of its diagram D . The (extremely rich) knot invariant thus obtained are called shadow quandlecocycle invariants , or simply shadow invariants .Another variation uses a positive quandle -cocycle of Q — that is, a map ω : Q × Q → A satisfying (5) and ω ( a, c ) + ω ( a ⊳ b, c ) = ω ( a ⊳ c, b ⊳ c ) + ω ( a, b ) + 2 ω ( b, c ) . (12) Definition 2.2.
The (positive) ω -weight of a Q -colored knot diagram ( D, C ) is thesum W posω ( D, C ) = X a b ε pos ( ) ω ( a, b ) , (13)where the sign ε pos ( ) is defined on Figure 3 (here the checkerboard coloring of thediagram’s regions is used, with the exterior region declared white). ε pos ( ) + − Figure 3: Positive quandle cocycle invariants
Theorem 2 ([CG14]) . For any positive quandle -cocycle ω of Q , the multiset { W posω ( D, C ) | C ∈ C Q ( D ) } yields a knot invariant. The knot invariant obtained is referred to as positive quandle cocycle invariant .In order to present the third generalization of quandle cocycle invariants we areinterested in, some technical but intuitive definitions are necessary.4 efinition 2.3. • The index i (∆) of a region ∆ in a knot diagram D is thealgebraic intersection number for D and a path γ connecting ∆ to the exteriorregion ∆ ex ( D ). Sign conventions are indicated in Figure 4 A ; γ and D aresupposed to have a finite number of simple transverse intersections. • The source region ∆( ) of a crossing in D is defined in Figure 4 B . • The index of a crossing is set to be i ( ) = i (∆( )).The index of a region/crossing does not depend on the choice of path γ satisfyingthe description above, and is thus well defined.∆∆ ex ( D ) γ D + D + D − i (∆) = + + − am m + 1 CFigure 4: Regions and indicesUntil the end of Section 4, suppose A to be a module over a commutative ring R ,and fix an α ∈ R ∗ . Take an α -twisted quandle -cocycle of Q — that is, a map ω : Q × Q → A satisfying (5) and ω ( a ⊳ c, b ⊳ c ) − αω ( a, b ) − ω ( a ⊳ b, c ) + αω ( a, c ) + (1 − α ) ω ( b, c ) = 0 . (14) Definition 2.4.
The (twisted, or α -twisted) ω -weight of a Q -colored knot diagram( D, C ) is the sum W twω ( D, C , α ) = X a b ε ( ) α − i ( ) ω ( a, b ) (15)taken over all crossings of D . (Recall that ε ( ) is the usual sign of , cf. Figure 2.) Theorem 3 ([CES02]) . For any α ∈ R ∗ and any α -twisted quandle -cocycle ω of Q , the multiset { W twω ( D, C , α ) | C ∈ C Q ( D ) } yields a knot invariant. This knot invariant is called ( α -)twisted quandle cocycle invariant .To prove Theorems 2 and 3, one could check directly that Reidemeister movesand induced coloring changes preserve the positive/twisted ω -weight, as it was donein [CG14, CES02]. Here we prefer a more conceptual solution: in the followingsections, the multisets from these theorems are presented as shadow invariants. The first ingredient we need is the Q -module Z with m ⊳ a = m + 1. For this Q -module, the region coloring rule from Figure 1 B specializes to the one fromFigure 4 C . Comparing it with the definition of the index of a region, one proves The term
Alexander number is sometimes used instead ([Ale23, CS98, CKS00]). In our diagrams, a solid crossing stands for a crossing of arbitrary sign. In other words, it is the region from which all normals to the arcs of point, hence the term. emma 3.1. Consider a ( Q, Z )-coloring of a knot diagram D such that the exteriorregion receives color 0. Then the color of any region ∆ coincides with i (∆).The region coloring from the lemma will be denoted by C ind : Regions ( D ) → Z .Remark that it is compatible with any arc coloring.Next, we show how to transform a twisted quandle 2-cocycle into a (non-twisted)quandle 2-cocycle, at the cost of introducing our non-trivial Q -module Z . Lemma 3.2.
A map ω : Q × Q → A is an α -twisted quandle 2-cocycle if and onlyif the map ω α : Z × Q × Q −→ A, ( m, a, b ) α − m ω ( a, b ) (16)is a shadow quandle 2-cocycle. Proof.
Conditions (5) for ω and (10) for ω α clearly coincide. Further, condition (9)for ω α reads α − m ω ( a, b ) + α − ( m +1) ω ( a ⊳ b, c ) + α − m ω ( b, c ) = α − ( m +1) ω ( a ⊳ c, b ⊳ c ) + α − m ω ( a, c ) + α − ( m +1) ω ( b, c ) , which, due to the invertibility of α , is the same as (14).We further compare α -twisted ω -weights with shadow ω α -weights: Lemma 3.3.
For any Q -colored diagram ( D, C ) and any α -twisted quandle 2-cocycle ω , one has W twω ( D, C , α ) = W shω α ( D, C , C ind ) . Proof.
It follows from the definitions and lemmas above.Putting everything together, one gets
Theorem 4.
For a knot diagram D and an α -twisted quandle -cocycle ω , themultiset { W twω ( D, C , α ) | C ∈ C Q ( D ) } of α -twisted ω -weights of D coincides withthe multiset { W shω α ( D, C , C sh ) | ( C , C sh ) ∈ C shQ, Z ( D ) , C sh (∆ ex ( D )) = 0 } of its Z -shadow ω α -weights with the exterior region colored by .Proof. According to Lemma 3.3, the first multiset coincides with { W shω α ( D, C , C ind ) |C ∈ C Q ( D ) } . But this is precisely our second multiset, since Lemma 3.1 allows toreplace any region Z -coloring C sh satisfying C sh (∆ ex ( D )) = 0 with C ind .This theorem yields the announced shadow interpretation of twisted quandlecocycle invariants. In particular, combined with Theorem 1, it immediately impliesTheorem 3. 6 Applications
We now turn to applications of our shadow interpretation of twisted quandle cocycleinvariants. Some results on shadow invariants are first recalled. Most of them seemto be folklore; their proofs are included for completeness. Their consequences in thetwisted setting are then discussed.
Lemma 4.1.
The shadow weights associated to a shadow quandle 2-coboundaryvanish.The proof we present is well adapted for higher-dimensional generalizations.
Proof.
Take a Q -module M , a map θ : M × Q → A , and a ( Q, M )-colored knotdiagram ( D, C , C sh ). Choose a point p on the knot, and examine the value ν ( γ ) = θ ( C sh (∆( γ )) , C ( γ )) as you travel along the knot, following its direction; here γ isthe arc you are on, and ∆( γ ) is the region to your right. You pass twice throughany crossing , and the sum of the increments of ν ( γ ) during these two passages isexactly the contribution of to W shdθ ( D, C ) (for a positive , it is shown on Figure 5;for a negative one things are similar). Returning to p , you recover the original value ν ( γ ). Considering such walks along each link component, one concludes that thetotal increment of ν ( γ ) — which is precisely W shdθ ( D, C ) — is zero. a bb a ⊳ b m m ⊳ am ⊳ b ( m ⊳ a ) ⊳ b ( θ ( m ⊳ b, a ⊳ b ) − θ ( m, a ))+ ( θ ( m, b ) − θ ( m ⊳ a, b ))= dθ ( m, a, b )Figure 5: The total cost of passing through a crossingAs a consequence, one obtains Proposition 4.2.
Cohomologous twisted quandle -cocycles yield the same weightsand identical twisted quandle cocycle invariants. In particular, one can talk about the weight W tw [ ω ] associated to the twisted quan-dle 2-cohomology class of ω . Proof.
To simplify notations, put d = d l − d r , d α = d l − αd r . By the linearity ofweights, it suffices to check the triviality of twisted d α θ -weights, with θ : Q → A . Aneasy direct verification gives ( d α θ ) α = αd ( θ α ) (we use notation (16) and its analogue θ α ( m, a ) = α − m θ ( a )). Lemma 3.3 then implies that W twd α θ ( D, C ) = W sh ( d α θ ) α ( D, C , C ind ) = W shαdθ α ( D, C , C ind ) = α W shdθ α ( D, C , C ind ) , which vanishes according to Lemma 4.1.As a second application, we explore an action of a quandle Q on ( Q, M )-coloringsand its effect on shadow and twisted quandle cocycle invariants.7 emma 4.3.
The set of (
Q, M )-colorings of a knot diagram D has the following Q -module structure: the coloring ( C , C sh ) ⊳ c is obtained from ( C , C sh ) by replacingeach employed color x with x ⊳ c . Moreover, for any M -shadow quandle 2-cocycle ω ,the ω -weight respects this Q -module structure — that is, for all c ∈ Q , one has W shω ( D, C , C sh ) = W shω ( D, ( C , C sh ) ⊳ c ) . (17) Proof.
We first check that ( C , C sh ) ⊳ c is indeed a ( Q, M )-coloring. For this, oneshould show that the coloring rules from Figure 1 remain valid when each color x isreplaced with x ⊳ c ; this follows from Axiom (1) for Q and for M . This axiom alsoimplies that one actually gets a Q -module structure.To show relation (17), rewrite the defining property (9) for ω as ω ( m, a, b ) − ω ( m ⊳ c, a ⊳ c, b ⊳ c ) =( ω ( m ⊳ a, b, c ) − ω ( m ⊳ b, a ⊳ b, c )) − ( ω ( m, b, c ) − ω ( m, a, c )) . The second line can be interpreted as dω c ( m, a, b ), where ω c ∈ C ( M, Q, A ) is definedby ω c ( m, a ) = ω ( m, a, c ), and d = d l − d r . Consequently, one has W shω ( D, C , C sh ) − W shω ( D, ( C , C sh ) ⊳ c ) = W shdω c ( D, C , C sh ) , which vanishes according to Lemma 4.1.This lemma leads to a better understanding of the dependence of shadow invari-ants from Theorem 1 on the choice of the exterior region color m ∈ M . We will usethe notion of orbits of a Q -module M , which are classes for the equivalence relationon M induced by m ∼ m ⊳ a , a ∈ Q . Proposition 4.4.
For any Q -module M , elements m and m lying in the sameorbit of M , and M -shadow quandle -cocycle ω of Q , the multisets { W shω ( D, C , C sh ) | ( C , C sh ) ∈ C shQ,M ( D ) , C sh (∆ ex ( D )) = m } and { W shω ( D, C , C sh ) | ( C , C sh ) ∈ C shQ,M ( D ) , C sh (∆ ex ( D )) = m } defining shadow invariants coincide.Proof. It suffices to check the assertion for m = m ⊳ c . In this case, the map( C , C sh ) ( C , C sh ) ⊳ c establishes a bijection between the ( Q, M )-colorings used fordefining the two multisets. Lemma 4.3 tells that the ω -weights of the correspondingcolorings are the same, so the multisets coincide.Returning to the twisted setting, one obtains several useful consequences. Proposition 4.5. An α -twisted quandle cocycle invariant does not change if all itselements are scaled by α . Orbits can also be thought of as the irreducible components of M . Every Q -module uniquelydecomposes as a disjoint union of irreducible Q -modules. roof. The invariants in question are multisets { W twω ( D, C , α ) | C ∈ C Q ( D ) } for α -twisted quandle 2-cocycles ω of Q . Theorem 4 identifies such a multiset with { W shω α ( D, C , C sh ) | ( C , C sh ) ∈ C shQ, Z ( D ) , C sh (∆ ex ( D )) = 0 } . According to Proposition 4.4, this multiset does not change if the color C sh (∆ ex ( D ))of the exterior region of D is imposed to be − Q -module Z consists of a single orbit. With this new condition, all region colors in C sh shouldbe reduced by 1. Recalling the definition (16) of the shadow quandle 2-cocycle ω α ,one sees that this corresponds to multiplying all the shadow ω α -weights by α . Thussuch a scaling does not change our invariant.This proposition can be used to show that certain twisted quandle cocycle in-variants are no stronger than usual quandle invariants. For instance, for a finitequandle Q and A = Z [ t ± ], t -twisted quandle cocycle invariants contain only zeros,since a non-zero weight would imply, by scaling by t , an infinity of pairwise distinctweights, whereas the total number of Q -colorings is finite. Proposition 4.6.
Suppose that for c , . . . , c k ∈ Q and ε , . . . , ε k ∈ {± } , relation ( · · · ( a ⊳ ε c ) ⊳ ε · · · ) ⊳ ε k c k = a (18) holds for all a ∈ Q ; we used notations ⊳ +1 = ⊳ , ⊳ − = e ⊳ . Then the multiplication by α P ki =1 ε i − annihilates all α -twisted weights.Proof. As usual, Lemma 3.3 allows us to work with shadow weights instead of α -twisted weights. A repeated application of Lemma 4.3 gives W shω α ( D, C , C ind ) = W shω α ( D, ( · · · (( C , C ind ) ⊳ ε c ) ⊳ ε · · · ) ⊳ ε k c k ) . Relation (18) implies ( · · · (( C , C ind ) ⊳ ε c ) ⊳ ε · · · ) ⊳ ε k c k = ( C , C ind + P ki =1 ε i ),where in the latter region coloring each color is increased by P ki =1 ε i . Repeating theargument from the proof of Proposition 4.5, one obtains W shω α ( D, C , C ind + P ki =1 ε i ) = α − P ki =1 ε i W shω α ( D, C , C ind ) . Now, combine everything to get W twω ( D, C , α ) = α − P ki =1 ε i W twω ( D, C , α ).For example, if Q contains a “central” element c satisfying a ⊳ c = a for all a ∈ Q , then scaling by α − α -twisted weights.The third application is a shadow enhancement of twisted quandle cocycle in-variants. To treat simultaneously the Q -module M we want to color regions withand the Q -module Z (with m ⊳ a = m + 1) used in our shadow interpretation oftwisted invariants, we use the following elementary observation: Lemma 4.7.
The direct product M × M ′ of Q -modules M and M ′ is also a Q -module, with the diagonal action ( m, m ′ ) ⊳ a = ( m ⊳ a, m ′ ⊳ a ).Now, repeating verbatim all the arguments from the previous and the beginningof the current sections for the Q -module M × Z instead of Z , one gets9 heorem 5. Take a Q -module M , a fixed m ∈ M , and an α -twisted M -shadowquandle -cocycle ω . The multiset of α -twisted M -shadow ω -weights { W tw,shω ( D, C , C sh , α ) | ( C , C sh ) ∈ C shQ,M ( D ) , C sh (∆ ex ( D )) = m } , where W tw,shω ( D, C , C sh , α ) = X a b ε ( ) α − i ( ) ω ( C sh (∆( )) , a, b ) , is a knot invariant . Moreover, cohomologous -cocycles yield identical invariants. The last bonus from our shadow interpretation of twisted quandle cocycle in-variants is their higher-dimensional generalization. Indeed, take a Q -colored k − D, C ) in R k , and an α -twisted quandle k -cocycle ω . De-fine the α -twisted ω -weight of ( D, C ) by a formula analogous to (15), using higher-dimensional versions of indices and signs for crossings of multiplicity k , and a rele-vant ordering of adjacent sheets. Lemmas 3.1-3.3 generalize directly. The multisetof α -twisted ω -weights is thus interpreted in terms of higher-dimensional shadow in-variants (for more details about the latter, see for instance [FR92, CKS04, PR13]).Moreover, this construction admits a shadow version, similar to that from Theo-rem 5. Hence any α -twisted (possibly shadow) quandle k -cocycle gives rise to a k − This section is devoted to the case α = −
1. For this α , the twisted quandle coho-mology theory is precisely the positive quandle cohomology theory from [CG14]. Inorder to show that positive and − Lemma 5.1.
For any Q -colored diagram ( D, C ) and any positive quandle 2-cocycle ω ,one has W twω ( D, C , −
1) = W posω ( D, C ). Proof.
Is is sufficient to show that for any crossing , the signs ε ( )( − − i ( ) and ε pos ( ) coincide. First, observe that, modulo 2, the coloring C ind reduces to thecheckerboard coloring, under the identification 0 = white , 1 = black . Using thisidentification, the definition of ε pos ( ) (Figure 3) can be restated as on Figure 6(note that we turned both diagrams so that the over-arc points south-west). Now,if the under-arc points south-east, then ε ( ) is 1, and i ( ) is the color of the regionto the west of ; one checks that the latter satisfies ( − − i ( ) = ε pos ( ), as desired.A change in the orientation of the under-arc changes ε ( ) and i ( ), but not ε pos ( ),so equality ε ( )( − − i ( ) = ε pos ( ) is preserved.0 011 + − Figure 6: Sign ε pos via region indices For notations, refer to Definition 2.3. − Theorem 6.
For a knot diagram D and a positive quandle -cocycle ω , the multiset { W posω ( D, C ) | C ∈ C Q ( D ) } of positive ω -weights of D coincides with the multiset { W shω − ( D, C , C sh ) | ( C , C sh ) ∈ C shQ, Z ( D ) , C sh (∆ ex ( D )) = 0 } of its Z -shadow ω − -weights with the exterior region colored by . Note that we replaced our Q -module Z with Z , with the same module operation m ⊳ a = m + 1; this is possible since ( − m is completely determined by m mod 2.The theorem interprets positive quandle cocycle invariants as shadow ones. Thisdirectly shows their invariance, and gives for free their shadow enhancement, answer-ing a question raised in [CG14]. Moreover, this implies analogs of Propositions 4.5and 4.6 in the positive setting: Proposition 5.2.
A positive quandle cocycle invariant is symmetric with respectto (that is, it does not change if all its elements change signs). Proposition 5.3.
Suppose that for some odd k and some c , . . . , c k ∈ Q and ε , . . . , ε k ∈ {± } , relation (18) holds for all a ∈ Q . Then all non-trivial positiveweights are of order in the abelian group A . For links with several components, twisted quandle cocycle invariants can be sharp-ened using a method developed in this section. In what follows, the diagram D represents a link with k ordered components, numbered 1 , . . . , k .First, we need a more subtle notion of index. Definition 6.1. • The j th index i j (∆) of a region ∆ in D is the index of ∆with respect to the j th component of D , in the sense of Definition 2.3. • The j th index of a crossing is set to be i j ( ) = i j (∆( )).For a quandle ( Q, ⊳ ), consider the equivalence relation induced by a ∼ a ⊳ b .Corresponding equivalence classes are called orbits , their set is denoted by Orb ( Q ),and the orbit of an a ∈ Q is denoted by O ( a ). A Q -coloring C of D assigns elementsof the same orbit to all the arcs belonging to the same component of D ; for the j thcomponent, this orbit is denoted by C ∗ ( j ).As before, let A be a module over a commutative ring R . Take a collection α = ( α O ∈ R ∗ ) O∈ Orb ( Q ) . Definition 6.2. • An α -twisted quandle -cocycle is a map ω : Q × Q → A satisfying (5) and α − O ( c ) ω ( a ⊳ c, b ⊳ c ) − ω ( a, b ) − α − O ( b ) ω ( a ⊳ b, c ) + ω ( a, c )+ ( α − O ( a ) − ω ( b, c ) = 0 . (19)11 The ( α -twisted) ω -weight of a Q -colored link diagram ( D, C ) is the sum W twω ( D, C , α ) = X a b ε ( ) α − i ( ) C ∗ (1) · · · α − i k ( ) C ∗ ( k ) ω ( a, b ) . (20) Theorem 7.
The multiset { W twω ( D, C , α ) | C ∈ C Q ( D ) } depends only on the un-derlying link (and not on the choice of its diagram D ), and only on the α -twistedquandle cohomology class [ ω ] of ω . The theorem is proved by interpreting the constructed multisets as shadow in-variants. The Q -module one should use here is the following one: Lemma 6.3.
The set M O∈ Orb ( Q ) Z e O with m ⊳ a = m + e O ( a ) is a Q -module. Proof.
One has to check that e O ( b ) + e O ( c ) = e O ( c ) + e O ( b ⊳ c ) , which is obvious since,by definition, b and b ⊳ c belong to the same orbit.As usual, our shadow interpretation gives for free shadow and higher-dimensionalgeneralizations of the theorem. Remark . Following [Ino13], the multiset from the theorem can be decomposedinto multisets { W twω ( D, C , α ) | C ∈ C Q ( D ) , C ∗ (1) = O , . . . , C ∗ ( k ) = O k } , indexed by k -tuples of orbits O , . . . , O k ∈ Orb ( Q ). The invariance of these smaller multisetsfollows from the fact that Reidemeister moves and induced local coloring changespreserve the orbit of the colors assigned to the arcs of a given link component. Acknowledgements.
The authors are grateful to the organizers of
Knots andLow Dimensional Manifolds (a satellite conference of Seoul ICM 2014), where theidea of this paper was born. They would like to thank Zhiyun Cheng for interest-ing discussions. This work was made possible thanks to JSPS KAKENHI Grants25 · References [Ale23] J.W. Alexander. A lemma on systems of knotted curves.
Proc. Nat. Acad.Science USA , 9:93–95, 1923.[CES02] J.S. Carter, M. Elhamdadi, and M. Saito. Twisted quandle homologytheory and cocycle knot invariants.
Algebr. Geom. Topol. , 2:95–135 (elec-tronic), 2002.[CG14] Z. Cheng and H. Gao. Positive quandle homology and its applications inknot theory.
ArXiv e-prints , March 2014.[CJK +
03] J.S. Carter, D. Jelsovsky, S. Kamada, L. Langford, and M. Saito. Quan-dle cohomology and state-sum invariants of knotted curves and surfaces.
Trans. Amer. Math. Soc. , 355(10):3947–3989, 2003.12CKS00] J.S. Carter, S. Kamada, and M. Saito. Alexander numbering of knottedsurface diagrams.
Proc. Amer. Math. Soc. , 128(12):3761–3771, 2000.[CKS01] J.S. Carter, S. Kamada, and M. Saito. Geometric interpretations of quan-dle homology.
J. Knot Theory Ramifications , 10:345–386, 2001.[CKS04] J. S. Carter, S. Kamada, and M. Saito.
Surfaces in 4-space , volume 142 of
Encyclopaedia of Mathematical Sciences . Springer-Verlag, Berlin, 2004.Low-Dimensional Topology, III.[CN11] W. Chang and S. Nelson. Rack shadows and their invariants.
J. KnotTheory Ramifications , 20(9):1259–1269, 2011.[CS98] J. S. Carter and M. Saito.
Knotted surfaces and their diagrams , volume 55of
Mathematical Surveys and Monographs . American Mathematical So-ciety, Providence, RI, 1998.[FR92] R. Fenn and C. Rourke. Racks and links in codimension two.
J. KnotTheory Ramifications , 1(4):343–406, 1992.[FRS95] R. Fenn, C. Rourke, and B. Sanderson. Trunks and classifying spaces.
Appl. Categ. Structures , 3(4):321–356, 1995.[Ino13] A. Inoue. Quasi-triviality of quandles for link-homotopy.
J. Knot TheoryRamifications , 22(6):1350026, 10, 2013.[Joy82] D. Joyce. A classifying invariant of knots, the knot quandle.
J. PureAppl. Algebra , 23(1):37–65, 1982.[Kam02] S. Kamada. Knot invariants derived from quandles and racks. In
Invari-ants of knots and 3-manifolds (Kyoto, 2001) , volume 4 of
Geom. Topol.Monogr. , pages 103–117 (electronic). Geom. Topol. Publ., Coventry, 2002.[Leb13] V. Lebed. Homologies of algebraic structures via braidings and quantumshuffles.
J. Algebra , 391:152–192, 2013.[Mat82] S.V. Matveev. Distributive groupoids in knot theory.
Mat. Sb. (N.S.) ,119(161)(1):78–88, 160, 1982.[PR13] J.H. Przytycki and W. Rosicki. Cocycle invariants of codimension 2-embeddings of manifolds.
ArXiv e-prints , October 2013.[Prz11] J.H. Przytycki. Distributivity versus associativity in the homology theoryof algebraic structures.
Demonstratio Math. , 44(4):823–869, 2011.[PS14] J.H. Przytycki and A.S. Sikora. Distributive products and their homology.
Comm. Algebra , 42(3):1258–1269, 2014.[RS00] C. Rourke and B. Sanderson. A new classification of links and somecalculations using it.