Cocycle invariants of codimension 2-embeddings of manifolds
aa r X i v : . [ m a t h . G T ] O c t COCYCLE INVARIANTS OF CODIMENSION2-EMBEDDINGS OF MANIFOLDS
J ´OZEF H. PRZYTYCKI, WITOLD ROSICKI
December 27, 2006 – October 11, 2013ABSTRACT. We consider the classical problem of a po-sition of n -dimensional manifold M n in R n +2 .We show that we can define the fundamental ( n +1)-cycleand the shadow fundamental ( n + 2)-cycle for a funda-mental quandle of a knotting M n → R n +2 .In particular, we show that for any fixed quandle, quan-dle coloring, and shadow quandle coloring, of a diagramof M n embedded in R n +2 we have ( n + 1)- and ( n +2)-(co)cycle invariants (i.e. invariant under Rosemanmoves). Contents
1. Introduction 21.1. Quandles and quandle homology 31.2. Presimplicial module and a weak simplicial module 71.3. Codimension 2 embedding, lower decker set 91.4. Quandle colorings, and quandle shadow coloring 91.5. ( n + 1) and ( n + 2)-chains of diagrams of knotting 142. ( n + 1) and ( n + 2) cycles for n -knotting diagrams 172.1. Shadow 3-cycle for n = 1 182.2. The general case of ( n + 1) and ( n + 2) cycles 203. Topological invariance of colorings 233.1. The fundamental rack and quandle of an n -knotting 233.2. Abstract definitions of a fundamental rack and quandle 254. Roseman moves are preserving homology classes offundamental cycles 284.1. Colorings and homology under Roseman moves 294.2. Cycle invariants of knottings 324.3. Cocycle invariants of knottings 33
5. Twisted (co)cycle invariants of knottings 376. General position and Roseman moves in codimension 2 396.1. General position 406.2. Arranging for moves 416.3. Listing of moves after Roseman 417. A knotting M n f → F n +1 × [0 , π → F n +1 Introduction
We consider the classical problem of a position of n -dimensional man-ifold M n , in an ( n + 2)-manifold W n +2 . The classical case deals with W n +2 = R n +2 , however our method is also well suited for a more gen-eral case of W n +2 being a product of an oriented ( n + 1)-manifold F n +1 and the interval or the twisted interval bundle over an unorientable( n + 1)-manifold F n +1 , as we have, in these cases, the natural projec-tion of W n +2 onto F n +1 (see Section 7).Historically, the main tool to study an n -knotting, f : M n → R n +2 ,was the fundamental group of the knotting complement in R n +2 . Thiswas greatly extended by applying quandle colorings and (co)cycle in-variants. We follow, to some extent, the exposition by Carter, Kamada,and Saito in [CKS-3], generalizing on the way the case of surfaces in R to general n -knottings. The paper is organized as follows: at thebeginning of the first section we recall the definition of a rack andquandle and their (co)homology. Then we give a short introduction todiagrams, D M , of knottings, and rack and quandle colorings of D M .Furthermore, we analyze shadow rack and quandle colorings. In thesecond part of the first section we define ( n + 1)- and ( n + 2)-chainsassociated to rack and quandle colorings. In the second section weprove that our chains are, in fact, cycles. In the third section we showthat the set of colorings by a given quandle is a topological invariant.The first step in this direction is given by comparing two definitions ofthe fundamental rack and quandle of a knotting (one from the diagramand one abstract). In the fourth section we show that the homologyclasses represented by cycles constructed for diagrams of knottings aretopological invariants. Here we carefully consider Roseman’s pass move ocycle invariants of codimension 2 embeddings (generalized third Reidemeister move). We offer various versions of cy-cle invariants of knottings in particular taking into account the factthat a quandle acts on the space of quandle colorings. We completeSection 4 by expressing invariants in the language of cohomology (co-cycle invariants). In the fifth section we generalize previous results totwisted homology and cohomology. In the sixth section we discuss indetail general position projection of n -knotting and Roseman moves;this is a service section to the previous considerations.Finally, in Section 7 we give a short overview of a knotting in F n +1 ¯ × [0 , Quandles and quandle homology.
We give here a short his-torical introduction to distributive structures and to homology basedon distributivity. The word distributivity was coined in 1814 by Fran-cois Servois. C.S. Peirce in 1880 emphasized the importance of (right)self-distributivity in algebraic structures [Peir]. The first explicit ex-ample of a non-associative self-distributive system was given by ErnstSchr¨oder in 1887 [Schr, Deh]. The detailed study of distributive struc-tures started with the 1929 paper by C. Burstin and W. Mayer [B-M].The first book partially devoted to distributivity is by Anton Sushke-vich, 1937 [Sus]. Definition 1.1.
Let ( X ; ∗ ) be a magma, that is a set with binary op-eration, then: (i) If ∗ is right self-distributive, that is, ( a ∗ b ) ∗ c = ( a ∗ c ) ∗ ( b ∗ c ) ,then ( X ; ∗ ) is called a RDS or a shelf (the term coined by AlissaCrans in her PhD thesis [Cra] ). (ii) If a shelf ( X ; ∗ ) satisfies the idempotent condition, a ∗ a = a for any a ∈ X , then it is called a right spindle, or just a spindle(again the term coined by Crans). Walter Mayer is well known for Mayer-Vietoris sequence and for being assistantto Einstein at Institute for Advanced Study, Princeton. The term coined in 1870 by Benjamin Peirce [Pei], the father of Charles SandersPeirce.
J´ozef H. Przytycki and Witold Rosicki (iii)
If a shelf ( X ; ∗ ) has ∗ invertible, that is the map ∗ b : X → X given by ∗ b ( a ) = a ∗ b is a bijection for any b ∈ X , then it iscalled a rack . (iv) If a rack ( X ; ∗ ) satisfies the idempotent condition, then it iscalled a quandle (the term coined in Joyce’s PhD thesis of 1979 [Joy-1] ). (v) If a quandle ( X ; ∗ ) satisfies ( a ∗ b ) ∗ b = a then it is called keior an involutive quandle. The term kei ( ) was coined in apioneering paper by M.Takasaki in 1942 [Tak] Three axioms of a quandle arise (see [Joy-2, Matv]) as an algebraicreflection of three Reidemeister moves on link diagrams. Idempotentcondition corresponds to the first move, invertibility to the second,and right self-distributivity to the third move. In Figure 1.2 we illus-trate how right self-distributivity is arising from the third Reidemeistermove, R when we color (label) arcs of the diagram by elements of X according to the following rule (Figure 1.1): * a bab b . Figure 1.1; magma coloring of a crossing If X is a set then the set Bin ( X ) of all binary operations on X forms a monoidwith composition ∗ ∗ given by a ∗ ∗ b = ( a ∗ b ) ∗ b and the identity element ∗ given by a ∗ b = a . Then the condition (ii) is equivalent to invertibility of ∗ in Bin ( X ). If ∗ is invertible, we write ¯ ∗ for ∗ − . The term wrack, like in “wrack and ruin”, of J.H.Conway from 1959, was modi-fied to rack in [F-R]). The main example considered in 1959 by Conway and Wraithwas a group G with a ∗ operation given by conjugation, that is, a ∗ b = b − ab [C-W]. Mituhisa Takasaki worked at Harbin Technical University in 1940, likely asan assistant to Kˆoshichi Toyoda. Both perished when Red army entered Harbinin 1945. Takasaki was considering keis associated to abelian groups, that is theTakasaki kei (or quandle) of an abelian group H , denoted by T ( H ) satisfies a ∗ b =2 b − a . ocycle invariants of codimension 2 embeddings *(a c)abc ca b (a b)b c ***abc c*b c R 3 *(a c) * (b c)** c Figure 1.2; Distributivity from R (Co)homology of racks was introduced by Fenn, Rourke and Sander-son between 1990 and 1995, [FRS-1, Fenn]. Quandle (co)homology wasconstructed by Carter, Kamada, and Saito (compare [CKS-3]). Theirmotivation was to associate to any link diagram an its quandle color-ing elements (cocycles) of quandle cohomology. In [CKS-3] it is donein details for knotting f : M n → R n +2 for n = 1 or 2, and we start ourpaper from the definition for any n (essentially following [CKS-3]).We give here the definition of rack, degenerate and quandle (co)homologyafter [CKS-3]. Definition 1.2. (i)
For a given rack X let C Rn ( X ) be the free abeliangroup generated by n -tuples ( x , x , ..., x n ) of elements of a rack X ; in other words C Rn ( X ) = Z X n = ( Z X ) ⊗ n . Define a bound-ary homomorphism ∂ : C Rn ( X ) → C Rn − ( X ) by: ∂ ( x , x , ..., x n ) = n X i =2 ( − i (( x , ..., x i − , x i +1 , ..., x n ) − ( x ∗ x i , x ∗ x i , ..., x i − ∗ x i , x i +1 , ..., x n )) . ( C R ∗ ( X ) , ∂ ) is called a rack chain complex of X . (ii) Assume that X is a quandle, then we have a subchain complex C Dn ( X ) ⊂ C Rn ( X ) generated by n -tuples ( x , ..., x n ) with x i +1 = x i for some i . The subchain complex ( C Dn ( X ) , ∂ ) is called adegenerated chain complex of a quandle X . (iii) The quotient chain complex C Qn ( X ) = C Rn ( X ) /C Dn ( X ) is calledthe quandle chain complex. We have the short exact sequenceof chain complexes: → C Dn ( X ) → C Rn ( X ) → C Qn ( X ) → . J´ozef H. Przytycki and Witold Rosicki (iv)
The homology of rack, degenerate and quandle chain complexesare called rack, degenerate and quandle homology, respectively.We have the long exact sequence of homology of quandles: ... → H Dn ( X ) → H Rn ( X ) → H Qn ( X ) → H Dn − ( X ) → ... R. Litherland and S. Nelson [L-N] proved that the short exact se-quence from (iii) splits respecting the chain maps. α : C Qn ( X ) → C Rn ( X ) is given, in the notation introduced in [N-P], by: α ( x , x , x , ..., x n ) = ( x , x − x , x − x , ..., x n − x n − ) . (recall that in our notation ( x , x − x , x − x , ..., x n − x n − ) = x ⊗ ( x − x ) ⊗ ( x − x ) ⊗ · · · ⊗ ( x n − x n − ) ∈ C Rn ( X )). In particular, α isa chain complex monomorphism and H Rn ( X ) = H Dn ( X ) ⊕ α ∗ ( H Qn ( X )).In a recent paper [P-P-2] it is demonstrated that degenerate homol-ogy of a quandle can be reconstructed from the quandle (normalized)homology of the quandle by a version of a K¨unneth formula.We define cohomology in a standard way (we follow [CKS-3]): Definition 1.3.
For an abelian group A define the cochain complexes C ∗ W ( X, A ) =
Hom ( C W ∗ , A ) . Here, W = D, R, Q so we describe allcases (degenerate, rack and quandle) . We define ∂ n : C n → C n +1 inthe usual way, that is for c ∈ C nW ( X ; A ) we have: ∂ n ( c )(( x , ..., x n , x n +1 ) = c ( ∂ n (( x , ..., x n , x n +1 )) . Cohomology groups are defines as usual as H nW ( X, A ) = ker∂ n /im ( ∂ n − ) . Another useful definition is that of right X -set, in particular theset of colorings of knotting diagram by a quandle X will be a right X -quandle-set. Definition 1.4.
Let E be a set, ( X ; ∗ ) a magma and ∗ : E × X → E an action of X on E (we can use the same symbol ∗ for operation in X and the action as it unlikely leads to confusion). Then (i) If ( X ; ∗ ) is a shelf and ( e ∗ x ) ∗ x = ( e ∗ x ) ∗ ( x ∗ x ) then E is a right X -shelf-set. (ii) If ( X ; ∗ ) is a rack and E a right X -shelf-set and additionallythe map ∗ b : E → E given by ∗ b ( e ) = e ∗ b is invertible then wesay that E is a right X -rack-set. In the case X is a quandle wewill say that E is a right X -quandle-set. The basic example of a quandle (resp. rack or shelf) right X -quandle- (resp. rack-, shelf-)-set is E = X n with ( x , ..., x n ) ∗ x =( x ∗ x, ..., x n ∗ x ). One should also add that for a given X -rack-set E one can define homology (as before) by assuming C n ( X, E ) = E × X n , ocycle invariants of codimension 2 embeddings and d ( ∗ ) i ( e, x , ..., x n ) = ( e ∗ x i , x ∗ x i , ..., x i − ∗ x i , x i +1 , ..., x n ). If E has one point, so the action is trivial, we reach exactly homology ofDefinition 1.2(i). Observation 1.5.
The chain groups C n ( X ) = ZX n of a rack chaincomplex are X -rack-sets. Furthermore, the action ∗ x : C n ( X ) → C n ( X ) , given by ∗ x ( x , ..., x n ) = ( x , ..., x n ) ∗ x = ( x ∗ x, ..., x n ∗ x ) ,is a chain map for any x , inducing the identity on homology. It iswell know but important fact (see e.g. [CJKS, N-P] ) and we use itin Theorems 4.4, 4.8, and 5.5. To prove this fact we use chain ho-motopy ( − n +1 h x : C n → C n +1 , where h x ( x , ..., x n ) = ( x , ..., x n , x ) .We check directly that ∂ n +1 ( − n +1 h x + ( − n h x ∂ n = Id − ∗ x . If weconsider ∂ T = t∂ ( ∗ ) − ∂ (( ∗ ) , as is the case in twisted rack or quandlehomology, we obtain that ∗ x induces t · Id on homology. We use it inDefinition 5.3. We need yet another observation that if X is a quandle and E is an X -quandle-set then X ∪ E has also a natural quandle structure. Observation 1.6.
Let ( X ; ∗ ) be a shelf and E an X -shelf-set (with aright action of X on E also denoted by ∗ , then X ⊔ E is also a shelfwith ∗ operation a ∗ y = a for any a ∈ X ⊔ E , and y ∈ E . Furthermore,if ( X ; ∗ ) is a rack (resp. spindle, or quandle) then ( X ⊔ E ; ∗ ) is also arack (resp. spindle, or quandle).Then we observe that a chain complex C n ( X, E ) is a subchain com-plex of C n ( X ⊔ E ) . Presimplicial module and a weak simplicial module.
Wefollow here [Lod, Prz-1] and introduce here the notion of a presimplicialand weak simplicial module. This will simplify our calculation andprovide a language for visualization.
Definition 1.7.
A weak simplicial module ( M n , d i , s i ) is a collectionof R -modules M n , n ≥ , together with face maps, d i : M n → M n − and degenerate maps s i : M n → M n +1 , ≤ i ≤ n , which satisfy thefollowing properties: (1) d i d j = d j − d i f or i < j. (2) s i s j = s j +1 s i , ≤ i ≤ j ≤ n, (3) d i s j = (cid:26) s j − d i if i < js j d i − if i > j + 1(4 ′ ) d i s i = d i +1 s i . ( M n , d i ) satisfying (1) is called a presimplicial module and leads to thechain complex ( M n , ∂ n ) with ∂ n = P ni =0 ( − i d i . J´ozef H. Przytycki and Witold Rosicki
If (4’) is replaced by a stronger condition(4) d i s i = d i +1 s i = Id M n then ( M n , d i , s i ) is a (classical) simplicialmodule.The following basic lemma will be used later: Lemma 1.8.
Let ( M n , d i ) be a presimplicial module then the map d d : C n → C n − is a chain map, chain homotopic to zero. In particular, if d d = 0 then ( − n d is a chain map.Proof. We think of d : C n → C n − as a chain homotopy and we have: d ∂ n + ∂ n − d = d n X i =0 ( − i d i + n − X i =0 ( − i d i d = d d + n X i =1 ( − i ( d d i − d i − d ) (1) = d d . In particular, if d d = 0 we have ( − n d ∂ n = ( − n − ∂ n − d . (cid:3) For us it is important that rack and quandle homology can be de-scribed in the language of weak simplicial modules:
Proposition 1.9. ( [Prz-1] ) (i) Let ( X ; ∗ ) be a rack, C n = ZX n , d ( ∗ ) i : C n → C n − is givenby d ( ∗ ) i ( x , ..., x n ) = ( x , ..., x i − , x i +1 , ..., x n ) , and d ( ∗ ) i : C n → C n − is given by d ( ∗ ) i ( x , ..., x n ) = ( x ∗ x i , ..., x i − ∗ x i , x i +1 , ..., x n ) ,and furthermore d i = d ( ∗ ) − d ( ∗ ) i , then ( C n ( X ) , d ( ∗ ) i ) , ( C n ( X ) , d ( ∗ ) i ) ,and ( C n ( X ) , d i ) are presimplicial modules. (we have here shiftby one comparing to Definition 1.7, that is we start from 1 notfrom 0, but it is not important in our considerations). (ii) Assume now that ( X ; ∗ ) is a quandle and degeneracy maps s i : C n ( X ) → C n +1 ( X ) is given, as before, by s i ( x , ..., x n ) =( x , ..., x i − , x i , x i , x i +1 , ..., x n ) . Then ( C n ( X ) , d ( ∗ ) i , s i ) , ( C n ( X ) , d ( ∗ ) i , s i ) ,and ( C n ( X ) , d i , s i ) are weak simplicial modules. Remark 1.10. (i)
The homology related to ( C n ( X ) , d ( ∗ ) i ) is calledone term distributive homology and it is studied in [Prz-1, P-S,P-P-1, P-P-2, CPP] . (ii) We define the trivial quandle ( X ; ∗ ) by a ∗ b = a . Then indeedwe have d ( ∗ ) i (( x , ..., x n ) = ( x ∗ x i , ..., x i − ∗ x i , x i +1 , ..., x n ) =( x , ..., x i − , x i +1 , ..., x n )(iii) Notice, that d ( ∗ )1 = d ( ∗ )1 so d = d ( ∗ )1 − d ( ∗ )1 = 0 , and this is thereason why we could start summation in Definition 1.2 from i = 2 (but ideologically it may be better to start summationfrom i = 1 ). ocycle invariants of codimension 2 embeddings (iv) Let γ n = d ( ∗ )0 : C n → C n − , that is γ ( x , x , ..., x n ) = ( x , ..., x n ) ,then ( − n γ n is a chain map in ( C n , ∂ n ) , by Lemma 1.8 (e.g. [CJKS, N-P] . Codimension 2 embedding, lower decker set.
We introducehere, following [CKS-3, Kam], the language needed to define quan-dle colorings and (co)cycle invariants in codimension 2. Let M = M n be a closed smooth n -dimensional manifold and f : M → R n +2 itssmooth embedding which is called a smooth knotting (or just knot-ting). Define π : R n +2 → R n +1 by π ( x , ...., x n +1 , x n +2 ) = ( x , ...., x n +1 )to be a projection on the first n + 1 coordinates. The projection ofthe knotting is the set M ∗ = πf ( M ). Crossing set (or singularityset) D ∗ of the knotting, is the closure in M ∗ of the set of all points x ∗ ∈ M ∗ such that ( πf ) − ( x ∗ ) contains at least two points (that is D ∗ = closure( { y ∈ R n +1 | | π − ( y ) ∩ M | > } )). We define the doublepoint set D = ( πf ) − ( D ∗ ) (or sometimes as ( π ) − ( D ∗ ) if we need it tobe a subspace of R n +2 ). Let f : M → R n +2 be a knotting which is ingeneral position with respect to the projection π : R n +2 → R n +1 . Theprecise definition is in Section 6 (Definition 6.1), here we only use thebasic notions: D ∗ is ( n − n − ).The crossing set D ∗ divides πf ( M ) into pieces. Each piece (connectedcomponent of πf ( M ) − D ∗ ) is an open n -manifold embedded in R n +1 consisting of regular points of πf ( M ), which is called open regularsheet. Regular sheets are 2-sided (even if we allow M to be nonori-entable [Kam]).The lower decker set D − is the closure of the subset of pure doublepoints which are lower in the projection (that is with respect to the lastcoordinate of R n +2 ). Similarly, the upper decker set D + is the closure ofthe subset of pure double points which are higher in the projection. M is cut by D − into the set of n -dimensional regions ( n -regions) denotedby R , that is R is the set of connected components of M − D − (noticethat πf restricted to M − D − is an embedding and that the image ofan n -region can contain several open regular sheets).The diagram D M of a knotting with a general position projection isthe knotting projection M ∗ together with “over under” information forthe crossing set. In other words, it is M ∗ with D + and D − given.1.4. Quandle colorings, and quandle shadow coloring.
We de-fine here, after [CKS-3], the notion of quandle coloring and quandleshadow coloring of diagrams of knottings ([CKS-3] gives only defini-tion in dimension n ≤ J´ozef H. Przytycki and Witold Rosicki in the work of Fenn, Rourke and Sanderson [F-R, FRS-2]). In [P-R] thecore coloring was considered for any n . We assume in the paper (un-less otherwise stated) that the considered n -dimensional manifold M is oriented thus the normal orientation (co-orientation) of every opensheet of M ∗ is well defined . Definition 1.11. (Magma coloring) Fix a magma ( X ; ∗ ) . Let f : M → R n +2 be an n -knotting, π : R n +2 → R n +1 a regular projection,and D M the knotting diagram. Let R be the set of n -regions of M cutby lower decker set. We define a magma coloring of a diagram D M (or a pair ( M, π ) ) as a function φ : R → X satisfying the followingcondition: if R and R are two regions separated by n -dimensionalupper decker region R and the orientation normal to R points from R to R , then φ ( R ) ∗ φ ( R ) = φ ( R ) ; compare Figure 1.3. Coloringof n -regions leads also to coloring of open sheets of the diagram D M ofthe knotting. Through the paper we often refer to this as coloring ofa knotting diagram. We denote by Col X ( D M ) the set of colorings of D M by X , and by col X ( D M ) its cardinality. Note that the definitionis not using any properties of ∗ ; only when we will demand invarianceof col X ( D M ) under various moves on D M , we will need some specificproperties of ∗ . R R R R (R ) *(R )= (R ) (R ) (R ) (R ) (R ) (R ) *(R ) R R (R )=
Figure 1.3; rules for magma (e.g. quandle) coloring for n = 1 , X ; ∗ ) is a shelf then the set Col X ( D M ) is a right X -shelf-space with an action of X on Col X ( D M ) given by ( φ ∗ x )( R ) = φ ( R ) ∗ x for any region R ∈ R . By the right self-distributivity of In the case of M unorientable, we can work with involutive quandle (kei) X and develop the theory of colorings and (co)-cycle invariants. ocycle invariants of codimension 2 embeddings ∗ we have ( φ ∗ x )( R ) ∗ ( φ ∗ x )( R ) = ( φ ( R ) ∗ x ) ∗ ( φ ( R ) ∗ x ) distr =( φ ( R ) ∗ φ ( R )) ∗ x = φ ( R ) ∗ x = ( φ ∗ x )( R ).Before we define shadow coloring it is useful to notice that rackcoloring of sheets of D M allows unique coloring of any closed path in R n +1 in a general position to D M , as long as a base point is colored(Lemma 1.12). In a preparation for the lemma we need the following:Fix a rack ( X, ∗ ) and an element q ∈ X . Let t < t < ... < t k < t k +1 be points on the line R . Each point t i (1 ≤ i ≤ k ) is equipped with a ± i t + , i t . Thenany function φ : { t , ..., t k } → X extends uniquely to the function˜ φ : [ t , t k ] → X with ˜ φ ( t ) = q by the following rule. If a ∈ [ t i − , t i ]and b ∈ [ t i , t i +1 ] then˜ φ ( b ) = (cid:26) ˜ φ ( a ) ∗ φ ( t i ) if the framing at t i is positive˜ φ ( a )¯ ∗ φ ( t i ) if the framing at t i is negativeIn particular, ˜ φ ( t k +1 ) = ( q ∗ φ ( t )) ∗ ... ∗ k φ ( t k ), where ∗ i = ∗ if theframing of t i is positive and ∗ i = ¯ ∗ if the framing of t i is negative.Finally we can apply the above to an arc α : [ t , t k +1 ] → R n +1 ina general position with respect to D M with some X -coloring φ , andwhich cuts D M at k points α ( t ) , ..., α ( t k ). The framing of points t i (1 ≤ i ≤ k ) is yielded by co-orientation of D M and points α ( t i ). Wecan identify φ ( t ) with φ ( α ( t i )), thus by above φ can be extended to thefunction ˜ φ : [ t , t k +1 ] → X . Now we are ready to prove that: Lemma 1.12. If α : [ t , t k +1 ] → R n +1 is a closed path, that is α ( t ) = α ( t k +1 ) , then ˜ φ ( α ( t )) = ˜ φ ( α ( t k +1 )) .Proof. Using the fact that R n +1 is simple connected, we can contract α to a base point α ( t ), and we can put contracting homotopy in ageneral position with respect to D M . The proof is by induction on thenumber of critical points of contracting homotopy. The critical pointsare either cancelling a piece of the path going for and back, or crossinga double point strata. In the first case if we start from the color a andcross color b , forth and back, thus we get a color ( a ∗ b )¯ ∗ b or ( a ¯ ∗ b ) ∗ b which is a by invertibility of ∗ . In the case when isotopy is crossinga double point set, we use the fact that double point crossing lookslike classical crossing multiplied by R n − , and the interesting case iswhen the closed path is below sheets it crosses. The situation can beillustrated by using classical crossing and coherence of coloring followsfrom right distributivity and invertibility of ∗ , see Figures 1.4 and 1.5. (cid:3) J´ozef H. Przytycki and Witold Rosicki (R) q = ( (t )) R (R) q * Figure 1.4; a path moving for and back through the n -sheet( q ∗ φ ( R ))¯ ∗ φ ( R ) = q q q q q * (t ) ( (t ))=q q = q * q q q q q * (q q ) * * q (t ) ( (t ))=q q = Figure 1.5; a path is crossing double point stratum; (( q ∗ q ) ∗ q )¯ ∗ q ) ∗ ¯ q =(( q ∗ q ) ∗ ( q ∗ q ))¯ ∗ q ) ∗ ¯ q = (( q ∗ q ) ∗ q )¯ ∗ q ) ∗ ¯ q = ( q ∗ q ) ∗ ¯ q = q Definition 1.13. (Magma shadow coloring) A shadow coloring of aknotting diagram (extending the given coloring φ ) is a function ˜ φ : ˜ R ∪R → X , where ˜ R is the set of ( n + 1) -dimensional regions(chambers )of R n +1 − πf ( M ) satisfying the following condition. If R and R are n + 1 regions (chambers) separated by n dimensional region (regularsheet) α where the orientation normal of α points from R to R , then ˜ φ ( R ) ∗ ˜ φ ( α ) = ˜ φ ( R ) and ˜ φ restricted to the set of n -dimensionalregions is a given coloring φ (compare [CKS-3] and Figure 1.6 for n =1 or ). Again the definition works for any binary operation but if ( X ; ∗ ) is a rack, then any coloring φ and a constant q chosen for afixed ( n +1) -chamber, R , yield the unique extension to shadow coloring We should appreciate here, not that accidental, analogy to Weyl chambers inrepresentation theory of Lie algebras. Thus we use the term chamber throughoutthe paper. ocycle invariants of codimension 2 embeddings ˜ φ so that ˜ φ ( R ) = q ; this follows from Lemma 1.12. . We denote by Col sh,X ( D M ) the set of all shadow colorings of ( R n +1 , D M ) by X andby col sh,X ( D M ) its cardinality. R R R (R ) (R ) (R ) *(R ) R (R ) (R ) (R ) *(R )= (R ) * (R ) (R ) (R ) *(R ) (R ) *(R ) * (R ) (R ) R (R ) (R ) (R )= * (R ) =* Figure 1.6; Quandle shadow coloring for n = 1 , X ; ∗ ) is a shelfthen the set Col sh,X ( D M ) is a right X -shelf-space with an action of X on Col sh,X ( D M ) given by ( ˜ φ ∗ x )( R ) = ˜ φ ( R ) ∗ x for any region orchamber R . Remark 1.14.
A shadow coloring of n -knotting diagram can be inter-preted as a special case of coloring in dimension n + 1 . To this aimwe consider the ( n + 1) dimensional manifold ˜ M = ( M n × R ) ⊔ R n +1 embedded in R n +3 as follows: Let ˜ f : ˜ M → R n +3 with ˜ f ( m, x ) =( f ( m ) , x ) and ˜ f ( x , ..., x n +1 ) = ( x , ..., x n +1 , h, , where for f ( m ) =( f ( m ) , ..., f n +2 ( m )) we assume h ≤ f n +2 ( m ) for any m ∈ M n (in otherwords, R n +1 is embedded below M n ). The projection ˜ π : R n +3 → R n +2 is defined as π × Id . We get the diagram D ˜ M = ˜ π ˜ f ( ˜ M ) = ( D M × R ) ∪ ( R n +1 × { } ) in R n +2 . The points of multiplicity n + 1 in D M givesrise to points of multiplicity n + 2 in D ˜ M , and each shadow coloringof R n +1 , D M ) gives rise to a coloring of D ˜ M (also the (shadow) chain c n +2 ( D M ) gives rise to the chain c n +2 ( D ˜ M as will be clear in the nextsubsection). As we do not use Remark 1.14 later, we should not worry We can also see this important property, as follows: consider a small trivialcircle T c in a chosen chamber R b of R n +1 − πf ( M ) let T be the boundary of aregular neighborhood of T c , thus T = S × S n − . We can extend coloring φ to M ∪ T by coloring T by a fixed color q . Then we isotope the circle T c (and T ) sothat it is always below f ( M ) part but it touches every chamber of R n +1 − πf ( M ).Now having initial chamber colored by q any other chamber is now colored by anappropriate color of part of T in the chamber. Unlike in approach using Lemma1.12, we use here the fact that Roseman moves on a diagram are preserving coloringsby a rack (see Theorem 3.4). J´ozef H. Przytycki and Witold Rosicki that the resulting manifold ˜ M is not compact (otherwise we need toconsider knotting up to isotopy with compact support). Observation 1.15.
As noted by S.Kamada, one can consider shadowcoloring ˜ φ also in the case of ( X ; ∗ ) a rack and E an X -rack-set. Inthis case we color the chambers of R n +1 − D M by elements of E witha natural convention that if R and R are chambers separated by n dimensional region α where the orientation normal of α points from R to R , then ˜ φ ( R ) ∗ ˜ φ ( α ) = ˜ φ ( R ) . One can also show, using Lemma1.12 or Footnote 8, that if we take φ ′ ( R ) = q for some q ∈ E then ˜ φ is uniquely extendable from coloring of D M . The reason is that inplace of X and E we can consider a rack X ∪ E as in Observation 1.6. n + 1) and ( n + 2) -chains of diagrams of knotting. We showin this part how to any knotting diagram D M given by πf : M n f → R n +2 π → R n +1 and a shelf ( X ; ∗ ) associate two chains of dimension( n + 1), and ( n + 2) in rack (and quandle) chain groups C Wn +1 ( X ) and C Wn +2 ( X ) respectively ( W = R or D or Q , that is Rack, Degenerate orQuandle).We start from the classical theory in dimensions n = 1.Carter-Kamada-Saito noticed in 1998 that if we color a classical ori-ented link diagram, D , by elements of a given quandle X and considera sum over all crossings of D of pairs in X , ± ( q , q ) according tothe convention of Figure 1.7 then the sum has an interesting behaviorunder Reidemeister moves. x * q q + q q x *
212 1 23 0 21 2 2 q q qq c (p)= −(q ,q ) c (p)= −(q ,q ,q )c (p)=(q ,q ,q )c (p)=(q ,q ) Figure 1.7; The contribution to the 2-chain is ( q , q ) for a positive crossing and − ( q , q ) for a negative crossing; in the case of the 3-chain for a shadow coloring,we have ( q , q , q ) and − ( q , q , q ), respectivelyThis lead them to define in 1998 a 2-cocycle invariant and relate itto rack homology defined between 1990 and 1995 by Fenn, Rourke, ocycle invariants of codimension 2 embeddings and Sanderson. Because the first Reidemeister move is changing thesum by ± ( x, x ) they were assuming that ( x, x ) should be equivalentto zero (so ( x, x ) should represent degenerate element). The 3-cocycleof the shadow X -colorings was motivated by [R-S] and developed in[CKS-1] (compare [CKS-3], page 154). In particular, they noted thatthe 3-chain constructed with convention of Figure 1.7, that is c ( D ) = P p ∈ crossings sgn ( p ) c ( p ), is a 3-cycle and Reidemeister moves preservehomology class of c ( D ).We define, in this subsection, the chains c n +1 ( D M , φ ) and c n +2 ( D M , ˜ φ )for any diagram D M of a knotting M and chosen colorings φ and ˜ φ (thefact that they are cycles is proven in Section 2, Theorem 2.1, and topo-logical invariance of their homology via Roseman moves is proven inSection 4, Theorem 4.1). To make the general definition we need someconventions and notation concerning a crossing of multiplicity ( n + 1)in a diagram of an n -knotting.The sign of a crossing of multiplicity ( n + 1) is chosen so that it agreeswith the definition of the sign of a crossing in a classical knot theory.We say that the sign of p is positive if n + 1 normal vectors to n + 1hyperplanes intersecting at p listed starting from the top (that is thenormal vector to the highest hyperplane is first) form a positive ori-entation of R n +1 ; otherwise the sign of p is equal to −
1. The sourcechamber R of R n +1 − πf ( M ) adjacent to p is the region from whichnormals of hyperplanes points (compare page 151 of [CKS-3]). Definition 1.16. ( ( n + 1 )-chain) Let f : M → R n +2 be an n -knotting, π : R n +2 → R n +1 a regular projection, and D M the knotting diagram. (i) Fix a quandle X and a coloring φ : R → X , Let p be a crossingof multiplicity n + 1 of D M . We define a chain c n +1 ( p, φ ) ∈ C n +1 ( X ) by c n +1 ( p ) = sgn ( p )( q , ..., q n +1 ) where ( q , ..., q n +1 ) is obtained as follows:We consider the source region, say R , around p and ( q , ..., q n +1 ) are colors of hyperplanes intersecting at p around R listed inthe order of hyperplanes from the lowest to the highest (see Fig-ure 1.8 for the case of n = 2 ). We usually write c n +1 ( D M ) for c n +1 ( p, φ ) if φ is fixed. (ii) The chain associated to the diagram and fixed coloring is thesum of above chains taken over all crossings of multiplicity n +1 of D M : c n +1 ( D M , φ ) = X p ∈ Crossings c n +1 ( p ) . J´ozef H. Przytycki and Witold Rosicki (iii)
Finally, if X is finite, we sum over all X colorings of D M sothe result is in the group ring over C n +1 ( X ) (in fact, it is inthe group ring of H n +1 ( X ) but this will be proven later). It isconvenient here to use multiplicative notation for chains so that c n +1 ( D M , φ ) = Π p ( q , ..., q n +1 ) sgnp and then c n +1 ( D M ) = X φ c n +1 ( D M , φ ) = X φ Y p ( q , ..., q n +1 ) sgn ( p ) . (iv) If X is possibly infinite, in place of a sum we consider the setwith multiplicity c setn +1 ( D M ) = { c n +1 ( D M , φ ) } φ ∈ Col X ( D M ) . Definition 1.17. ( n +2 (shadow) chain for D M ) Here we generalize theprevious definition to construct ( n + 2) -chains from shadow coloringsrelated to link diagram. We color not only regions of M (as in Defini-tions 1.11, 1.16) but also ( n + 1) -chambers of R n +1 cut by πf ( M ) . Asbefore we take the product of signed chains associated to every multi-plicity ( n + 1) -crossing point, p and sum these products over all shadowcolorings. Thus we start from c n +2 ( p ) = sgn ( p )( q , q , ..., q n +1 ) , where q is the color of the source chamber R . In effect, if X is finite then: c n +2 ( D M ) = X ˜ φ c n +2 ( D M , ˜ φ ) = X ˜ φ Y p ( q , q , ..., q n +1 ) sgn ( p ) . If X is possibly infinite, in place of a sum we consider the set withmultiplicity c setn +2 ( D M ) = { c n +2 ( D M , ˜ φ ) } ˜ φ ∈ Col sh,X ( D M ) . We illustrate the case of n = 1 in Figure 1.7, and the case of n = 2 in Figure 1.8. ocycle invariants of codimension 2 embeddings q =( * ) * q q q( * ) * ( * )
21 3 q q q q( )( * * ) * q q q qnormal vectors q q q Figure 1.8; multiplicity three point in M knotting and quandle coloring;the point yields the 3-chain ( q , q , q ) and the 4-chain ( q , q , q , q );normal vectors yield here positive orientationFor M which is not connected we can use a trick of [CENS] to havemore delicate chains. They behave nicely under Roseman moves butthey are not cycles. Thus we can use them to produce cocycle invariantsbut not cycle invariants of knottings (Theorem 4.8(iii)). Definition 1.18.
Let M = M ∪ M ∪ ... ∪ M k . We define an ( n + 1) -chain c n +1 ( D M , φ, i ) by considering only those crossings of multiplicity ( n + 1) whose bottom sheet belongs to M i . We denote the set of such ( n + 1) -crossings by T i . Then we define c n +1 ( D M , φ, i ) = X p ∈T i c n +1 ( p, φ ) .
2. ( n + 1) and ( n + 2) cycles for n -knotting diagrams We show in this section the two chains c n +1 ( D M , φ ) and c n +2 ( D M , ˜ φ )constructed in Subsection 1.5 are, in fact, cycles. Theorem 2.1.
The chains c n +1 ( D M , φ ) and c n +2 ( D M , ˜ φ ) are cycles in C Qn +1 ( X ) and C Qn +2 ( X ) respectively. The main idea of the proof is to analyze points of multiplicity n + 1and n + 2 in M ∗ ∈ R n +1 , and associated ( n + 1)- and ( n + 2)-chains in Z X n +1 and Z X n +2 , and to identify face maps of the chains, d ( ∗ ) i and J´ozef H. Przytycki and Witold Rosicki d ( ∗ ) i , as associated to arcs of multiplicity n . Then by Roseman theory(see Section 6), such an arc starts at a point of multiplicity n + 1 (say p ) and ends either at another point of multiplicity n + 1 (say p ) orat a singular point (of multiplicity less than n ). In the first case, weprove that there are proper cancellations of face maps associated to thearc. In the second case (which may happen for n > C Qn +1 ( X ) and C Qn +2 ( X ). Details are given in the following subsections, starting fromthe classical case of n = 1.2.1. Shadow 3-cycle for n = 1 . We start from the known case of n = 1 but present our proof in a way which will allow natural gen-eralization for any n . We show in detail in this subsection that the3-chain constructed with convention of Figure 1.7, that is c ( D ) = P p ∈ crossings sgn ( p ) c ( p ), is a 3-cycle.We start from 2 crossings p and p connected by an arc colored bythe pair q sh ( arc ) = ( q , b ) in our convention (that is the color of thearc is b and the shadow color of the source region (chamber) close tothe arc is q ; see Definition 2.2). In our examples the horizontal lineis first above and then below the other arcs, and it will be denotedas pair of type (2 ,
1) later in generalization. For a reader who wouldlike to visualize here the general case, we stress that the arc connectingcrossings p and p will be an arc of points of multiplicity n (intersectionof n hyperplanes in R n +1 ) connecting points of multiplicity n + 1.Now consider our four cases. q a c+ +b q’’ q’’ q *c=a+ − cb=w c* wq’’ q’’ q *c= q’ q’a − cb=w c* w − = *a q’ q q’*aa c − = b + Figure 2.1; In all case the connecting arc has label q sh ( arc ) equal to ( q , b );the multi-labelling q sq is explained in full generality in Definition 2.2 ocycle invariants of codimension 2 embeddings To describe precisely the outcome of our pictures we denote theshadow 3-chain of our diagram by c ( D ), the contribution of the firstcrossing p by c ( p ), the contribution of the second crossing, p , by c ( p ), and c ( p , p ) = c ( p ) + c ( p ) denotes the contribution of bothcrossings. We also use the notation of Figure 2.2 which is the specialcase of Definition 2.2. q q b*b arc sh q (arc)=(q ,b) Figure 2.2; Convention for arc coloring, q sh ( arc ) = ( q , b )Thus in the first case we have: c ( p ) = ( q , a, b ) , c ( p ) = ( q , b, c ) , and c ( p , p ) = ( q , a, b )+( q , b, c ) . We have then: d ( ∗ )2 ( c ( p )) = d ( ∗ )2 (( q , a, b ) = d ( ∗ )3 (( q , b, c ) = ( q , b ) = q sh ( arc ) . Thus proper pieces of ∂ ( p ) and ∂ ( p ) cancel out in c ( p , p ). This willbe the case always, and below we shortly analyze other cases:In the second case we have c ( p ) = ( q , a, b ) , c ( p ) = − ( q ′′ , w, c ) , and c ( p , p ) = ( q , a, b ) − ( q ′′ , w, c ) . Where b = w ∗ c . We have then: d ( ∗ )2 (( q , a, b ) = d ( ∗ )3 (( q ′′ , w, c ) = ( q , b ) = q sh ( arc ) . In the third case we have: c ( p ) = − ( q ′ , a, b ) , c ( p ) = − ( q ′′ , w, c ) , and c ( p , p ) = − ( q ′ , a, b ) − ( q ′′ , w, c ) . We have then d ∗ (( q ′ , a, b ) = d ∗ (( q ′′ , w, c ) = ( q , b ) = q sh ( arc ) . Finally, in In the fourth case we have: c ( p ) = − ( q ′ , a, b ) , c ( p ) = +( q , b, c ) , and c ( p , p ) = − ( q ′ , a, b )+( q , b, c ) . We have then: d ( ∗ )2 (( q ′ , a, b ) = d ( ∗ )3 (( q , b, c ) = ( q , b ) = q sh ( arc ) . J´ozef H. Przytycki and Witold Rosicki
This proves that c ( D ) is a cycle in C ( X ), as ∂ = ∂ ( ∗ ) − ∂ ( ∗ ) = P i =2 ( − i d ( ∗ ) i − P i =2 ( − i d ( ∗ ) i . We didn’t consider all cases as wewill argue in the general case that all cases follows at once, howeverwe illustrate one more case, of type (2 , c ( p ) = − ( q ′ , a, b ) , c ( p ) = − ( q , c, b ) , and c ( p , p ) = − ( q ′ , a, b ) − ( q , c, b ) . We have then: d ( ∗ )2 (( q ′ , a, b ) = d ( ∗ )2 (( q , c, b ) = ( q , b ) = q sh ( arc ); as illustrated in Figure 2.3 . q’ q q’*aa c − = b − Figure 2.3; two crossings of type (2 , T in place of a link diagram D ; then c ( T ) is not an absolute cycle, but we can work in the settingof relative chain (of ( T, ∂T )). We do not follow this idea here but itmaybe useful in many situations.2.2.
The general case of ( n + 1) and ( n + 2) cycles. For the generalcase we need a notation for coloring of strata of a neighborhood of acrossing of multiplicity ( n +1) in R n +1 generalizing coloring and shadowcoloring.For a given vector w in R n +1 let V w be an ( n + 1)-dimensional linearsubspace orthogonal to w . For basic vectors e = (1 , , ..., , . . . e i =(0 , .... , , , ..., , . . . e n +1 = (0 , ..., ,
1) we write V i for V e i . We have T n +1 i =1 V i = (0 , ...,
0) = p and it is our “model” singularity (crossingof multiplicity n + 1). In a standard way we associate to this cross-ing the signum +1. If our system of hypersurfaces S V i is a part ofa knotting diagram, the sign +1 would correspond to the situation V > V > ... > V n +1 , that is V i above V i +1 at the crossing. However,our convention, motivated by right self-distributivity, makes more con-venient assumption V < V < ... < V n +1 and then with our conventionthe crossing p has the signum sgn ( p ) = ( − n ( n +1) / . ocycle invariants of codimension 2 embeddings We introduce two labelings (generalizing colorings and shadow col-orings):(i) q : S V i → X ∪ X ∪ ... ∪ X n +1 ;(ii) q sh : R n +1 → X ∪ X ∪ ... ∪ X n +2 The strata of the labeling of a point x depends on the “order of singular-ity” that is q ( x ) (respectively q sh ( x )) is an element of X k , (respectively X k +1 ) where X is a fixed shelf and k = k ( x ) is the number of hy-perplanes V i to which x belongs (if x ∈ R n +1 − S V i then k ( x ) = 0).Both colorings are coherent because of right self-distributivity law in X . The idea of coloring is that we choose a color, say q for a sourceregion, R , of R n +1 − S n +1 i =1 V i (this will be part of q sh coloring, that is q sh ( x ) = q for x ∈ R ). Furthermore, we choose colors q , q , ..., q n +1 and if x ∈ R ( s ) i where R ( s ) i is the source sheet of V i , then q ( x ) = q i and q sh ( x ) = ( q , q i ) ∈ X .With an assumption that V i is always below V i +1 in our considera-tions. we propagate our colors according to our rules of coloring. Noticethat we use only ∗ (never ¯ ∗ ) in our coloring, so assumption that X isa shelf suffices here.We describe this idea formally below. Definition 2.2.
Choose a shelf X and n +2 elements ( q , q , ..., q n , q n +1 ) in X . (i) For x = ( x , ..., x n , x n +1 ) ∈ R n +1 − S V i the label q sh ( x ) is de-fined to be q ∗ q i ∗ ... ∗ q i s where < i < ... < i s are preciselythese indexes for which x i j > . In particular, if for all i , x i < (i.e. x is in the source sector), then q sh ( x ) = q . (ii) If x = ( x , ..., x n , x n +1 ) has only one coordinate, say i th, equalto zero (that is x ∈ V i but x / ∈ V j for j = i ) then q ( x ) is definedto be q i ∗ q i ∗ ... ∗ q i s where i < i < ... < i s are precisely theseindexes for which i j > i and x i j > . In particular, if for all j such that j > i we have x j < , then q ( x ) = q i . (iii) Let x = ( x , ..., x n , x n +1 ) belongs to exactly k hyperplanes, x ∈ T kj =1 V i j , then q ( x ) = ( q ( i ) , ..., q ( i k ) ) ∈ X k , where q i j ( x ) = q ( x i j ) where x i j is obtained from x by replacing all coordinatesequal to , but x i j , by − (recall that x i = x i = ... = x i k = 0 and other coordinates are different from ). (iv) If x ∈ S V i then q sh ( x ) is obtained from q ( x ) by q sh ( x ) =( q sh ( x ′ ) , q ( x )) where x ′ is obtained from x = ( x , ..., x n ) by re-placing all in the sequence by − (i.e. x ′ is a point in a sourcechamber). In particular, if q ( x ) = q i then q sh ( x ) = ( q , q i ) . J´ozef H. Przytycki and Witold Rosicki
We can complete with this notation, the proof that c n +1 ( D M ) and c n +2 ( D M ) are cycles.Very schematic visualization of the general case is shown in Figure2.4 where the vertical lines represent n dimensional sheets of M ∗ = f π ( M n ) with source colors a and c respectively and horizontal linerepresenting the line of intersection of n sheets (in R n +1 ) with the col-oring b = q ( arc ) = ( q , ..., q n ). q p V vp b=(q ,...,q ) a c p V vp Figure 2.4; connecting arc has label q sh ( arc ) = ( q , q ( arc )) = ( q , b ) = ( q , q , ..., q n )We denote the position of the first vertical sheet as i ( p ) and the secondvertical sheet by i ( p ). The n + 2 chains corresponding to p and p depend on direction of vertical vectors, n p and n p to vertical sheets V vp and V vp . We write ǫ ( p , p ) = 1 if the vector n p points from p to p and 0 otherwise. Similarly ǫ ( p , p ) = 1 if the vector n p pointsfrom p to p , and it is 0 otherwise. This notation is used to identifyoperation ∗ and ∗ = ∗ . Then we have: d ( ∗ ǫ ( p ,p ) i ( p ) ( q , q , ..., q i ( p ) − , a, q i ( p ) , ..., q n ) = d ( ∗ ǫ ( p ,p ) i ( p ) ( q , q , ..., q i ( p ) − , a, q i ( p ) , ..., q n ) = d ( ∗ ǫ ( p ,p ) i ( p ) ( q , q , ..., q i ( p ) − , b, q i ( p ) , ..., q n )Thus sgn ( p )( − i ( p ) = − sgn ( p )( − i ( p ) and in c n +2 ( p , p ) the terms d ( ∗ ǫ ( p ,p ) i ( p ) and d ( ∗ ǫ ( p ,p ) i ( p ) cancel out. In conclusion, the ( n + 2)-chain c n +2 ( D M , ˜ φ ) is a cycle.The similar proof works for c n +1 ( D M , φ ). The fact that c n +1 ( D M , φ )is a cycle follows also from the following observation: Observation 2.3.
Consider the map γ n : C n ( X ) → C n − ( X ) given bycutting the first coordinate, that is γ n ( x , x , ..., x n ) = ( x , ..., x n ) . Thenby Lemma 1.8 (compare Remark 1.10(iv)) the map ( − n γ is a chainmap for ∂ ( ∗ ) and ∂ ( ∗ ) thus also for ∂ = ∂ ( ∗ ) − ∂ ( ∗ ) . From definitionsof c n +1 and c n +2 , we have c n +1 ( D M , φ ) = γ n +2 c n +2 ( D M , ˜ φ ) . From this ocycle invariants of codimension 2 embeddings we conclude that if c n +2 ( D M , ˜ φ ) is an ( n + 2) -cycle then c n +1 ( D M , φ ) is an ( n + 1) -cycle. In the next two sections we show that our cycles (and their sums)are topological invariants. We will start from the fact that the spacesof colorings (and shadow colorings) are topological invariants.3.
Topological invariance of colorings
The logic of the section is as follows: We introduce here, following[Joy-2, F-R], two definitions of a fundamental rack or quandle of aknotting, the abstract one and the concrete definition. The abstractdefinition of the fundamental rack or quandle of a knotting is indepen-dent on any projection in the similar way, as the fundamental group.In the concrete definition, for a given projection, we get the concretepresentation of a fundamental rack or quandle from the diagram usinggenerators and relations in a way reminiscent of the Wirtinger presen-tation of the fundamental group of a classical link complement. It wasobserved in [F-R, FRS-2] that these definitions are equivalent using ageneral position argument. As a consequence we get that a concreterack and quandle colorings are topological invariants (independent ona diagram) because in both cases abstract and concrete colorings areobtained from a homomorphism from the fundamental object to X .3.1. The fundamental rack and quandle of an n -knotting. Thefirst definition, we give, uses a knotting diagram. We follow Definitions1.11 and 1.13, except that in place of concrete chosen ( X ; ∗ ) we build auniversal (called fundamental) object (magma, shelf, rack or quandle). Definition 3.1. (Fundamental Magma of a knotting diagram) Let f : M → R n +2 be a knotting, π : R n +2 → R n +1 a regular projection and D M related knot diagram. (i) The fundamental magma X ( D M ) = ( X ; ∗ ) is given by the fol-lowing finite presentation. The generators of X are in bijectionwith the set of regions of M cut by lower decker set. Relationsin ( X ; ∗ ) are given as follows: if R and R are two regionsseparated by n -dimensional upper decker region R (with vari-ables, respectively, q , q and q ) and the orientation normal to R points from R to R , then q ∗ q = q . Notice that however our definition of X ( D M ) is not using any properties of ∗ but still for n > q ∗ q ) ∗ q =( q ∗ q ) ∗ ( q ∗ q )). J´ozef H. Przytycki and Witold Rosicki (ii) If ( X ; ∗ ) is required to be a shelf we get a fundamental shelf of D M . (iii) If ( X ; ∗ ) is required to be a rack we get a fundamental rack of D M . (iv) If ( X ; ∗ ) is required to be a quandle we get a fundamental quan-dle of D M . (v) Fix a quandle ( X ; ∗ ) and the knotting f : M → R n +2 , withregular projection π : R n +2 → R n +1 . The quandle (resp. rack)coloring of a diagram D M is a quandle homomorphism from thefundamental quandle (resp. rack) X ( D M ) to X . The quandle (or rack) coloring described in Definition 3.1 is equiva-lent to Definition 1.11.The Fundamental shadow magma, shelf, rack and quandle of a knot-ting diagram are defined analogously to that of coloring; we give a fulldefinition so it is easy to refer to it.
Definition 3.2. (Fundamental shadow magma of a knotting diagram)Let f : M → R n +2 be a knotting, π : R n +2 → R n +1 a regular projectionand D M related knot diagram. (i) The fundamental shadow magma X sh ( D M ) = ( X ; ∗ ) is given bythe following finite presentation. The generators of X are inbijection with the set R ∪ R cha where R is the set of regions of M cut by lower decker set and R cha is the set of chambers of R n +1 − πf ( M ) . Relations in ( X ; ∗ ) are given as follows: if R and R are two regions separated by n -dimensional upper deckerregion R (with variables, respectively, q , q and q ) and theorientation normal to R points from R to R , then q ∗ q = q . Furthermore, if ˜ R and ˜ R are n + 1 chambers separatedby n dimensional region α where the orientation normal of α points from ˜ R to ˜ R , and ˜ q , ˜ q , ˜ q are colors of ˜ R , ˜ R and α ,respectively, then ˜ q ∗ ˜ q = ˜ q . (ii) If ( X ; ∗ ) is required to be a shelf we get a fundamental shelf of D M . (iii) If ( X ; ∗ ) is required to be a rack we get a fundamental rack of D M . (iv) If ( X ; ∗ ) is required to be a quandle we get a fundamental quan-dle of D M . Notice that however our definition of X sh ( D M ) is not using any properties of ∗ but still if there is a double crossing in the projection then colors involved in thecrossing satisfy right self-distributivity (see Figure 1.6). ocycle invariants of codimension 2 embeddings (v) Fix a quandle (or a rack) ( X ; ∗ ) and the knotting f : M → R n +2 , with regular projection π : R n +2 → R n +1 . The quan-dle (resp. rack) shadow coloring of a diagram D M is a quan-dle homomorphism from the fundamental quandle (resp. rack) X sh ( D M ) to X . Again, the quandle (or rack) shadow coloring described in Definition3.2 is equivalent to Definition 1.13.
Remark 3.3.
If we assume that X ( D M ) is the fundamental rack (orquandle) of a diagram D M , then the presentation of the fundamentalshadow rack X sh ( D M ) can be obtained by adding one generator andno new relations (except that of rack (or quandle) relations). That is: X sh ( D M ) = { X ( D M ) , w | } . The new generator w is a color of anarbitrary, but fixed, chamber of R n +1 − D M . The presentation can bejustified using Lemma 1.12 or by a method described in Footnote 8 toDefinition 1.13. We recall in the next subsection a projection free approach to thefundamental rack and quandle of a knotting and use it to notice that X ( D M ) and X sh ( D M ) are independent on the concrete diagram.3.2. Abstract definitions of a fundamental rack and quandle.
Joyce, Fen, and Rourke [Joy-2, F-R] gave an abstract definition of thefundamental rack of a knotting, independent on a projection and theynoted that it is equivalent to the concrete definition given in Subsection3.1.We follow here [F-R] in full generality, however we are concernedmostly with the case of of the ambient manifold W = R n +2 .(i) Let L : M → W be a knotting (codimension two embedding).We shall assume that the embedding is proper at the boundaryif ∂M = ∅ , that W is connected and that M is transverselyoriented in W . In other words we assume that each normaldisk to M in W has an orientation which is locally and globallycoherent. The link is said to be framed if there is given crosssection (called framing) λ : M → ∂N ( M ) of the normal diskbundle (the total space of the bundle is a tubular neighborhoodof L ( M ) in W ). Denote by M + the image of M under λ . Wecall M + the parallel manifold to M .(ii) We consider homotopy classes Γ of paths in W = closure( W − N ( M ) from a point in M + to a base point. During the homo-topy the final point of the path at the base point is kept fixedand the initial point is allowed to wander at will in M + . J´ozef H. Przytycki and Witold Rosicki (iii) The set Γ is a right π ( W )-group-set, that is the fundamentalgroup of a knotting complement acts on Γ as follows: let γ be a loop in W representing an element g of the fundamentalgroup. If a ∈ Γ is represented by the path α then define a · g to be the class of the composition path α ◦ γ . We can use thisaction to define a rack structure on Γ. Let p ∈ M + be a pointon the framing image. Then p lies on a unique meridian circleof the normal circle bundle. Let m p be the loop based at p which follows round the meridian in a positive direction. Let a, b ∈ Γ be represented by the paths α, β respectively. Let ∂ ( b )be the element of the fundamental group determined by theloop ¯ β ◦ m β ◦ β . (here ¯ β represents the reverse path to β and m β is an abbreviation for m β (0) the meridian at the initial pointof β .) The fundamental rack of the framed link L is defined tobe the set Γ = Γ( L ) of homotopy classes of paths as above withoperation a ∗ b = a · ∂ ( b ) = [ α ◦ ¯ β ◦ m β ◦ β ] . (iv) A rack coloring, by a given rack ( X ; ∗ ) is a rack homomorphism f : Γ( L ) → X . In a case of W = R n +2 we give also down toearth definition from the link projection (initially depending onthe projection) (see Definition 1.11).(v) If L is an unframed link then we can define its fundamentalquandle : let Γ q = Γ q ( L ) be the set of homotopy classes of pathsfrom the boundary of the regular neighborhood ( N ( M )) to thebase point where the initial point is allowed to wander duringthe course of the homotopy over the whole boundary. The rackstructure on Γ q ( L ) is defined similarly to that of Γ( L ). Thusthe fundamental quandle of L is the quotient of the fundamentalrack of L by relations generated by idempotency x ∗ x = x .(vi) A quandle coloring, by a given quandle ( X ; ∗ ), is a quandlehomomorphism f : Γ q ( L ) → X . Fiber of a normaldisk bundle
Figure 3.1; Composition a · g where a is a class of an arc α and g is a class of a loop γ ocycle invariants of codimension 2 embeddings For W = R n +2 (or S n +2 ) our two definitions of the fundamentalrack (or quandle) are equivalent. If D M is a diagram of a knotting L : M → R n +1 with the regular projection π : R n +2 → R n +1 then wehave a natural epimorphism F q : X q ( D M ) → Γ q ( L ) given as follow:Let x H be a generator of X q ( D M ) corresponding to a sheet (region) H of D M . Choose a base point of R n +1 − πL ( M ) very high (call it ∞ ) and project it to a point of H cutting ∂V M at some point m H ,then we define F q ( x H ) to be the class of a straight line from m H to ∞ . Similarly we define a rack epimorphism F : X ( D M ) → Γ( L ), byextending F q ( x H ) by starting from the point of M + being on the samefiber disk of V M as m H and connecting along the boundary of the diskto m H . Theorem 3.4. ( [F-R, FRS-2] ) (i) Two definitions of a fundamental rack (resp. quandle) of a(framed) n -link L : M → R n +2 coincide, the map F : X ( D M ) → Γ( L ) is a quandle isomorphism. In particular, Definition 3.1(and equivalent 1.11) for racks and quandles are independenton regular projection and give a finite presentation of the fun-damental rack Γ( L ) (resp. quandle Γ q ( L ) ). (ii) The fundamental shadow rack (resp. quandle) is independenton regular projection thus it is well defined for a knotting n -link L : M → R n +2 ; we denote it by Γ s h ( L ) (resp. Γ sq h ( L ) ). (iii) The sets ( X -quandle-sets) Col X ( D M ) a Col
X,sh ( D M ) do notdepend on the diagram of a given linking M .Proof. The statement and a sketch of a proof is given in [F-R] (Re-marks(2) p. 375 ) and [FRS-2](Lemma 3.4; p.718). One can give alsoa proof using Roseman moves (starting with the pass move S ( c, n +2 , X sh ( D M ) (in fact, X quandle set)of shadow colorings is a topological invariant. We use here Remark In quandle case we consider knottings up to (smooth) ambient isotopy (equiv-alently up to Roseman moves), and in a rack case up to framed (smooth) ambientisotopy. Fenn and Rourke write in Remarks(2):
A similar analysis can be carried outfor an embedding of M n in S n +2 : we obtain a “diagram” by projecting onto R n +1 in general position and regarding top dimensional strata ( n -dimensional sheets) as“arcs” to be labelled by generators and ( n − -dimensional strata (simple doublemanifolds) as “crossings” to be labelled by relators. In general position a homotopybetween paths only crosses the ( n − -strata and a proof along the lines of thetheorem can be given that this determines a finite presentation of the fundamentalrack. J´ozef H. Przytycki and Witold Rosicki X ( D M ) is a topological invariant, so is X sh ( D M ) = { X ( D M ) , w | } . (cid:3) Remark 3.5.
Theorem 3.4 should be understood as follows: If R isa Roseman move on a diagram of n -knotting D M resulting in RD M ,then there are natural X -rack isomorphisms (i.e. bijections preserv-ing right action by X ), R : Col X ( D M ) → Col X ( RD M ) and ˜ R : Col
X,sh ( D M ) → Col
X,sh ( RD M ) . Natural means here that outside a ball(tangle) in which the move R takes place, the bijections R and ˜ R areidentity. This raises an interesting question: assume that after usinga finite number of Roseman moves we come back to the diagram D M .What automorphism of X -quandle-sets Col X ( D M ) and Col
X,sh ( D M ) we performed? Is it always an inner automorphism . When it is theidentity? Corollary 3.6.
The homology, H W ∗ ( X ( D M )) and H W ∗ ( X sh ( D M )) ofthe fundamental rack and the fundamental shadow rack are topologicalknotting invariants. M.Eisermann proved that in the classical case aknot is nontrivial if and only if H Q ( X ( K )) = Z , [Eis-1] . We can askwhat we can say in a general case about H Qn +1 ( X ( D M )) . The presentation of X ( D M ) gives the coloring of D M by the quandle X ( D M ); we call this the fundamental coloring and denote by φ fund .Similarly, The presentation of X sh ( D M ) gives the shadow coloring of D M by the quandle X sh ( D M ); we call this the fundamental shadowcoloring and denote by ˜ φ fund ( D M ). Corollary 3.7.
The homology classes of cycles c n +1 ( D M , φ fund ) (resp. c n +2 ( D M , ˜ φ fund ) are knotting invariants up to isomorphism of homologygroups generated by an automorphism of a fundamental quandle (resp.fundamental shadow quandle). M.Eisermann proved that in the classi-cal case, the homology class of the fundamental cycle of a nontrivialknot is a generator of H Q ( X ( K )) = Z , [Eis-1] . Roseman moves are preserving homology classes offundamental cycles
We deal in this section with the main result of our paper, about(co)cycle invariants of knottings. We start by describing precisely the If we change a base point in the definition of the fundamental rack we makean inner automorphism on it (i.e. generated by X action) reflecting similarity withfundamental group), [Joy-2]. This class is called in [Eis-1] the orientation class of K . ocycle invariants of codimension 2 embeddings case of a pass move, R , (generalization of the third Reidemeister move),that is a move of type S ( c, n + 2 ,
0) in notation of [Ros-1, Ros-2]; seeSection 6 (we write R ∈ S ( c, n + 2 , Colorings and homology under Roseman moves.
Fix a quan-dle ( X ; ∗ ). We already established bijection, for any Roseman move R between sets of colorings of D M and the set of colorings of RD M , The-orem 3.4. We denote this bijection by R , so R : Col X ( D M ) → Col X ( RD M ). In fact, R is X -quandle-sets isomorphism, that is itpreserves the right multiplication by elements of X ( i.e. R ( φ ∗ x ) = R ( φ ) ∗ x ). . Theorem 4.1.
For a fixed φ ∈ Col X ( D M ) the cycles c n +1 ( D M , φ ) and c n +1 ( RD M , R ( φ )) are homologous in H Qn +1 ( X ) . Similarly for ˜ φ ∈ Col
X,sh ( D M ) , the cycles c n +2 ( D M , ˜ φ ) and c n +2 ( RD M , R ( ˜ φ )) arehomologous in H Qn +2 ( X ) . The main, and, as we see later, essentially the only one nontrivial tocheck is the pass move R of type S ( c, n + 2 , Lemma 4.2.
The pass move R ∈ S ( c, n + 2 , preserves the homologyclass of c n +1 ( L ) and c n +2 ( L ) . To be precise let φ be a fixed coloringof D M and R ( φ ) the corresponding coloring of RD M then the cycles c n +1 ( D M , φ ) and c n +1 ( RD M , R ( φ )) are homologous in H Qn +1 ( X ) . Sim-ilarly for ˜ φ ∈ Col sh,X ( D M ) , the cycles c n +2 ( D M , ˜ φ ) and c n +2 ( RD M , R ( ˜ φ )) are homologous in H Qn +2 ( X ) .Proof. We show the result for all possible types of pass moves (includingall possible co-orientation of sheets) at once. We start from n +2 sheets(hypersurfaces) in R n +2 with arbitrary co-orientation, intersecting in apoint p , and the direction of time ~t in general position to co-orientationvectors.Before we give technical details, we first use a simple visualization ofour proof:On each side of the move, say for t = − t = 1 we have n + 2 cross-ings ( p , ..., p n +2 ) and ( p ′ , ..., p ′ n +2 ) respectively. Each crossing repre-sents the intersection of n +1 sheets in R n +1 that is p i is the intersectionof all sheets V , ..., V n +2 but V i at t = − p ′ i is the intersection ofall sheets but V i at t = 1. Furthermore, we have sgn ( p i ) = sgn ( p ′ i )for any i ≥
1. We concentrate on the case of a shadow coloring˜ φ of D M (the non shadow case being similar). The weights associ-ated to p i and p ′ i are sgn ( p i ) d ( ∗ ) i q sh ( p ) and sgn ( p ′ i ) d ( ∗ ) i q sh ( p ) (the or-der depends on co-orientation ~n i of V i ). Furthermore, the sign of p i J´ozef H. Przytycki and Witold Rosicki is ( − n − i sgn ( ~n i · ~t ) that is the sign depend on whether ~n i agrees ordisagrees with ~t (that is the scalar product ~n i · ~t is positive or nega-tive). In effect c n +2 ( p i ) − c n +2 ( p ′ i ) = ǫ ( − − i ( d ( ∗ ) i − d ( ∗ ) i )( q sh ( p ), where ǫ = ±
1, and in effect c n +2 ( D M ) − c n +2 ( RD M ) = ± ∂ n +3 ( q sh ( p )). There-fore c n +2 ( D M ) − c n +2 ( RD M ) is homologous to zero in H n +2 ( X ) asneeded . Similarly c n +1 ( D M , φ ) − c n +1 ( RD M , R ( φ )) = ± ∂ n +2 ( q ( p )).This follows also directly by using Observation 2.3. (cid:3) qq q q * q q q * q p p q p q q q q * q p’ p’ p’ q q q * q * q * q q * q q q * q q * q t n i t sh 0 q (p)= (q ,q ,q ,q ) Projections before and after move normal coorientation vectorsand time vector
Figure 4.1; From isotopy to pass move ( n = 1 case) c ( p , p , p ) = ( q , q , q ) − ( q , q , q ) + ( q , q , q ) = ∂ ( ∗ ) ( q , q , q , q ) c ( p ′ , p ′ , p ′ ) = ( q ∗ q , q , q ) − ( q ∗ q , q ∗ q , q ) + ( q ∗ q , q ∗ q , q ∗ q ) = ∂ ( ∗ ) ( q , q , q , q ) We were informed by Scott Carter that this observation was crucial in the defi-nition of rack homology by Fenn, Rourke and Sanderson. In particular, the relationof the generalized Reidemeister move can be read from the boundary of singularityof one dimension higher. We deal then with a point ˆ p of multiplicity n + 2 and wechoose any time vector ~t in general position to normal vectors of n + 1-dimensionalhyperplanes. We shadow color neighborhood of ˆ p so that q sh (ˆ p ) = ( q , q , ..., q n +2 ).Then we analyze face maps d ∗ i ( q sh (ˆ p )) and d ∗ i ( q sh (ˆ p )) and recognize q sh of points p ,..., p n +2 and p ′ , ..., p ′ n +2 at crossection at t = − t = 1. Figure 4.1 illustrateit for n = 1. ocycle invariants of codimension 2 embeddings Now we can complete the proof of Theorem 4.1:We use Roseman moves in more substantial way then before. Detailsof Roseman theory is given in Section 6 were we follow [Ros-1, Ros-2,Ros-3]. Here we give a short description referring often to that section.D. Roseman proved that for any n there is a finite number of moveson link diagrams in R n +1 so that if two diagrams F n and F n representambient isotopic links in R n +2 then our diagrams are related by a finitenumber of Roseman moves. For n = 1 , n = 1 these are classical Reidemeister moves).We are showing that any Roseman move is preserving the homologyclass of c n +1 ( D M ) and c n +2 ( D M ). Because only crossings of multiplicity n +1 are contributing to cycles c n +1 ( D M ) and c n +2 ( D M ), thus is sufficesto consider only those Roseman moves which involve singularities ofmultiplicity n + 1 before or after the move. A precise definition ofRoseman moves and their properties is given in Section 6 and herewe need only the fact that there are exactly three types of moves ofinterest:(i) A move of type S ( c, n + 2 ,
0) which we analyzed in Lemma 4.2called the pass move or maximal crossing move or the general-ized third Reidemeister move.(ii) A move of type S ( c, n + 1 ,
0) (or its inverse a move of type S ( c, n + 1 , n = 2 , e ) in [Ros-1]).In the isotopy the arc of points of multiplicity n + 1 joinsthese crossing points, and they have opposite signs. Further-more, up to sign, this crossings have the same contributions to c n +1 ( D M , φ ) (and c n +2 ( D M , ˜ φ )). Thus in the state sum of Def-initions 1.16 and 1.17 they do cancel.Notice that we can interpret our situation as a special caseof considerations in Subsection 2.2 (consider Figure 2.4 with V vp = − V vp and a = c as describing the knotting diagrambefore the move).(iii) A move of type S ( m, (1 , n − , , p ) (with p = 0 or 1), where oneside of the isotopy has a point of multiplicity n +1 (compare themove (f) in the case of M in R and the move ( ℓ ) in the case J´ozef H. Przytycki and Witold Rosicki of M in R [Ros-1]). Then the branch set B is the boundaryof the lower decker set D − so the lower decker set does notseparate the regions; thus both sides of this set have the samecolor. Therefore the chains corresponding to the crossing ofmultiplicity n +1 are degenerate in C Rn +1 ( X ) and C Rn +2 ( X ), thusthese chains do not contribute to quandle homology H Qn +1 ( X )and H Qn +2 ( X ), respectively.This complete our proof of Theorem 4.1If M = M ∪ M ∪ ... ∪ M k we can generalize Theorem 4.1 for anon-shadow coloring of D M (we use notation of Definition 1.18). Ourproof of Theorem 4.1 also work in this case: Corollary 4.3.
For a fixed φ ∈ Col X ( D M ) and a Roseman move R the difference of chains before and after the move, c n +1 ( D M , φ, i )) − c n +1 ( RD M , R ( φ, i )) is a boundary (so homologically trivial). Cycle invariants of knottings.
To obtain invariants of knot-tings using Theorem 4.1 we can either sum over all colorings of thecycles c n +1 ( D M , φ ) or take them as a set with multiplicity (in ordernot to loose an information that some colorings have the same cycle): Theorem 4.4.
Let ( X ; ∗ ) be a fixed quandle, f : M → R n +2 be an n -knotting, π : R n +2 → R n +1 a regular projection, and D M the knottingdiagram. We use the notation [ c ] for a homology class of a cycle c . (1) Let [ c n +1 ( D M ) , φ ] denotes the homology class of the cycle c n +1 ( D M , φ ) .For a finite X the state sum, defined below [ c n +1 ( D M )] = X φ ∈ Col X ( D M ,φ ) [ c n +1 ( D M ) , φ ] in the group ring ZH n +1 ( X ) is a topological invariant of a knot-ting M . Thus we can denote this invariant by c n +1 ( M ) and callit the (non-shadow) cycle invariant of a knotting f : M → R n +2 (or shortly of M ). (2) The reduced (non-shadow) cycle invariant of the knotting f : M → R n +2 c redn +1 ( M ) = [ c redn +1 ( D M )] = X φ ∈ Col red,X ( D M ) [ c n +1 ( D M , φ )] is a topological invariant. Notation is explained as follows. Wesum here over smaller number of colorings using the fact thatset of colorings Col X ( D M ) is an X -quandle-set and as proven in There is a misprint in [Ros-1] page 353; it should be S ( m, (1 , , ,
0) or S ( m, (1 , , ,
1) in place of S ( m, (1 , , ,
0) or S ( m, (1 , , , ocycle invariants of codimension 2 embeddings Observation 1.5 c n +1 ( D M ) , φ ) is homologous to c n +1 ( D M ) , φ ∗ x ) ,for any x ∈ X . Thus we take Col red,X ( D M ) to be the subset ofall X coloring, one coloring from every orbit. Even if we canhave various choices for Col red,X ( D M ) the resulting c redn +1 ( M ) =[ c redn +1 ( D M )] is well defined. (3) Let [ c n +2 ( D M ) , ˜ φ ] denotes the homology class of the cycle c n +2 ( D M , ˜ φ ) .For a finite X the state sum [ c n +2 ( D M )] = X ˜ φ ∈ Col sh,X ( D M ) [ c n +2 ( D M , ˜ φ )] in the group ring ZH n +2 ( X ) is a topological invariant of a knot-ting M . Thus we denote this invariant by c n +2 ( M ) and call itthe shadow cycle invariant of a knotting M . (4) The reduced shadow cycle invariant of the knotting M is a topo-logical invariant c redn +2 ( M ) = [ c redn +2 ( D M )] = X ˜ φ ∈ Col red,sh,X ( D M ) [ c n +2 ( D M , ˜ φ )] . (5) If in any sum of (1)-(4) we replace sum by a set with multi-plicity, we obtain topological invariants, c setn +1 ( M ) , c red,setn +1 ( M ) , c setn +2 ( M ) , and c red,setn +2 ( M ) respectively (we allow X to be infinitehere). Cocycle invariants of knottings.
In this subsection we refor-mulate our main result in the language of cocycles and cohomology. Fora fixed quandle ( X ; ∗ ) and fixed cocycles in C n +1 ( X, A ) and C n +2 ( X, A )we obtain directly cocycle invariants of n -knotting. It generalizes thecase of n = 1 , n = 3checked in [Rosi-2]. We start from the definition which involves dia-grams. Definition 4.5.
Let ( X ; ∗ ) be a fixed quandle, f : M → R n +2 an n -knotting, π : R n +2 → R n +1 a regular projection, and D M the knottingdiagram. (1) For a fixed coloring φ ∈ Col X ( D M ) and ( n + 1) -cocycle Φ : Z X n +1 → A we define the value Φ( D M , φ ) ∈ A by Φ( D M , φ ) = Φ( c n +1 ( D M , φ )) = X p Φ( c n +1 ( p, φ )) , where Φ( c n +1 ( p, φ )) is a Boltzmann weight of the crossing p ofmultiplicity n + 1 , and the sum is taken over all crossings of D M . J´ozef H. Przytycki and Witold Rosicki (2)
We can also take into account the fact that M is not necessaryconnected (following [CENS, CKS-3] in the case n = 1 ). Thatis if M = M ∪ M ∪ ... ∪ M k , we can take the sum from (1) notover all crossings p of multiplicity n + 1 but only those whichhave M i on the bottom of the crossings. Let us denote such aset of crossings by T i . Then we define Φ( D M , φ, i ) = X p ∈T i Φ( c n +1 ( p, φ ))(3) For a fixed shadow coloring ˜ φ ∈ Col
X,sh ( D ( M )) and ( n + 2) -cocycle ˜Φ : Z X n +1 → A , we define the value ˜Φ( D M , ˜ φ ) ∈ A bythe formula: ˜Φ( D M , ˜ φ ) = ˜Φ( c n +2 ( D M , ˜ φ )) = X p ˜Φ( c n +2 ( p, ˜ φ )) . Theorem 4.6. (Cocycle invariants) Consider a knotting f : M → R n +2 and fix a quandle ( X ; ∗ ) and ( n + 1) -cocycle Φ : Z X n +1 → A and ( n + 2) -cocycle ˜Φ : Z X n +2 → A . (1) For a fixed colorings φ ∈ Col X ( D M ) , the element Φ X ( D M , φ ) ∈ A is preserved by any Roseman move R that is Φ X ( D M , φ ) =Φ X ( RD M , R ( φ )) in A . (2) If M = M ∪ M ∪ ... ∪ M k then the conclusion of (1) holds alsofor Φ( D M , φ, i ) , that is Φ X ( D M , φ, i ) = Φ X ( RD M , R ( φ ) , i ) in A . (3) For a fixed shadow coloring ˜ φ ∈ Col
X,sh ( D M ) , the element ˜Φ X ( D M , ˜ φ ) ∈ A is preserved by any Roseman move, that is ˜Φ X ( D M , ˜ φ ) = ˜Φ X ( RD M , R ( ˜ φ )) in A .Proof. Theorem 4.6 follows directly from the analogous result for ho-mology. We should stress that we do not need here the property that c n +1 ( D M , φ, i ) are cycles, as we can evaluate a cocycle on any chain.Furthermore, we can work also with tangles not only with knottingdiagrams. (cid:3) To produce invariant of a knotting we should make our invariants ofdiagram independent on a choice of a coloring (and use Theorems 4.4and 4.6). As before, two natural solutions are to take a set with mul-tiplicity of invariants over all colorings or, for finite X , sum invariantsover all colorings as usually is done in statistical mechanics. Definition 4.7.
Let ( X ; ∗ ) be a fixed quandle, f : M → R n +2 be an n -knotting, π : R n +2 → R n +1 a regular projection, and D M the knottingdiagram. ocycle invariants of codimension 2 embeddings (1) Let Φ be a fixed cocycle in C n +1 Q ( X ) . Then: (i)Φ X ( D M ) = X φ ∈ Col X ( D M ) Φ X ( D M , φ ) = X φ Y p Φ( c n +1 ( p, φ )) , where X is a finite quandle. Here we have classical cocycleinvariant in ZA , written as a state sum ( A n a multi-plication notation) and generalizing cocycle invariants of [CKS-3] . (ii) Φ X ( D M , i ) = X φ ∈ Col X ( D M ) Φ X ( D M , φ, i ) , where M = M ∪ M ∪ ... ∪ M k and we consider only thecrossings which have M i on the bottom of the crossings.For n = 1 , this invariant of M with ordered componentswas described in [CENS, CKS-3] . (iii) Φ redX ( D M ) = X φ ∈ Col
X,red ( D M ) Φ X ( D M , φ ) . Here we use the fact that X acts on Col X ( D M ) and we canchoose in the sum one representative from any orbit. Anychoice is good (see Observation 1.5) and we write for thechosen subset Col
X,red ( D M ) , (iv) Φ redX ( D M , i ) = X φ ∈ Col
X,red ( D M ) Φ X ( D M , φ, i ); here we reduce crossings as in (ii) and colorings as in (iii). (v) Without restriction to finite X we can repeat all defini-tions of (i)-(iv) by considering, in place of the sum overcolorings, the set of invariants indexed by colorings (thuswe have a set with multiplicities, or better cardinalities ofelements if X is infinite). We get Φ set ( D M ) , Φ set ( D M , i ) , Φ redset ( D M ) , and Φ redset ( D M , i ) , respectively. (2) For a fixed ( n + 2) -cocycle ˜Φ : Z X n +2 → A , we define: For example, for the trivial knotting S n ⊂ R n +2 , finite X and any cocycle,we have Φ X ( S ) = | X | · redX ( S ) = |O r | ·
1, where O r is the set of orbit ofthe action of X on X on the right. Furthermore, in our notation 1 is a zero of anabelian group written multiplicatively, and |O r | · Z ( ZX n +1 ). J´ozef H. Przytycki and Witold Rosicki (i)˜Φ X ( D M ) = X ˜ φ ∈ Col
X,sh ( D M ) ˜Φ X ( D M , ˜ φ ) = X ˜ φ Y p ˜Φ( c n +2 ( p, ˜ φ )) , where X is a finite quandle. Here we have classical shadowcocycle invariant in ZA (ii) ˜Φ X,red ( D M ) = X ˜ φ ∈ ˜ Col
X,red ( D M ) ˜Φ X,red ( D M , ˜ φ ) . (iii) Without restriction on X to be finite, we can repeat defini-tions of (i) and (ii) by considering, in place of the sum overcolorings, the set with multiplicity of invariants indexed bycolorings. We get ˜Φ set ( D M ) and ˜Φ set,red ( D M ) . Theorem 4.8. (Cocycle invariants) Consider a knotting f : M → R n +2 and for a fixed quandle ( X ; ∗ ) , quandle cocycles Φ : Z X n +1 → A and ˜Φ : Z X n +2 → A . Then: (1) If X is finite then Φ X ( M ) Φ redX ( M ) , ˜Φ X ( M ) , and ˜Φ redX ( M ) aretopological invariants of the knotting (i.e. independent on adiagram, invariant under Roseman moves). They are calledcocycle, and shadow cocycle invariants of a knotting M . (2) For any X , Φ set ( M ) , Φ redset ( M ) , ˜Φ set ( M ) , and ˜Φ redset ( M ) are topo-logical invariants of the knotting M . (3) If M = M ∪ ... ∪ M n and X is finite then Φ X ( M, i ) and Φ redX ( M, i ) are topological invariants of the knotting M with or-dered components. Similarly, for any X the sets with multiplic-ity Φ setX ( M, i ) and Φ setX,red ( M, i ) are topological invariants of theknotting M with ordered components. (4) We can make invariants of (3) to be independent on the orderof components if we take the set with multiplicity of invariantsover all i .Proof. It follows directly from Theorems 4.4 and 4.6. Notice here thatfor shadow coloring the idea of considering M = M ∪ M ∪ ... ∪ M k andonly crossings where M i is on the bottom will not work as d ( ∗ )2 usuallydiffers from d ( ∗ )2 , thus c redn +2 ( p i ) − c redn +2 ( p ′ i ) is not necessary homologicalto zero.. In the non-shadow case we only needed d ( ∗ )1 = d ( ∗ )1 . (cid:3) ocycle invariants of codimension 2 embeddings Twisted (co)cycle invariants of knottings
Twisted homology (and cohomology) was introduced in [CENS].Most of the results of the paper generalize, without much changes tothe twisted case so we give a concise explanation.
Definition 5.1. (i)
The twisted chain complex of a shelf ( X ; ∗ ) isgiven by the chain modules C Tn ( X ) = Z [ t ± ] X n (that is a freemodules with basis X n and with coefficients in a ring of Laurentpolynomials in variable t ), and the chain map ∂ T = t∂ ∗ − ∂ ∗ .Recall that: ∂ ∗ ( x , ..., x n ) = n X i =1 ( − i ( x , ..., x i − , x i +1 , ..., x n ) , and ∂ ∗ ( x , ..., x n ) = n X i =1 ( − i ( x ∗ x i , ..., x i − ∗ x i , x i +1 , ..., x n ) . (ii) If ( X ; ∗ ) is a spindle (e.g. a quandle) we define as in the un-twisted case the degenerate and quandle homology. Thus as be-fore we consider H T Wn ( X ) for W = R, D and Q . (iii) The cohomology H nT W ( X, A ) are defined in a standard way with A being an Z [ t ± ] -module. The theory of cocycle invariants, for n=1 or 2, was introduced in[CES-1] for n = 1 ,
2. We give definition for any n -knotting below.Our description follow [CKS-3], the important tool we use is the clas-sical Alexander numbering of chambers in ( R n +1 , πf ( M )) (see [CKS-0,CKS-3]). Our version of the definition refers to shadow colorings byan (extended) shift rack structure on integers with infinity ( Z ∪ ∞ ; ∗ s )where a ∗ s b = a + 1 (in particular ∞ ∗ b = ∞ ). Definition 5.2. (i)
Let X be a set and f : X → X a bijection witha fixed point b . We define a rack ( X ; ∗ f ) by a ∗ f b = f ( a ) . Thenfor a given knotting diagram D M the shadow rack coloring ofchambers of the knottings, is called the generalized Alexandernumbering. More precisely, we color regions of the diagramtrivially by b , choose one chamber and color it by an elementof X − b and the resulting shadow coloring of chambers is ageneralized Alexander numbering. (ii) The Alexander coloring of Chambers (e.g. [CKS-3] ) starts fromthe rack ( Z ∪ ∞ ; ∗ s ) and the unbounded chamber is colored by . Definition 5.3. (Twisted chains of knotting) Let f : M → R n +2 be an n -knotting, π : R n +2 → R n +1 a regular projection, and D M the knotting J´ozef H. Przytycki and Witold Rosicki diagram. Furthermore, fix a rack or quandle X , a coloring φ : R → X ,and a shadow coloring ˜ φ . (1) If p is a crossing of multiplicity ( n + 1) then we define thechain (twisted Boltzmann weight) associated to p as c Tn +1 ( p, φ ) = t − k ( R ) c n +1 ( p, φ ) , where k ( R ) is the Alexander numbering ofthe source region in the neighborhood of p and c n +1 ( p, φ ) is theuntwisted Boltzmann weight. (2) In the case ˜ φ is the shadow coloring we define a twisted shadowBoltzmann weight by: c Tn +2 ( p, ˜ φ ) = t − k ( R ) c n +2 ( p, ˜ φ ) , where c n +2 ( p, ˜ φ ) is the untwisted shadow Boltzmann weight associated to p . (3) The twisted chain associated to the diagram D M is c Tn +1 ( D M , φ ) = X p ∈ Crossings c Tn +1 ( p, φ ) . (4) Finally, we sum over all X colorings of D M so the result isin the group ring over C Tn +1 ( X ) (in fact, it is in the group ringof H Tn +1 ( X ) ) It is convenient here to use multiplicative notationfor chains so that c Tn +1 ( D M , φ ) = (Π p ( q , ..., q n +1 ) sgnp ) t − k ( R andthen c Tn +1 ( D M ) = X φ c Tn +1 ( D M , φ ) . (5) We define the twisted shadow chain associated to the diagram D M in an analogous manner: c Tn +2 ( D M , ˜ φ ) = X p ∈ crossings c Tn +2 ( p, ˜ φ ) . Then we sum over all colorings to get: c Tn +2 ( D M ) = X ˜ φ c Tn +2 ( D M , ˜ φ ) . (6) As in untwisted version we can consider smaller sum by takinginto account only one element from each orbit of action by X onthe space of colorings. However we should be careful here aboutwhich action we consider because the action ( x , ..., x n ) ∗ x isequal on homology to t · Id according to Observation 1.5 Thuswe should change this action to ( x , ..., x n ) → t − ( x , ..., x n ) ∗ x .We obtain then reduced versions of (4) and (5). (7) Each of the above has its cocycle version as long as we choose atwisted ( n + 1) − and ( n + 2) -cocycles in C n +1 T ( X ) and C n +2 T ( X ) ,respectively. ocycle invariants of codimension 2 embeddings Most of the results as in Theorems 4.1, 4.6, and 4.8 generalizes with-out any problem to twisted (co)homology. We give two examples below.
Theorem 5.4.
For a fixed φ ∈ Col X ( D M ) the chain c Tn +1 ( D M , φ ) is acycle and it is homologous to c Tn +1 ( RD M , R ( φ )) in H T Qn +1 ( X ) , where R is any Roseman move on a diagram D M . Similarly for a fixed shadowcoloring ˜ φ ∈ Col
X,sh ( D M ) , the chain c Tn +2 ( D M , ˜ φ ) is a cycle and it ishomologous to c Tn +2 ( RD M , R ( ˜ φ )) in H T Qn +2 ( X ) . The main, nontrivial Roseman move to check is the pass move R oftype S ( c, n + 2 , Theorem 5.5. (Twisted cocycle invariants) Consider a knotting f : M → R n +2 and fix a quandle ( X ; ∗ ) and ( n + 1) -twisted cocycle Φ T : Z [ t ± ] X n +1 → A and ( n + 2) -cocycle ˜Φ T : Z [ t ± ] X n +2 → A where A isa [ t ± ] -module. (1) For a fixed colorings φ ∈ Col X ( D M ) , the element Φ X ( D M , φ ) =Φ T ( c Tn +1 ( D M , φ ) ∈ A is preserved by any Roseman move R thatis Φ X ( D M , φ ) = Φ X ( RD M , R ( φ )) in A . (2) For a fixed shadow coloring ˜ φ ∈ Col
X,sh ( D M ) , the element ˜Φ X ( D M , ˜ φ ) = ˜Φ T ( c Tn +2 ( D M . ˜ φ ) ∈ A is preserved by any Rose-man move, that is ˜Φ X ( D M , ˜ φ ) = ˜Φ X ( RD M , R ( ˜ φ )) in A . (2) We can sum now over coloring of a finite quandle X , or sumover reduced colorings, or just take a set over coloring, to gettwisted cocycle invariants of a knotting. Remark 5.6.
If we work with racks and rack (or degenerate) homol-ogy we cannot ignore degenerate elements, so the Roseman move oftype S ( m, (1 , n − , , p ) (generalized first Reidemeister move) cannotbe performed on the diagrams without possibly changing (co)homologyclass of (co)cycles. For other Roseman moves however all our re-sults work well. Thus we have (co)cycle invariants of diagrams of n -knottings up to all Roseman moves except moves of type S ( m, (1 , n − , , p ) , for any rack. General position and Roseman moves in codimension 2
An important tool our work is given by the work of Roseman ongeneral position of isotopy on moves in co-dimension two and the moveshe developed. The next subsections follow [Ros-1, Ros-2, Ros-3]. Wehave used these notion in the paper; here is more formal development.Before we can define Roseman moves we need several definitions. J´ozef H. Przytycki and Witold Rosicki
General position.
Let M = M n be a closed smooth n -dimensionalmanifold and f : M → R n +2 its smooth embedding which is called asmooth knotting. Define π : R n +2 → R n +1 given by π ( x , ...., x n +1 , x n +2 ) =( x , ...., x n +1 ) to be a projection on the first n + 1 coordinates. The pro-jection of the knotting is the set M ∗ = πf ( M ). Crossing set D ∗ of theknotting, is the closure in M ∗ of the set of all points x ∗ ∈ M ∗ suchthat ( πf ) − ( x ∗ ) contains at least two points.We define the double point set D as D = ( πf ) − ( D ∗ ). The branch set B of f is the set of all points x ∈ M such that πf is not an immersionat x . In general, if A ⊂ M then A ∗ denote πf ( A ). Definition 6.1.
Let f : M → R n +2 be a smooth knotting with branchset B and double point set D . We say that f is in general position withrespect to the projection π if the following six conditions hold:(1) B is a closed n − dimensional submanifold of M .(2) D is a union of immersed closed ( n − -dimensional submanifoldsof M n with normal crossings. Denote the set of points of D wherenormal crossings occur as N and call this the self-crossing set of D .(3) B is a submanifold of D and for any b ∈ B there is a small ( n − -dimensional open sub-disk V with b ∈ V , V ⊆ D such that V − B hastwo components V and V , each of which is an ( n − -disk which isembedded by the restriction of π ◦ f but with V ∗ = V ∗ (Figure 2.1).(4) B meets N transversely.(5) ( π ◦ f ) | B is an immersion of B with normal crossings.(6) The crossing set of B ∗ is transverse to the crossing set of ( D − B ) ∗ . b * B * V * b VV VB = V * Figure 6.1; projecting (folding) of V = V ∪ V ∪ ( B ∩ V ) onto( V − B ) ∗ = V ∗ = V ∗ Theorem 6.2.
Given a knotting f : M n → R n +2 we may isotope f toa map which is in general position with respect to the projection π . Similarly we define what it means for an isotopy F : M × I → R n +2 × I to be in general position with respect to the projection π ′ = π × Id . ocycle invariants of codimension 2 embeddings It is just the previous definition for general position of a codimensiontwo knotting except that B and D may have nonempty boundary. Inparticular, F = F/ ( M × { } ), F = F/ ( M × { } ) : M → R n +2 aresmooth knottings in general position.6.2. Arranging for moves.
We put on our isotopy additional condi-tions called arranging for moves [Ros-3]. Roughly speaking, we filtrate D ∗ of F : M × I → R n +2 × I in such a way that the projection p : R n +1 × I → I restricted to any component of each stratum, Q ( i ) , isa Morse style function. Definition 6.3 (Roseman) . Let q be a proper immersion of a manifold Q in R n +1 × I . We say that q ( Q ) is immersed in Morse style if pq isa Morse function, where p : R n +1 × I → I . We assume that a Morsefunction has critical points on different levels. For details see [Ros-3]. Here we just mention that Q (0) is the crossingset of B ∗ , Q (1) = B ∗ , the projection of the branch set of F . Q (2) = D ∗ ,generally, Q ( k ) , k > D ∗ such that F ′ = π ′ F is at least k to 1. Roseman proves: Theorem 6.4 (Roseman) . Any isotopy F : M × I → R n +2 × I can bearranged for moves. Listing of moves after Roseman.
The standard set of moves M n is described as follows:Fix a dimension n and suppose we are given an isotopy F : M n × I → R n + × I which is arranged for moves. This gives a sequence ofelementary singularities. Each singularity will correspond to a standardlocal knot move in our collection M n .In the notation which follows, we consider three general types ofpoints:(1) branch type: critical points of B ∗ and self-crossing points of B ∗ for which we use the letter b (2) crossing type: critical points of D ∗ and the crossing set of D ∗ which do not belong to B ∗ for which we use the letter c .(3) mixed type: critical points which are in the crossing set of D ∗ and are in B ∗ , a “mixed” type for which we use the letter m .The first collection of branch type points is denoted { S ( b, k, p, q ) } . If x ∗ ∈ D ∗ is such a singular point, where D ∗ is the crossing set of anisotopy F : M n × I → R n +2 × I , let k denote the number of points of F ′− ( x ∗ ). In our case the branch point set B ∗ of F , is codimension 2in M n × I that is it is of dimension n − J´ozef H. Przytycki and Witold Rosicki
If in projection this branch set intersects itself generically, the self-intersection set will have dimension n −
4. It follows that 1 ≤ k ≤ n − p is the index of the singularity. The integer q has range0 ≤ q ≤ k and might be called transverse index of this critical point.This is defined as follows. If x ∈ B consider a curve δ in D transverseto B (recall that B has codimension one in D ) so that δ ∗ is, exceptfor the point x , the two-to-one image of δ . In the I direction, theimage of this curve has a local maximum or a local minimum at b .Now suppose b ∗ is a k -fold point of B ∗ then we have k such curves toconsider. The number q is the number of those curves for which wehave a local maximum. Of course, it follows that k − q of the curveshave a local minimum.The next collection of crossing type singularities is denoted by { S ( c, k, p ) } .If x ∗ is such a singularity, k denotes the cardinality of F ′− ( x ∗ ). Thus k is an integer 2 ≤ k ≤ n + 2. Furthermore, on this set of points, where F ′ is k -to-one, x ∗ is a critical point in the I direction, of index p . Asingle point has index 0 by convention.Finally S ( m, ( i, j ) , p, q ) denotes a mixed singularities. Such a sin-gularity x ∗ has F ′− ( x ∗ ) consisting of i + j points, where exactly i ofthese points are in B . Again p is the index of the singularity and q isan integer, 0 ≤ q ≤ i which is the number of local maxima we get bylooking at those i arcs transverse to B at the points of F ′− ( x ∗ ) ∩ B .7. A knotting M n f → F n +1 × [0 , π → F n +1 The Roseman (local) moves can be used for any n -knotting f : M n → W n +2 by the following classical PL-topology result following from The-orem 6.2 in [Hud] (we will use it in a smooth case which can be derivedusing Whitehead results on triangulation of smooth manifolds). Lemma 7.1. If C is a compact subset of a manifold W and F : W × I → W is the isotopy of W then there is another isotopy ˆ F : W × I → W such that F = ˆ F , F /C = ˆ F /C and there exists a number N such that the set { x ∈ W | ˆ F / { x } × ( k/N, ( k + 1) /N ) is not constant } sits in a ballembedded in W . Let f : M n → F n +1 ¯ × [0 ,
1] be an n -knotting where F n +1 is an ( n +1)-dimensional manifold and F n +1 ¯ × [0 ,
1] is an [0 , F n +1 (trivial bundle if F n +1 is oriented and the twisted [0 , F n +1 if F n +1 is unorientable. In both cases the manifold is oriented).Let π : F n +1 ¯ × [0 , → F n +1 . By Lemma 7.1 an embedding f can beassumed to be in general position with respect to π and every ambient ocycle invariants of codimension 2 embeddings isotopy of a knotting can be decomposed into Roseman moves (on D M ). If π ( F n +1 ) = 0 then essentially all results of the paper can bealso proven for the knotting (we need W = F n +1 ¯ × [0 ,
1] to be simpleconnected in Lemma 1.12, Remark 3.3, and Theorem 3.4).
Remark 7.2.
If we do not assume that π ( F n +1 ) = 0 in the case of W n +2 = F n +1 × [0 , , we can still develop the theory of (co)cycle invari-ants by following [FRS-2] where the notion of a reduced fundamentalrack is developed (essentially one kills the action of π ( F n +1 ) ). Thenthe reduced fundamental rack (or quandle) is, according to Corollary 3.5of [FRS-2] , the same as the fundamental rack (or quandle) obtained byrack (or quandle) abstract coloring of any diagram of the knotting. Speculation on Yang-Baxter homology and invariantsof knottings
Yang-Baxter operator can be thought as a direct generalization ofright self-distributivity when we go from the category of sets to thecategory of k -modules.We follow here [Leb-1, Leb-2, Prz-1, Prz-2] describing the classicalcase n = 1.First we note how to get Yang-Baxter operator from a right self-distributive binary operation. Let ( X ; ∗ ) be a shelf and kX be a freemodule over a commutative ring k with basis X (we can call kX a linear shelf ). Let V = kX , then V ⊗ V = k ( V ) and the operation ∗ yields a linear map Y = Y ( X ; ∗ ) : V ⊗ V → V ⊗ V given by Y ( a, b ) =( b, a ∗ b ). Right self-distributivity of ∗ yields the equation of linear maps V ⊗ V ⊗ V → V ⊗ V ⊗ V as follows:(1) ( Y ⊗ Id )( Id ⊗ Y )( Y ⊗ Id ) = ( Id ⊗ Y )( Y ⊗ Id )( Id ⊗ Y ) . In general, the equation of type (1) is called a Yang-Baxter equationand the map Y a Yang-Baxter operator. We also often require that Y is invertible. For example if Y is given by invertible ∗ , then Y ( X ; ∗ ) isinvertible with Y − X ; ∗ ) ( a, b ) = ( b ¯ ∗ a, a ).In our case Y ( X ; ∗ ) permutes the base X × X of V ⊗ V , so it is called apermutation or a set theoretical Yang-Baxter operator. Our distribu-tive homology, in particular our rack homology ( C n , ∂ Y = ∂ ( ∗ ) − ∂ ( ∗ ) )can be thought of as the homology of Y . It was generalized fromthe Yang-Baxter operator coming from a self-distributive ∗ to any settheoretical Yang-Baxter operator (coming from biracks or biquandles),[CES-2]. For a general Yang-Baxter operator, there is no general ho-mology theory (however, compare [Eis-1, Eis-2]). The goal/hope is to J´ozef H. Przytycki and Witold Rosicki define homology for any Yang-Baxter operator and develop the homo-logical invariants of n -knottings (it is done for n = 1 and a set theo-retical Yang-Baxter equation in [CES-2]). The simple visualization ofthe distributive face map d ( ∗ ) i from Figure 8.1, observed by I.Dynnikovduring Przytycki’s talks in Moscow in May 2012 (and slightly earlierby Victoria Lebed when she was writing her PhD thesis [Leb-1]), easilygives a hint to homology of set theoretical Yang-Baxter homology, and,partially, to general Yang-Baxter homology (this is studied in [Prz-2]).The homology invariants of n -knotting should follow, and combiningthe method of this paper with [Leb-2, Prz-2] looks rather promising. q iq i−1q 1 q i+1q i*i−1q q nq i+1q iq 1* q n i( )* ... Diagramatic realization of a face map d
Figure 8.1; Diagrammatic visualization of a face map gives hint to Yang-Baxter homology.For a right self-distributive ∗ we have a face map d ( ∗ ) i ( q , ..., q n ) = ( q ∗ q i , ..., q i − ∗ q i , q i +1 , ..., q n ) .We can also interpret the picture to be applicable to Yang-Baxter theoryby using Yang-Baxter operator at each crossing9. Acknowledgments
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Department of Mathematics,The George Washington University,Washington, DC 20052e-mail: [email protected] ,University of Maryland CP,and University of Gda´nskInst.. of Mathematics, University of Gda´nsk,e-mail:,University of Maryland CP,and University of Gda´nskInst.. of Mathematics, University of Gda´nsk,e-mail: