Parity, Skein Polynomials and Categorification
aa r X i v : . [ m a t h . G T ] O c t Parity, Skein Polynomials and Categorification
Aaron KaestnerNorth Park [email protected] H. KauffmanUniversity of Illinois at Chicagokauff[email protected] 15, 2018
Abstract
We investigate an application of crossing parity for the bracket expan-sion of the Jones polynomial for virtual knots. In addition we consider anapplication of parity for the arrow polynomial as well as for the categori-fications of both polynomials. We present a number of examples foundthrough our calculations. We provide tables of calculations for these in-variants on virtual knots with at most 4 real crossings.
Key Words:
Jones Polynomial, Arrow Polynomial, Parity, Categorifica-tion, Khovanov Homology, Virtual Knot, Flat Virtual Knot, Graphical JonesPolynomial, Graphical Arrow Polynomial, Graphical Khovanov Homology
Since the introduction of virtual knot theory, crossing parity has provided avaluable resource for creating invariants. (See Definition 3.1 in the present pa-per for our definition of crossing parity.) For instance, given a virtual knotthe odd writhe [23] (i.e. sum of the signs of the odd crossings) is an easilycomputable invariant. Recently, Manturov introduced a parity version of thebracket polynomial [31] and described how this construction can pass to Kho-vanov homology via a filtration on the space of virtual knots. Our goal hereis to investigate these constructions and show how we can apply a similar con-struction to the arrow polynomial. Along the way we show how known factsabout minimal surface genus for virtual knots extend to the categorical setting.The bracket polynomial (and Jones polynomial) as well as the arrow polyno-mial have a rich history. For an introduction to virtual knot theory we recom-mend [19]. For an introduction to the bracket polynomial we point the reader1o [21] and [22]. Similarly, for the arrow polynomial we recommend the paper[18] as well as the work of Dye and Kauffman in [8] [9]. Note that the arrowpolynomial is equivalent to the Miyazawa polynomial as constructed in [35] and[36]. We use the term arrow polynomial as our definition follows the construc-tion introduced by Dye and Kauffman.The Jones polynomial was categorified by Khovanov in [26] and for an in-troduction to Khovanov homology we point the reader to Bar-Natan [4], [5] andKauffman [20]. Since Khovanov’s seminal work categorification in classical knottheory has been a fruitful topic of research. Notably, Rasmussen [37] used aspectral sequence introduced by Lee [29] to produce a lower bound on the slicegenus for a classical knot. Recently it was also announced by Kronheimer andMrowka [28] that Khovanov homology detects the unknot by showing there isa spectral sequence from (reduced) Khovanov homology to instanton Floer ho-mology. This is exciting news as it provides support for the similar conjecturefor the Jones polynomial.The study of Khovanov homology in relation to virtual knot theory is stillrather new. The introduction of the virtual crossing brings with it a new dia-gram for the unknot which has the property d = 0 (using Khovanov’s originaldefinition) only for coefficients in Z (see Figure 6 of [30]). A wonderful expla-nation of how far we can take Khovanov’s original work is given by Viro in [39].To work around this new problem methods have been introduced by Asaeda,Przytycki, and Sikora [1] as well as Manturov [32]. The Asaeda-Przytycki-Sikoraapproach requires not only a diagram but an embedding in a (fixed) thickenedsurface. On the other hand, Manturov’s version introduces a signed, orientedand ordered basis on the states of the cube complex to make all faces commute.Other than its introduction in [10] and recent review in [19], little has beenwritten about the categorifications of the arrow polynomial. In this regard weproduce lower bounds for the homological width of the fully-graded categorifi-cation and we as extend bounds for the surface genus to the categorical setting.In the Appendix we provide a collection of programs for computing all of theinvariants discussed in this paper. We hope these programs will increase theawareness and interest in the study of these categorifications. In [24] Kauffman introduced virtual knots and links as a natural extensionof classical knots and links. Virtual knot theory can be thought of both as(1) equivalence classes of embeddings of closed curves in a thickened surface(possibly non-orientable) up to isotopy and handle stabilization on the surfaceand (2) the completion of the signed oriented Gauss codes (i.e. an arbitraryGauss code corresponds to a virtual knot while not every Gauss code correspondsto a classical knot.) 2e recall in Figure 1 the Reidemeister Moves for classical knot diagrams.Figure 2 displays the additional Virtual Reidemeister Moves for the theory interms of planar diagrams. Here we have introduced the virtual crossing whichis neither an under-crossing or over-crossing. We represent the virtual crossingby two arcs which cross and have a circle around the crossing point.Figure 1: Reidemeister MovesFigure 2: Virtual Reidemeister MovesNote that the move in Figure 3 is not an equivalence relation for diagrams ofvirtual knots and links. It can be shown that adding this relation and its mirrorimage to the Virtual Reidemeister Moves allows one to unknot any virtual knot.For this reason we refer to this move as the forbidden move . Adding just one ofthese two forbidden moves yields a nontrivial theory called welded knots [12].3igure 3: The “Forbidden Move”
For completeness we recall the construction of the bracket polynomial intro-duced by Kauffman in [21].
Definition 1.1.
Given a diagram D for a virtual knot K the bracket polynomialof K is defined by the relations in Figures 4.
Figure 4: Bracket Polynomial Skein RelationsTo be more precise, suppose D is an n-crossing diagram for the (virtual)knot K. We generate the polynomial by smoothing every crossing in each ofthe two ways possible. The result of each smoothing at a particular crossing ismultiplication by the term, A or A − , following the conventions of Figure 4, andwe take the sum of these two weighted smoothings. For a particular crossingwe call these smoothings the A-smoothing and
B-smoothing respectively.Next let s = ( s , . . . , s n ) where s i ∈ { , } . Define the state of D correspond-ing to s to be the result of applying an B-smoothing at crossing i when s i = 0and an A-smoothing at crossing i when s i = 1. Furthermore define h D | s i tobe the product of A ’s and A − ’s that label the state s multiplied by the loopvalue d k s k where d = ( − A − A − ) and k s k denotes the number of loops in thestate. Here each state is obtained by a choice of smoothing for each crossingand is labeled with the type of smoothing at each of the smoothing sites. Remark 1.1.
This definition of the bracket polynomial is shifted from the stan-dard definition. Here the value of a single loop is d = ( − A − A − ) as opposed to in the standard definition. This has the effect that our definition is d × hh K ii , where hh K ii is the standard definition of the bracket polynomial as in [21]. Definition 1.2.
Let S be the collection of all states of a diagram D for a knotK. We may define the bracket polynomial of K to be h K i = X s ∈S h D | s i
4t follows from this definition that the state sum is well defined and theexpansion identities in Figure 4 follow from the state sum definition.
Definition 1.3.
Given a virtual knot K with diagram D, the normalized bracketpolynomial of K is given by f A ( K ) = f A ( D ) = ( − A ) − ω ( D ) h D i where h D i is the bracket polynomial of D and ω ( D ) = writhe ( D ) = ( positive crossings in D ) − ( negative crossings in D ) . Theorem 1.1.
The normalized bracket polynomial is an invariant of (virtual)knots.Proof.
Here each loop, regardless of virtual crossings, evaluates as the loop value d . See [21] Similarly we recall the construction of the arrow polynomial. Given a diagramD for a virtual knot K the (un-normalized) arrow polynomial of K is defined bythe smoothing relations in Figure 5 analogous to the prior construction of thebracket polynomial and reduction relations in Figure 6.Figure 5: Arrow Polynomial Crossing RelationsNotice that the smoothing relations for the arrow polynomial depend on thesign of the crossing. While we still have an A-smoothing and B-smoothing andwill refer to these choices, we also differentiate the smoothings by whether theyagree with the original (pre-smoothed) orientation of the knot diagram. If theorientations agree we call these oriented smoothings else they are disorientedsmoothings . The disoriented smoothings create cusps in the state which satisfythe relations in Figure 6. Moreover, these cusps introduce an infinite familyof variables { K n } , n ∈ N . That is, we cancel consecutive cusps pointing in thesame direction (locally both inward or outward) and resolve virtual crossingsfollowing the rules in Figure 6. The remaining 2 n alternately-oriented cusps ona loop are counted and we assign the loop the value { K n } when n >
0. We referto these variables as arrow numbers [9] [18].5igure 6: Arrow Polynomial Reduction Relations
Definition 1.4.
Given a virtual knot K with diagram D, the un-normalizedarrow polynomial of K is given by h D i A = X s ∈S h D | s i A where S is the collection of all states of D. Definition 1.5.
Given a virtual knot K with diagram D, the normalized arrowpolynomial of K is given by AP ( K ) = AP ( D ) = ( − A ) − ω ( D ) h D i A where h D i A is the arrow polynomial of D and ω ( D ) = writhe ( D ) = ( positive crossings in D ) − ( negative crossings in D ) . Theorem 1.2.
The normalized arrow polynomial is an invariant of virtualknots.Proof.
See [18].
Remark 1.2.
The Jordan Curve Theorem implies that the arrow polynomial isequivalent to the normalized bracket polynomial for classical knots.
Recall that virtual knots are in 1-1 correspondence with equivalence classes ofknots in thickened oriented surfaces modulo 1-handle stabilization and Dehntwists. This raises the question, given a knot, what is the minimal genus forthis embedding. 6 efinition 1.6.
Given a virtual knot K , the (orientable) surface genus of K , s ( K ) , is the minimal genus of the surface S g such that S ֒ → S g × I correspondsto K . Theorem 1.3. (Theorem 4.5 of [9]) Let K be a virtual knot diagram with arrowpolynomial h K i A . Suppose that h K i A contains a summand with the monomial K e i K e i · · · K e n i n where i j = i k for all j, k in the set { , , . . . , n } . Then n deter-mines a lower bound on the genus g of the minimal genus surface in which Kembeds. That is, if n ≥ then the minimum genus is 1 or greater and for g ≥ if n > g − then s ( K ) > g .Proof. The proof follows from showing that non-zero arrow numbers correspondto essential curves on the surface. The bounds follow from considering themaximal number of non-intersecting essential curves on the surface which donot bound annuli.
Remark 1.3.
In the g = 1 case it can be shown that if there is more than1 non-intersecting essential curve then they must bound an annulus. Hence if h L i A contains a summand with a monomial of the form K i K j with i = j thenthe minimal surface genus is at least 2. Figure 7: Virtual Knot 3.1Following the naming conventions in Jeremy Green’s virtual knot tables [13]we consider virtual knot 3.1 as shown in Figure 7. Figures 8 and 9 show how oneuses the state-sum formulas from the previous sections to arrive at the respectivepolynomials. Note that by the previous theorem the arrow polynomial gives thatthe surface genus of virtual knot 3.1 is at least one. The diagram above hassurface genus 2. 7igure 8: Virtual Knot 3.1 Bracket Polynomial State-Sum
For completeness we recall the definition of Khovanov homology [26] [27]. Ourconstruction follows closely that of Dror Bar-Natan [4] and Kauffman [19]. Forother descriptions of the construction we point the reader to Khovanov [26][27], Wehrli [40], Viro [39], Shumakovitch [38], Elliott [11], Kauffman[19], andManturov [32]. For technical reasons involving the construction of the categori-fication of the arrow polynomial we will take coefficients over the field Z . Itshould be noted that the construction of Khovanov homology for virtual knotscan be extended to arbitrary coefficients following the construction of Man-turov in [32]. As one would expect, Manturov’s definition is equivalent to thatof Khovanov for classical knots.Before we recall the construction, we first rewrite the normalized bracketpolynomial in order to simplify the definition. For a given virtual knot or linkK with corresponding diagram D, let c ( K ) denote the crossing number of D.Sending h K i to A − c ( K ) h K i and A to − q − we get the following definition of the8igure 9: Virtual Knot 3.1 Arrow Polynomial State-Sumbracket polynomial: h Ø i = 1 ; h(cid:13) K i = ( q + q − ) h K i ; h i = h i − q h i And, as pointed out in Bar-Natan [4] we can summarize the Khovanovbracket via the axioms: J Ø K = 0 → Z → J (cid:13) K K = V ⊗ J K K ; J K = Cone ( J K d → J K { } )where V = Z [ X ] / ( X ), { } is the “degree shift by one” operation on thequantum grading and Cone is the mapping cone over the differential d whichwe have yet to define. Note that we will use the enhanced state definitioncommon in the literature, where each enhanced state corresponds to a labelingof the circle by 1 or X. This has the correspondence1 ⇔ q +1 and X ⇔ q − . Remark 2.1.
For convenience we will continue to use the terms A-smoothingand B-smoothing as defined earlier.
Kho-vanov complex C • , • by setting C i,j to be the linear span of states s ∈ S where i = n B ( s ) = “the number of B-smoothings in s” and j = j ( s ) = n B ( s ) + λ ( s )where λ ( s ) = “the number of loops in s labeled 1 minus the number of loopslabeled X”. We will refer to i as the homological grading and j as the quantumgrading .For any two states s, s ′ that differ by replacing an A-smoothing by a B-smoothing at a single site respectively we define a local differential , d s,s ′ , suchthat the homological grading is increased by 1 and the quantum grading ispreserved. Once this is defined, we then have the differential d : C i,j → C i +1 ,j defined by d ( s ) = X s ′ d s,s ′ All that remains is to determine the possible values for d s,s ′ . Since we areonly concerned with resmoothing at a single site there are 3 possible scenariosrelating s and s ′ in the setting of virtual knots. Circle AnnihilationCircle CreationSingle-Cycle Smoothing
For any circles in state s not involved in resmoothing we set d s,s ′ to act asthe identity on the enhanced states. For the enhanced circles involved in theresmoothing we define m, ∆ and η as follows:10 : ⊗ → ⊗ X → XX ⊗ → XX ⊗ X → : (cid:26) → ⊗ X + X ⊗ X → X ⊗ Xη : (cid:26) → X → Khovanov homology of a knot or linkK , H ( K ) = J K K [ − n − ] { n + − n − } where [ l ] is the shift operator on the homological grading and { l } is the shift op-erator on the quantum grading. Moreover we can define the Khovanov invariant to be the Poincar´e polynomial: Kh ( K ) := X i,j t i q j dim H i,j ( K ) Example 2.1.
Consider the virtual knot 3.1 in Figure 7. The Khovanov com-plex for the unenhanced states is shown in Figure 10. It is a small exercise toshow Kh ( V K . ) = q + q − Figure 10: Virtual Knot 3.1 Khovanov Homology ComplexWe will omit the well known proof that d = 0, which amounts to checkingall cases in the virtual setting, as well as the proof of invariance under the Rei-demeister Moves. These details can be found in [26], [4], [5], [32], and [39].11f not for the self-imposed Z setting, we could also discuss applications ofLee’s spectral sequence [29], [37] in the virtual setting. We plan to return tothis subject in a future paper. In 2009 Dye, Kauffman and Manturov [10] introduced two categorifications ofthe arrow polynomial for virtual knots. Both constructions are homology the-ories defined over Z and agree with Khovanov homology over Z for classicalknots. We remark that this construction is similar in flavor to the constructionof Khovanov homology described in the previous section. The major differencein the constructions presented here are the considerations of additional gradingsarising from the arrow numbers. We will use the same renormalization for cat-egorification of the arrow polynomial as we did in Khovanov homology, namelysending h K i to A − c ( K ) h K i and A to − q − , where c ( K ) is the crossing numberfor the diagram of K.We first recall the construction introduced in [10] introducing the multi-ple grading and vector grading. Consider the collection of enhanced statesS arising from applying an A-smoothing and B-smoothing at each crossing,where A- and B-smoothings are defined as in Figure 5, and furthermore la-beling each of the resulting circles by either 1 or X. Define the arrow complex C • , • , • , • A by setting C i,j,m,vA to be the linear span of enhanced states s ∈ S where i = n B ( s ) = “the number of B-smoothings in s” and j = j ( s ) = n B ( s ) + λ ( s )where λ ( s ) = “the number of loops in s labeled 1 minus the number of loopslabeled X”. We will refer to i as the homological grading and j as the quantumgrading .Given a state s define the multiple grading of s , mg ( s ), to be the set of arrownumbers of s.Given an enhanced state s, consider the collection Λ s of enhanced circlescarrying nonzero arrow numbers. For a circle c ∈ Λ with arrow number p letthe order of c , o ( c ), be the value of k such that p = 2 k − ∗ l with gcd (2 , l ) = 1.Define the function vg ( c ) by vg ( c ) = e o ( c ) if c is labeled by X and vg ( c ) = − e o ( c ) if c is labeled by 1, where e , e , . . . is the standard basis for R ∞ . Then the vectorgrading , vg ( s ), is given by vg ( s ) = X c ∈ Λ vg ( c ) Example 2.2.
The state s = has mg ( s ) = { K , K } and vg ( s ) = ( − , , , , , ... )As before, for any two states s, s ′ that differ by an A- to B- resmoothing at asingle site it remains to define a local differential d s,s ′ such that the homological12rading is increased by 1 and the quantum grading, multiple grading and vectorgrading are all preserved. Once this is defined, we have the differential d : C i,j,m,vA → C i +1 ,j,m,vA defined by d ( s ) = X s ′ d s,s ′ Finally, The local differential d s,s ′ is defined by d s,s ′ = ˜ ∂ s,s ′ ◦ ∂ s,s ′ where ∂ s,s ′ is the Khovanov local differential between the corresponding states and˜ ∂ s,s ′ is the projection map preserving the multiple grading and vector grading.It is a short exercise to show that this satisfies the requisite d = 0 throughchecking all possible cases. A key observation for the proof is noting that forcircle annihilation and circle creation arrow numbers are ± -additive while asingle cycle resmoothing causes the arrow number to change by 1 as is shownin Figure 11.Figure 11: The Effect of Resmoothing on Arrow NumbersWe are now in a position to define the homology for the fully-graded categori-fication of the arrow polynomial of a knot or link K, H A ( K ), by renormalizinganalogously to Khovanov homology. Moreover we can define the fully-gradedarrow categorification invariant to be the Poincar´e polynomial: AKh ( K ) := X i,j ∈ Z ,m ∈ M ( D ) ,v ∈ V ( D ) m × v × t i × q j × dim H Ai,j,m,v ( K )where S ( D ) = { enhanced states of D } , M ( D ) = [ s ∈ S mg ( s ) and V ( D ) = [ s ∈ S vg ( s )for a diagram D of K. 13 xample 2.3. Consider the virtual knot 3.1 in Figure 7. The cube complex forthe unenhanced states is shown in Figure 12. It is a small exercise to show
AKh ( V K . ) = vg (2 , K [2] q t + vg (1 , K [1] q + vg (2 , − K [2] qt + q vg (1 , − K [1]+ 2 K [1] q Remark 2.2.
To translate the polynomial into the form of the definition oneonly has to see how to read the multiple grading and vector grading for a givenmonomial. The multiple grading is simply the collection of coefficients of theform K [ i ](= K i ) . The vector grading is obtained as follows. Each coefficientof the form vg ( i, a ) corresponds to having coefficient a i = a when the vectorgrading is written as X i ∈ N a i e i . The product of vector gradings corresponds to thesum of the individual gradings. For example vg (2 , vg (1 , − corresponds to thevector grading ( − , , , , , . . . ) . Figure 12: Virtual Knot 3.1 Arrow Polynomial Categorification ComplexWe will again omit the proof of invariance under the Reidemeister Moves,which is similar to the equivalent proof for Khovanov homology. Similarly weleave out most of the proof that d = 0 other than to point out why we workover Z . As with Khovanov homology, the proof follows by showing that thedifferential commutes for every possible face with every possible grading con-figuration in the cube complex. Much of this is follows from the additivityrelations for arrow numbers under resmoothing. However, the face in Figure 13is an example of a general type necessitating working over Z .14igure 13: An example of the necessity of working over Z In [10] a simpler categorification for the arrow polynomial is introduced. Wecan arrive at this construction through a simple modification of the differential˜ ∂ s,s ′ . Suppose we represent the vector grading as vg ( s ) = P i ∈ N a i s ∗ e i where e , e , . . . is the standard basis for R ∞ . Rather than preserving the multiplegrading and vector grading, ˜ ∂ s,s ′ is defined by˜ ∂ s,s ′ = (cid:26) a s ≡ a s ′ mod 20 , if a s a s ′ mod 2We have constructed a Mathematica program to calculate all of the categori-fications mentioned in this section for knots with at most 6 classical crossingsbased on Jeremy Green’s table [13]. We have been unable to find two virtualknots that are distinguished by the fully-graded categorification and not by thesimpler categorification. Additionally, we have no examples of knots which aredistinguished from the unknot by the categorification and not by the arrowpolynomial. We may extend Theorem 1.3 on surface genus bounds produced by the arrowpolynomial to the fully-graded categorification as follows.
Theorem 2.1.
Let K be a virtual knot diagram with fully-graded arrow cat-egorification invariant
AKh ( K ) . Suppose that AKh ( K ) contains a summandhaving multiple grading M , a nonempty set of arrow numbers, with |M| = n .Then n determines a lower bound on the genus g of the minimal genus surfacein which K embeds. That is, if n = 1 then the minimum genus is at least 1, if n = 2 then the minimum genus is 2 or greater and for n ≥ if n > g − then ( K ) > g . We add a bit more detail to the earlier sketch to highlight why we only getthe extension in the fully-graded categorification. The proof relies on the fol-lowing fact from [15].
Lemma 2.2.
Consider a collection A of non-intersecting essential curves (i.e.not contractible) on an orientable surface S g of genus g no pair of which co-bound an annulus. If g = 1 then |A| ≤ and if g ≥ then |A| ≤ g − .Proof of 2.1: The proof follows by the above lemma once we show thatmultiple grading corresponds to n non-intersecting essential curves of which nopair co-bound an annulus. Since the multiple grading is an invariant of the knotwe have that any embedding into S g × I for K must contain a state s with mg ( s ) = A .To see that each element of A is an essential curve suppose for contradiction K i l ∈ A bounds a disk. By the disoriented smoothing relation each cusp in K i l is paired with another cusp somewhere in s corresponding to the other half of thesmoothing. Since K i l bounds a disk (in the projection to S g ) the Jordan CurveTheorem implies the interior and exterior cusps cannot be paired. If we consideronly the internal cusps in K i l they too cannot be paired with one-another (forodd arrow numbers this follows from parity and for even arrow numbers thisfollows from orientation.) Thus an inner cusp of K i l must be paired with thecusp either of another K i j or of circle with an even number of canceling cusps.In either case we produce another cusps with which to repeat the argument.Since our knot has a finite number of crossing (hence a finite number of cusps)this is a contradiction. A similar argument shows that given K i l , K i j ∈ A with i l = i j they cannot co-bound an annulus. The subject of the width of Khovanov homology for various classes of knotshas been of interest since Khovanov’s seminal work [26]. Here we produce adefinition of width for the virtual setting, recall known results in the classicalsetting and give some basic results in the virtual knot setting.In the classical case it was noticed early on that when plotting the homologi-cal degree versus the quantum degree for the support of the Khovanov invariantthe majority of small knots were supported on 2 diagonals corresponding to thesignature of the knot ±
1. In the classical case we say that a knot is
H-thin if its Khovanov homology is supported on 2 diagonals corresponding to lines t − q = constant , else it is called H-thick . The thickness of a number of classesof classical knots is known and a recent summary of the known results can befound in [11]. In the case of alternating knots Lee [29] proved that they are16-thin. Since all alternating knots are of even parity (which we will defineshortly), Lee’s proof extends to alternating virtual knots as was pointed our byViro [39].In the virtual knot setting we need to re-examine our definition of H-thickand H-thin. If we wish to use the thickness of the homology to determine if theKhovanov homology holds additional information over the Jones polynomial wequickly run into trouble in the virtual case. It is no longer enough to determine ifthe homology is supported on 2 diagonals to determine if the homology containsmore information than the polynomial (not to mention that the signature is notwell-defined). For instance virtual knot 2.1 has bracket polynomial( A + A − A )( − A − A − )and Khovanov invariant( q + q ) t + ( q + q ) t + q + q. By the classical definition virtual knot 2.1 is H-thick as it is supported on 4diagonals. However, the Khovanov homology does not hold any additional in-formation.Two good questions to ask are (1) for a given virtual knot, on how manydiagonals is the homology H ( K ) supported and (2) what is the maximal widthbetween the diagonals (i.e. c max − c min for the supported diagonals t − q = c .)We call the solution to the first the thickness of the homology and denote itby T h ( H ( K )) for a given knot K. The second is referred to as the width of thehomology and denoted by W ( H ( K )).More can be said about knots with orientable atoms. We recall from Man-turov [33] that for a given knot K, an atom for K is a pair ( M, Γ) where M is asurface without boundary (not necessarily connected or orientable) and Γ is a4-valent graph on M such that ( M, Γ) admits a checkerboard coloring. We sayan atom is orientable if M is orientable.In [34] Manturov proves that for knots with orientable atoms we have:
Theorem 2.3.
For a knot K with orientable atom,
T h ( Kh ( K )) ≤ g ( K ) + 2 where g ( K ) is the Turaev genus (or atom genus) of the knot. The proof requires a careful examination of the interplay between the Turaevgenus of the knot and the number of crossings in the knot diagram. Note thatthe orientable condition is important. Virtual knot 2.1 has
T h ( Kh ( K )) = 4(as shown above), however g ( K ) = 1 / Theorem 2.4.
Given a virtual knot K suppose the arrow polynomial h K i A contains a monomial with non-zero arrow number K e i . . . K e n i n then W ( AKh ( K ))) ≥ X i =1 ,...,n e i ) Proof.
This is an immediate consequence of categorification. Since we have cat-egorified up to multiple grading and vector grading the monomial with non-zeroarrow number K e i . . . K e n i n corresponds to a term in AKh ( K ). If we considerthe vector gradings related to the corresponding unenhanced state in the cubecomplex we see the the largest and smallest correspond to the “all 1” label-ing and the“all X” labeling. The corresponding width between these states is2( X i =1 ,...,n e i ) giving the theorem.The above proof actually gives more if we consider all possible labelings. Theorem 2.5.
Given a virtual knot K suppose the arrow polynomial h K i A contains a monomial with non-zero arrow number K e i . . . K e n i n , then T h ( AKh ( K ))) ≥ X i =1 ,...,n e i Inspired by Dror Bar-Natan’s program for Khovanov homology [4] we have cre-ated a collection of Mathematica programs which calculate the categorificationsof the arrow polynomial mentioned above. More information on these programscan be found in the Appendix. The program are also available from the firstauthors website. The following examples have been computed using the list ofknots available at Jeremy Green’s Knot Tables [13] as well as those in LinKnot[16].
Virtual knots 5.129 in Figure 14 and 5.267 in Figure 15 bothhave Khovanov invariant Kh (5 . Kh (5 . q t + 1 q t + 1 q t + q t + 1 q + 1 qt + q + 1 q + t + 118igure 14: Virtual Knot 5.129Figure 15: Virtual Knot 5.267 and arrow polynomial AP (5 . AP (5 . − A + A − A K2 − A K1 − A K1 + K1 A +2 A − K2However,
AKh (5 . vg (2 , K [2] q t + 2 vg (1 , K [1] q + q t vg (1 , − K [1] + vg (1 , K [1] q + qt vg (2 , − K [2] + vg (2 , − K [2] qt + t vg (2 , K [2] q + 2 q vg (1 , − K [1] + t vg (1 , K [1] + vg (1 , − K [1] + 4 K [1] q + 1 q t + 1 q t + 2 q t + 2 qt + qt + tq + q + 1 q nd AKh (5 . vg (2 , K [2] q t + 2 vg (1 , K [1] q + q t vg (1 , − K [1] + vg (1 , K [1] q + qt vg (2 , − K [2] + vg (2 , − K [2] qt + t vg (2 , K [2] q + 2 q vg (1 , − K [1] + t vg (1 , K [1] + vg (1 , − K [1] + 4 K [1] q + 1 q t + 1 q t + 2 q t + 2 qt Lee [29] showed that the Khovanov homology of an alternating knot is com-pletely determined by its Jones polynomial. Recall that every classical rationalknot is isotopic to an alternating knot (see Theorem 3.5 of [25]). Hence theKhovanov homology of every classical rational knot is completely determinedby its Jones polynomial.This result of Lee is not the case for virtual knots and the categorificationsof the arrow polynomial. The following examples were found with the help ofSlavik Jablan and the program LinKnot [16].Figure 16: Virtual Knots with Equivalent Polynomials but Distinguished by aCategorificationBoth of the knots in Figure 16 have identical normalized bracket polynomial1 A + 1 A − A − A + 1 A − A − A − A and normalized arrow polynomial A + A K1 − A − A K1 − K1 A + 1 A + 2K1The knot on the left hand side has Khovanov invariant20 t + q t + q t + q t + q t + 2 q t + q t + 2 q t + 2 q t + 2 q t + q t + q + q t + q and fully-graded arrow categorification invariant q t vg(1 , − K [1] + q t vg(1 , K [1]+2 q t vg(1 , − K [1] + 2 q t vg(1 , K [1]+2 q t vg(1 , − K [1] + 2 q t vg(1 , K [1]+ q t vg(1 , − K [1] + q t vg(1 , K [1] + q t + q t + q t + q t + q + q while the knot on the right hand side has Khovanov invariant q t + q t + q t + q t + q t + 2 q t + q t + q t + q t + 2 q t + 2 q t + q t + 2 q t + q t +2 q t + q t + q t + q t + q + q t + q and fully-graded arrow categorification invariant q t vg(1 , − K [1] + q t vg(1 , K [1]+2 q t vg(1 , − K [1] + 2 q t vg(1 , K [1]+2 q t vg(1 , − K [1] + 2 q t vg(1 , K [1]+ q t vg(1 , − K [1] + q t vg(1 , K [1] + q t + q t + q t + q t + q t + q t + q t + 2 q t + q t + q t + q t + q + q Given a diagram D for a knot K label each crossing uniquely 1 through n , where n is the total number of crossings in D . Let P an arbitrary base-point on theknot. Starting at P and following the orientation of the knot we can construct asequence of length 2 n with terms corresponding to each crossing we encounter.Each term is a 3-tuple of the form ( O or U , Crossing Number, ± ) where O or U corresponds to an over or under-crossing respectively and ± corresponds to thesign of the crossing. The resulting code is referred to as the (signed, oriented) Gauss code for the diagram D of the knot K .The Gauss code can be represented diagrammatically as follows. Given acircle (often refereed to as the core circle ) place upon it in a counterclockwise21ashion 2 n points where each point is labeled by a crossing name (an integerbetween 1 and n) in the cyclic order corresponding to the Gauss code. Betweenthe two occurrences of a crossing on the core circle, place a signed, orientedchord where the sign corresponds to the crossing sign and the orientation goesfrom the over crossing to the under crossing. We call this the chord diagram for D [14], [24]. For example, the knot 3 . O − , O − , U − , O , U − , U ′′ and chord diagram as in Figure 17.Figure 17: Chord Diagram for Virtual Knot 3.1 Definition 3.1.
Given a diagram D for a knot K we can label each crossing aseven or odd in the following manner. For each crossing v locate the 2 occur-rences of v in the Gauss code for D. If the number of crossing labels between thetwo occurrences of v is even then label the crossing even. Else it is labeled odd. Remark 3.1.
This parity is well-defined for a 1-component links (i.e. knots)as the number of crossing labels in the Gauss code is n where n is the numberof crossings. It is important to notice how parity behaves under the classical Reidemeistermoves. Note that virtual Reidemeister moves do not change the Gauss code orchord diagram and thus do not affect parity. • Reidemeister I
A first Reidemeister move is always even, as is shown in Figure 18Figure 18: Reidemeister I equivalence for flat chord diagram • Reidemeister II
The two crossings involved in a second Reidemeister move are either both22ven or both odd. To see this, note that in Figure 19 if the number ofcrossings before the second Reidemeister move is n + 2 and a and b denotethe number of markings on the core circle as labeled in the figure then a + b = 2 n is even. Hence either a and b are both even or both odd.Figure 19: Reidemeister II equivalence for flat chord diagram • Reidemeister III
In a third Reidemeister move either all crossings are even or two are evenand one is odd. To see this note that in Figure 20 if the number ofcrossings not involved in the third Reidemeister move is n and a, b and c denote the number of markings on the core circle as labeled in the figurethen a + b + c = 2 n is even. Hence either a, b and c are all even or two areodd and one is even.Figure 20: Reidemeister III equivalence for flat chord diagram Manturov [31] introduced the following graphical modification for the bracketpolynomial.
Definition 3.2.
The parity bracket polynomial of a virtual knot K is definedby the relations in Figure 21.
Definition 3.3.
Given a virtual knot K with diagram D, the normalized paritybracket polynomial of K is given by
P F A ( K ) = P F A ( D ) = ( − A ) − ω ( D ) h D i P where h D i P is the parity bracket polynomial of D and ω ( D ) = writhe ( D ) = ( positive crossings in D ) − ( negative crossings in D ) . Theorem 3.1.
The parity bracket polynomial is an invariant of virtual knots.Proof.
We give an outline of the proof. The majority of this proof follows fromKauffman’s proof of invariance for the bracket polynomial [21]. You can find asimilar proof by Manturov in [31].1. Reidemeister I follows from the writhe normalization.2. Reidemeister II follows for even crossing as in the classical case and forodd crossing by the reduction relations.3. Reidemeister III follows by applying the usual trick at a single even cross-ing. (Note in the mixed case there is only 1 even crossing to choose.)
Example 3.1.
Figure 22 displays the calculation of the parity bracket polyno-mial for virtual knot 3.1 in Figure 7.
Notice that the parity bracket polynomial contains graphical coefficients.Since all of the remaining crossings are virtual or graphical (i.e. coming from anodd crossing) the only skein relations we may apply to the graphical coefficientsare the graphical Reidemeister II move as well as the graphical detour move.
Given a graphical coefficient D the minimal surface genus s ( D ) is the minimal genus for an orientable surface S g such that there is anembedding D → S g . Given the product of graphical coefficients D D · · · D n thesurface genus s ( D D · · · D n ) is the minimal genus for an orientable surface S g such that there exist disjoint embedding D i → S g for i ∈ { , . . . , n } . Remark 3.2.
Note that in the case of the parity bracket polynomial the minimalsurface genus for a graphical coefficient is the same as the minimal surfacegenus for the underlying flat virtual knot. Hence for a graphical coefficient D , s ( D ) ≥ s ( K ) where K is any virtual knot arising from D by resolving all crossingin any manner. Theorem 3.2.
Given a knot K, the parity bracket polynomial gives a lowerbound on the surface genus of K, s ( K ) . More precisely, if P F K ( A ) contains amonomial with graphical coefficients D D · · · D n then s ( D D · · · D n ) ≤ s ( K ) Proof.
Suppose K is given by an embedding S ֒ → S g × I and s ′ is the state of theparity bracket polynomial corresponding to the term with graphical coefficients D D · · · D n . Projecting s down onto S g × { } , we see that the minimal surfacegenus of s ′ is at least s ( D D · · · D n ). Since the polynomial is an invariant ofthe knot this holds for every projection. Example 3.2.
Let K be virtual knot 4.72 in Figure 23. It is a short exerciseto show h K i A = 1 and that h K i P = − A − A + D [1] − − A − A here D [1] = . Note that s ( D [1]) = 1 [8] and hence s ( K ) ≥ . Note that the diagram in Figure 23 has genus 2. It is not currently known tous if the minimal surface genus is 1 or 2. Similarly the diagram in Figure 23has virtual crossing number 3. It is unclear if there is a diagram for this knotwith a lower virtual crossing number. Figure 23: Virtual Knot 4.72
We use Manturov’s idea of graphical coefficients [31] to extend the arrow poly-nomial via parity as follows.
Definition 3.5.
The (un-normalized) parity arrow polynomial of a virtual knotK is defined by the relations in Figures 24 and 25. We expand as usual on theeven crossing and make a graphical vertex for odd crossings.
Figure 24: Parity Arrow Polynomial Crossing Skein Relations
Definition 3.6.
Given a virtual knot K with diagram D, The normalized parityarrow polynomial of K is given by
P AP A ( K ) = P AP A ( D ) = ( − A ) − ω ( D ) h D i P A where h D i P A is the parity arrow polynomial of D and ω ( D ) = writhe ( D ) = ( positive crossings in D ) − ( negative crossings in D ) . Theorem 3.3.
The normalized parity arrow polynomial is an invariant of vir-tual knots.Proof.
As in the case of the parity bracket polynomial much of the proof is thesame as in the non-parity version.1. Reidemeister I follows from the writhe normalization.2. Reidemeister II follows for even crossing by the equivalent proof for thearrow polynomial and for odd crossing by the reduction relations.3. Reidemeister III follows by applying the usual trick at a single even cross-ing in conjunction with the cusped ‘Reidemeister II’-like relation. (Notein the mixed case there is only 1 even crossing to choose.) See Figure 26for one of diagramatic proofs. The others follow similarly.
Example 3.3.
The normalized parity arrow polynomial for virtual knot 4.70 inFigure 27 is equal D [3] A + 2 K A − A − A where D [3] is the graphical coefficient . We should note that theparity bracket polynomial of virtual knot 4.70 also contains a graphical coeffi-cient. Remark 3.3.
In [18] Kauffman introduced a polynomial called with extendedbracket polynomial which is a generalization of the arrow polynomial. One keydifference between these two polynomials is that the cusps in the extended bracketpolynomial are maintained in associated pairs. One can extend the parity arrowpolynomial in a similar fashion by replacing the mixed ‘Reidemeister II’-likerelation with a similar relation .
Similar to the parity bracket polynomial we have the following lower bound onthe surface genus.
Theorem 3.4.
Given a knot K, the parity bracket polynomial gives a lowerbound on the surface genus of K, s ( K ) . More precisely, if h D i P A contains amonomial with graphical coefficients D D · · · D n then s ( D D · · · D n ) ≤ s ( K )28 roof. This is identical to the proof for the parity bracket polynomial.
Example 3.4.
Virtual knot 5.5 in Figure 28 has arrow polynomial A K1 − A K1 − A K1 + 3 A K1 + A − A − A K2 − A K1 + K1 A + 2 A and parity arrow polynomial − A D [1] − A − A − A Where D [1] = Note that by [9] the arrow polynomial gives aminimal surface genus, s ( K ) ≥ . However, in [8] Dye and Kauffman showthat a Kishino knot lying under D [1] (in the sense of resolving graphical nodesinto knot crossings) has surface genus 2. Hence the parity arrow polynomialgives the lower bound s ( K ) ≥ . Figure 28: Virtual Knot 5.5
When computing the normalized bracket polynomial for virtual knots the rela-tion of Z-Equivalence, depicted in Figure 29, goes undetected. Passing to theparity bracket polynomial we have the option to add the corresponding graph-ical relation to the coefficients (i.e. including the relation in Figure 29 whenthe classical crossings are replaced by graphical nodes.) We call the resultingpolynomial the z-parity bracket polynomial
Going a step further, we can choose to completely ignore the graphical co-efficients by sending the graphical nodes to virtual crossings. This leads to thefollowing forgetful parity bracket polynomial .29igure 29: Z-Equivalence and the Bracket Polynomial
Lemma 3.5.
The parity bracket polynomial is strictly stronger than the z-paritybracket polynomial which in turn is strictly stronger than the forgetful paritybracket polynomial.Proof.
For the parity bracket polynomial and z-parity bracket polynomial thefollows immediately from the definition. For the forgetful parity bracket polyno-mial and z-parity bracket polynomial, notice that the action of swapping an oddcrossing for a virtual crossing in the Z-Equivalence relation yields an identity.
In [31] Manturov introduced the following descending filtration on the categoryof virtual knots. Given a virtual knot K with chord diagram D. Let D = D bethe equivalence class of D up to Reidemeister moves and define D i +1 to be theequivalence class of D i after removing all odd chords (or equivalently, turningodd crossings into virtual crossings.)This is well-defined since parity is invariant under the Reidemeister movesas shown above, most notably this filtration does not introduce the forbiddenmove from Figure 3 when applied to the mixed-parity Reidemeister III move.Hence for 2 representatives D i , D ′ i ∈ [ D i ] := Equivalence Class of D i we have[ D i +1 ] ≡ [ D ′ i +1 ].Diagrammatically this filtration is described by the map sending odd cross-ing to virtual crossing and hence the forgetful parity polynomials are preciselyan application of the respective polynomial on the filtration. Theorem 3.6.
1. For any virtual knot this filtration is finite. (i.e. thereexists n such that D n = D i for n ≤ i .)2. For any classical knot this filtration is of the form D = D = . . . , that isall levels of the filtration are identical.Proof.
1. This follows from the finiteness of crossings.2. Every classical knot is equivalent to a knot with all even crossings.It is not difficult to construct a family of virtual knots for which given an n , D n = D i for n ≤ i . Consider the family in Figure 30. Here the knot F is the2-crossing virtual knot (hence not equivalent to the unknot.) Each additional30ember of the family is produced from its predecessor by the addition of twoodd positive crossing arcs and which become the only odd arcs in the new dia-gram. Since we are unable to cancel the two new odd crossings with one anotherwe see each new virtual knot created in this way is unique from its predecessor.Moreover for F n it is easy to verify that for D n = D i = Unknot for n ≤ i .Figure 30: A Parity Filtration Family For this subsection we will take a wider view and consider the space of virtuallinks (2 or more components). One should note that our definition of even andodd parity does not naturally extend. For example, the links in Figure 31 illus-trate some of the difficulty in the natural extension.Figure 31: Examples of Difficulty in Extending the Definition to LinksOmitting signs, the left link in 31 has Gauss code “ O , U , O U
2” whilethe other has Gauss code “ U O , O U Example 3.5.
The link in Figure 32 has Gauss code “ O , O , O , U , U , U , O U , O , U , U , O , O , U ′′ Crossings 1, 2, 4 and 5 are odd, crossings 3, and 6 are even and crossing 7 is alink crossing.
As we did with odd crossings, we investigate the invariance of crossings be-tween links to provide the framework for generalizing the parity polynomials.We will call a crossing where both arcs involved are in the same link componenta self-crossing while a crossing whose arcs are in separate components a link-crossing. • Reidemeister I
In a Reidemeister I move only a single link component is involved henceis always an even self-crossing. • Reidemeister II
The two arcs involved in a second Reidemeister move are either bothin the same component or each is in a different component. Thus eitherboth crossings above are self-crossings or both crossings are link-crossings. • Reidemeister III
In a third Reidemeister move either all strands involved are in one com-ponent, or two in one component and one in another or all three in sepa-rate components. Thus either all crossings are self-crossings, or one self-crossing and two link-crossings or three link-crossings respectively.This motivates the following definitions for link parity polynomials.
Definition 3.7.
Given a diagram D for a virtual knot K the (link) paritybracket polynomial of K is defined by the relations in Figures 33 and 34.
Definition 3.8.
Given a virtual link K with diagram D, The normalized (link)parity bracket polynomial of K is given by
P F A ( K ) = P F A ( D ) = ( − A ) − ω ( D ) h D i P where h D i P is the parity bracket polynomial of D and ω ( D ) = writhe ( D ) = ( positive crossings in D ) − ( negative crossings in D ) . Theorem 3.7.
The (link) parity bracket polynomial is an invariant of virtualknots.Proof. RI:
A crossing involved in a Reidemeister I move is always aneven self-crossing, hence invariance follows from the writhe normalizationas in the normalized bracket polynomial.2.
RII:
Reidemeister II follows for even self-crossing as in the classical caseand for odd self-crossing and link-crossings by the reduction relations.33.
RIII:
Here we have three cases to consider:(a) Reidemeister III for three self-crossings follows by applying the usualtrick at a single even crossing.(b) For a Reidemeister III involving one self-crossings and two link cross-ings, if the self-crossing is even then the usual trick with the RII-likemove for link crossings gives invariance. If the self-crossing is oddthen the result is immediate by the reduction relations.(c) Reidemeister II for three link-crossings also follows immediately fromthe reduction relations.4.
Mixed Move:
For an even self-crossing this is the standard proof and foran odd self-crossing or link-crossing it follows from the reduction relations.
Example 3.6.
Figure 35 displays the calculation for the (link) parity bracketpolynomial for the given 2-component link.
Figure 35: Graphical Link Parity Bracket Polynomial ExampleSimilarly for the parity arrow polynomial we have:
Definition 3.9.
Given a diagram D for a virtual link K the (link) parity arrowpolynomial of K is defined by the relations in Figures 33 and 34.
Definition 3.10.
Given a virtual link K with diagram D, The normalized (link)parity arrow polynomial of K is given by
P AP A ( K ) = P AP A ( D ) = ( − A ) − ω ( D ) h D i P A where h D i P A is the parity arrow polynomial of D and ω ( D ) = writhe ( D ) = ( positive crossings in D ) − ( negative crossings in D ) . Theorem 3.8.
The (link) parity arrow polynomial is an invariant of virtualknots. roof.
This proof is nearly identical to the one above. The only difference isthe proof for the Reidemeister III move involving an even self-crossing and twolink crossings. Here we use a cusped RII-like move for link crossing in the usualtrick for RIII invariance.
Remark 3.4.
We have used a similar extension to links in [17] to generalizethe construction of Parity Biquandles.
We would like to categorify the parity polynomials in an analogous manner to theoriginal polynomials by adding an additional grading, similarly to the construc-tion of the categorification of the arrow polynomial, based on the equivalenceclasses of graphified flat knot diagrams. However, it is fairly simple to constructan example showing this to be naive. For instance, consider the virtual knot inFigure 38. Figure 38: Kishino KnotPerforming the available Reidemeister II move, the resulting diagram is oneof three virtual knots often referred to as Kishino knots. Figure 39 shows thatboth the parity bracket polynomial and parity arrow polynomial of the knot isthe graphified version of the diagram as there are no graphical Reidemeister IImoves or detour moves available.Figure 39: Graphical Parity Bracket Polynomial of a Kishino KnotHowever, when we consider the Khovanov complex (the arrow polynomialcategorifications have equivalent complexes) as in Figure 40 we can see that d =0. In particular, the all-A and all- A − states are both graphically equivalentto two circles while in the middle we have the top state graphically equivalentto a graphified Kishino knot and the bottom state graphically equivalent to36 circles. Hence, as shown in the figure, the upper differentials are both the0-map. Considering the element ( x ⊗ d ( x ⊗
1) = (1 ⊗ m ) ◦ (∆ ⊗ x ⊗
1) = (1 ⊗ m )( x ⊗ x ⊗
1) = x ⊗ x = 0Figure 40: Graphical Parity Kishino ComplexHowever, this does not prevent us from applying Manturov’s parity filtrationalong with the forgetful version of the parity categorifications. In doing so weclearly lose some of the power of the parity polynomial (for instance the Kishinoknot is no longer detected) but we still retain an invariant which is capable ofdetecting non-classicality. We can define the parity Khovanov homology (re-spectively, parity arrow categorification ) to be the homology theory producedby first applying Manturov’s parity filtration to the given knot and then com-puting the Khovanov homology (respectively, the arrow categorification) of theresulting knot. Similarly, define the parity Khovanov invariant (respectively, parity arrow invariant ) to be the resulting Poincar´e polynomial as producedpreviously.For instance, consider the knot in Figure 41. Applying the parity Khovanovhomology we have that the original knot has Khovanov invariant1 q t + 1 q t + 1 q t + 1 q t + 1 q t + 1 q t + 1 q t + 1 q + 1 q t + 1 q Applying the filtration and turning crossings 1 and 4 into virtual crossingswe have that the underlying knot at this level of the filtration is the two crossingvirtual knot. Hence virtual knot 4.9 has parity Khovanov invariant1 q t + 1 q t + 1 q t + 1 q + 1 q t + 1 q and moreover is non-classical.Using Jeremy Green’s virtual knot table [13] we have been able to calculatethe parity categorifications on knots with at most 6 real crossings. Table 1 isa collection of planar diagram codes for 8 knots which are not distinguished37igure 41: Virtual Knot 4.9from the unknot via the bracket polynomial, arrow polynomial or their cate-gorifications, but are distinguished from the unknot via the parity arrow cate-gorification. Please see the Appendix for an explanation of our planar diagramconventions. The top four knots have parity arrow invariantvg(2 , K [2] q t + vg(1 , K [1] q + vg(2 , − K [2] qt + q vg(1 , − K [1] + 2 K [1] q while the lower four have parity arrow invariant q t vg(2 , − K [2] + q vg(1 , − K [1] + qt vg(2 , K [2] + vg(1 , K [1] q + 2 qK [1] Following the construction presented in Section 3.6 we can extend the graphicalpolynomials to links. However, the graphical coefficients for links suffer a simi-lar problem to that of knots when categorified. As with knots we map use theforgetful map to send the graphical link coefficients to virtual crossings. Theeffect of this for links is rather unfortunate as it reduces a link to the disjointunion of its components. (This is easiest to see by thinking of the chord dia-gram.) Hence it reduces the link parity categorification back to the knot paritycategorification setting.
Using Jeremy Green’s tables [13] we have calculated the above invariants asalong with the Sawollek polynomial and z-parity Sawollek polynomial([17]) forknots with at most 6 real crossings. The knots in Figure 42 and Figure 4338.5508 PD[X[4, 2, 5, 1], X[7, 4, 8, 3], X[10, 6, 11, 5], Y[12, 3, 1, 2],Y[9, 7, 10, 6], Y[8, 12, 9, 11]]6.5627 PD[X[4, 2, 5, 1], X[7, 4, 8, 3], X[10, 6, 11, 5], Y[12, 3, 1, 2],Y[9, 7, 10, 6], Y[11, 9, 12, 8]]6.7613 PD[X[4, 2, 5, 1], X[7, 4, 8, 3], X[9, 7, 10, 6], Y[12, 3, 1, 2],Y[10, 6, 11, 5], Y[8, 12, 9, 11]]6.7701 PD[X[4, 2, 5, 1], X[7, 4, 8, 3], X[9, 7, 10, 6], Y[12, 3, 1, 2],Y[10, 6, 11, 5], Y[11, 9, 12, 8]]6.24828 PD[X[6, 4, 7, 3], X[9, 5, 10, 4], X[11, 8, 12, 7], Y[12, 3, 1, 2],Y[1, 11, 2, 10], Y[8, 6, 9, 5]]6.37012 PD[X[6, 4, 7, 3], X[8, 6, 9, 5], X[11, 8, 12, 7], Y[12, 3, 1, 2],Y[1, 11, 2, 10], Y[9, 5, 10, 4]]6.60677 PD[X[3, 7, 4, 6], X[9, 5, 10, 4], X[11, 8, 12, 7], Y[12, 3, 1, 2],Y[1, 11, 2, 10], Y[8, 6, 9, 5]]6.65816 PD[X[3, 7, 4, 6], X[8, 6, 9, 5], X[11, 8, 12, 7], Y[12, 3, 1, 2],Y[1, 11, 2, 10], Y[9, 5, 10, 4]]Table 1: Undistinguished from Unknot by Categorification but Distinguishedby Parityare special in that they are not distinguished from the unknot via any of theinvariants. Knot 6 . . A Appendix
A.1 Computational Results
A.1.1 Planar Diagram Conventions
The following examples and programs compute the associated invariant basedon a planar diagram code for a diagram for the given knot whose arcs have beenconsecutively labeled. Our conventions are listed in Figure 44.Figure 44: Planar Diagram Code Conventions
Example A.1.
Virtual knot 3.1 as labeled in Figure 45 has planar diagramcode:
P D [ X [1 , , , , X [5 , , , , Y [6 , , , .1.2 Parity Polynomials The following table displays the parity bracket polynomial and parity arrowpolynomial for virtual knots with at most four crossings. Naming conventionsare as in [13]. Graphical coefficients are labeled by D [1] , D [2] , D [3] and D [1] , . . . , D [4] corresponding to the following diagrams. D [1] = D [2] = D [3] = D [1] = D [2] = D [3] = D [4] =It has been shown by Dye and Kauffman (Theorem 4.1 of [8]) that graphicalcoefficients D [1] and D [2] have surface genus s ( D [1]) = 2 and s ( D [2]) = 2.Using this fact we can see that the parity polynomials are able to give a betterbound on the genus than that of the arrow polynomial [9] for certain knots. Inparticular we have s ( K ) ≥ K ∈ { . , . , . , . , . , . , . , . , . , . } .41 able 2: Parity Bracket and Parity Arrow Polynomial CalculationsKnot Parity Bracket Parity Arrow2 . − A − A − A − A . A − D [1] A + A A − D [1] A + A . − A − A − A − A . A − D [1] A + A A − D [1] A + A . − D [1] A + A + A − D [1] A + A + A . A − A − A − A K A − A − K A − A . A − A − A − A A − A − A − A . − A − A − A + K A − A − K A . [1] D [1]4 . [2] D [2]4 . − A − A − A − A . [1] D [1]4 . [2] D [2]4 . − A − A − A − A . [2] D [2]4 . [1] D [1]4 . [1] A − A − A − A − A D [2] A + 2K A − A − A . A − A − A − A − K A + K A − A − A . − A − A + D [1] − − A − A K A − A + D [3] + K A − A . − A − A − A − A . − A − A − A − A . D [1] A − A − A − A − A A + D [2] A − A − A . A − A − A − A − K A + K A − A − A .
16 D [1] A − A − A − A − A D [2] A + 2K A − A − A . − A − A + D [1] − − A − A K A − A + D [3] + K A − A . − A − A − A − A . − A − A + D [1] − − A − A K A − A + D [2] + K A − A . − A − A − A + A K A − K A − A − A . D [1] A − A − A − A − A A + D [2] A − A − A . − A − A − A + A K A − K A − A − A . A − A − A − A − K A + K A − A − A . − A − A − A + A K A − K A − A − A . − A − A − A − A .
26 D [3] D [3]4 . − A − A − A − A .
28 D [4] D [4]4 .
29 D [1] A − A − A − A − A D [3] A + 2K A − A − A . − A − A + D [1] − − A − A K A − A + D [2] + K A − A . A − A − A − A − K A + K A − A − A . − A − A − A − A . − A − A + D [1] − − A − A K A − A + D [2] + K A − A . D [1] A − A − A − A − A A + D [2] A − A − A . − A − A − A − A . − A − A − A + A K A − K A − A − A not Parity Bracket Parity Arrow4 . A − A − A − A − K A + K A − A − A . − A − A − A − A . − A − A − A − A . − A − A − A + A K A − K A − A − A . A − A − A − A − K A + K A − A − A . − A − A − A − A . − A − A − A − A . − A − A − A − A .
45 D [3] D [3]4 . − A − A − A − A .
47 D [4] D [4]4 .
48 D [1] A − A − A − A − A D [2] A + 2K A − A − A . − A − A + D [1] − − A − A K A − A + D [3] + K A − A . A − A − A − A − K A + K A − A − A . − A − A − A − A . D [1] A − A − A − A − A A + D [2] A − A − A . − A − A − A − A . − A − A − A − A .
55 D [1] D [1]4 .
56 D [2] D [2]4 .
57 D [1] A − A − A − A − A D [3] A + 2K A − A − A . − A − A + D [1] − − A − A K A − A + D [2] + K A − A . − A − A + D [1] − − A − A K A − A + D [2] + K A − A . − A − A − A + A K A − K A − A − A . A − A − A − A − K A + K A − A − A . − A − A − A − A . − A − A − A − A . − A − A − A + A K A − K A − A − A . A − A − A − A − K A + K A − A − A . − A − A − A − A . − A − A − A − A . − A − A − A + A K A − K A − A − A . A − A − A − A − K A + K A − A − A .
70 D [1] A − A − A − A − A D [3] A + 2K A − A − A . − A − A + D [1] − − A − A K A − A + D [2] + K A − A . − A − A + D [1] − − A − A K A − A + D [2] + K A − A . − A − A − A − A . − A − A − A − A . − A − A − A − A .
76 D [2] D [2]4 .
77 D [1] D [1]4 . A − A − A − A − K A + K A − A − A . A − A − A − A − K A + K A − A − A .
80 D [3] D [3]4 .
81 D [4] D [4]4 . − A + D [1] A − A − D [1] A − A + D [1] A − A − D [1] A not Parity Bracket Parity Arrow4 . − A − A − A − A . − A + D [1] A − A − D [1] A − A + D [1] A − A − D [1] A . − A − A − A + K A − A − K A . − A − A − A − A + K A − A − K A + A . − A + D [1] A − A − D [1] A − A + D [1] A − A − D [1] A . − A + D [1] A − A − D [1] A − A + D [1] A − A − D [1] A . A − A − A − A − A + 2K A − A − A + A + A − A . − A − A − K A + A + K A − A + K A − A − K A + A . − A − A − A − A . − A − A − A − A . − A + D [1] A − A − D [1] A − A + D [1] A − A − D [1] A . − A − A − A − A . − A − A − A − A . − A + D [1] A − A − D [1] A − A + D [1] A − A − D [1] A . − A − A − A − A . − A − A − A − A . − A − A − A − A . − A − A − A − A . − A − A − A − A . − A − A − A − A . − A + D [1] A − A − D [1] A − A + D [1] A − A − D [1] A . − A − A − A − A . A − A − A − A A − A − A − A . − A − A − A + K A − A − K A . − A − A − A − A . − A − A − A − A A.2 A Mathematica Program
A.2.1 A Parity Categorification
The following program for a categorification of the arrow polynomial and the for-getful parity version is based on Dror Bar-Natan’s construction [4] for Khovanovhomology. Here we implement a version of Gaussian elimination for computinghomology with coefficients over Z that was pointed out to us by Marc Cullerand implemented by Baldwin and Gillam for computation of Heegaard-Floerknot homology in [2]. This can be described graphically as in Figure 46 wherewe reduce based on the chosen marked edge. This is equivalent to applyingGaussian Elimination to the chain complex as in Figure 47 where we assume φ (the equivalent of the selected edge) is invertible. Maps denoted by • are arbi-trary and inconsequential in the final result. For more on Gaussian Eliminationand homotopy equivalence we point the reader to [7]. Remark A.1.
A similar program for Khovanov homology and the forgetful arity version with Z coefficients is available on the first authors website. Figure 46:Figure 47:45 p[L_PD] := Count[L, _X];nm[L_PD] := Count[L, _Y];SetAttributes[del, Orderless]np and nm count the number of positive and negative crossings for a given planardiagram respectively. We set del to be orderless to reduce the number of necessaryrelations.The following lines of program perform the forgetful mapping on the odd crossings. EvenParityPD[L_PD] :=Sort[L /. {X[i_, j_, k_, l_] :>Odd[i, j, k, l] /; OddQ[i - j]} /. {Y[i_, j_, k_, l_] :>Odd[i, j, k, l] /; OddQ[i - j]}]EvenParity performs a simple check to determine if a given crossing is even or odd.For an odd crossing it replaces the head X or Y with Odd . Note this subroutine assumesthat arcs of a given knot diagram are labeling consecutively.
OddIdentities[L_PD] :=ReplacePart[ ReplacePart[Reverse[Sort[Flatten[ReplacePart[Take[EvenParityPD[L],Length[EvenParityPD[L]] - (np[EvenParityPD[L]] +nm[EvenParityPD[L]])],0 :> List] /. {Odd[i_, j_, k_, l_] :> {del[i, k],del[j, l]}}]]],0 :> Times] //. {del[a_, b_] del[b_, c_] :> del[a, c]}, 0 :> List]OddIdentities creates a collection of arc relations displayed in terms of Kroneckerdeltas based on the odd crossings.
EvenCross[L_PD] :=Drop[EvenParityPD[L],Length[EvenParityPD[L]] - (np[EvenParityPD[L]] +nm[EvenParityPD[L]])]EvenCross collects the even crossings from a given PD code.
PDReduction[L_PD, d_del] := (mm := Min[d[[1]], d[[2]]];nn = Max[d[[1]], d[[2]]]; L /. {nn :> mm})PDReduction turns a single identity produced by
OddIdentities into a reduction re-lation and applies this relation.
ForgetfulEvenParityPD[L_PD] := (RedPD = EvenCross[L];Do[RedPD = PDReduction[RedPD, OddIdentities[L][[i]]], {i,Length[OddIdentities[L]]}]; RedPD) orgetfulEvenParityPD applies PDReduction for all of the relations in
OddIdentities and returns the resulting PD code. rule2 = {del[a_, b_][m_] del[a_, b_][m_] :> del[a, a][m],del[a_, b_][m_] del[b_, c_][n_] :> del[a, c][Min[m, n]],del[a_, b_][m_] led[b_, c_][n_] :> led[a, c][Min[m, n]],del[a_, b_][m_] led[c_, b_][n_] :> led[c, a][Min[m, n]],led[a_, b_][m_] del[b_, c_][n_] :> led[a, c][Min[m, n]],led[a_, b_][m_] del[c_, b_][n_] :> led[a, c][Min[m, n]],led[a_, b_][m_] led[b_, c_][n_] :> del[a, c][Min[m, n]]};rule3 = {del[a_, a_][m1_] :> c[m1, 0, 0],del[a_, a_][m1_]^_ :> c[m1, 0, 0],led[a_, b_][m1_]^2 :> c[m1, 1, 1],led[a_, b_][m1_] led[a_, b_][m2_] :> c[Min[m1, m2], 1, 1],led[a_, b_][m1_] led[x_, b_][m2_] led[x_, d_][m3_] led[a_, d_][m4_] :> c[Min[m1, m2, m3, m4], 2, 2],led[a_, b_][m1_] led[x_, b_][m2_] led[x_, d_][m3_] led[e_, d_][m4_] led[e_, f_][m5_] led[a_, f_][m6_] :>c[Min[m1, m2, m3, m4, m5, m6], 3, 1],led[a_, b_][m1_] led[x_, b_][m2_] led[x_, d_][m3_] led[e_, d_][m4_] led[e_, f_][m5_] led[g_, f_][m6_] led[g_, h_][m7_] led[a_, h_][m8_] :>c[Min[m1, m2, m3, m4, m5, m6, m7, m8], 4, 3],led[a_, b_][m1_] led[x_, b_][m2_] led[x_, d_][m3_] led[e_, d_][m4_] led[e_, f_][m5_] led[g_, f_][m6_] led[g_, h_][m7_] led[y_, h_][m8_] led[y_, z_][m9_] led[a_, z_][m10_] :>c[Min[m1, m2, m3, m4, m5, m6, m7, m8, m9, m10], 5, 1]};rule2 and rule3 are reduction relations used by S . rule2 joins arcs and cusps while rule3 produces the labeled circles for a basic (unenhanced) state. We follow theconvention c[m, p,k] is circle m with arrow number p and dot of order k . Where p = l*2^(k - 1) for l odd. ruleStar = {v___c u___ X[i_, j_, k_,l_] :> ((u del[i, j][Min[i, j]] del[k, l][Min[k, l]] //.rule2 //.rule3) -> (u led[l, i][Min[l, i]] led[j, k][Min[j, k]] //.rule2 //. rule3)),v___c u___ Y[i_, j_, k_,l_] :> ((u led[l, i][Min[l, i]] led[j, k][Min[j, k]] //.rule2 //.rule3) -> (u del[i, j][Min[i, j]] del[k, l][Min[k, l]] //.rule2 //. rule3))};ruleStar is a reduction relations used by S which produces notation corresponding toa bifurcation on an edge denoted by * on the cube complex. S[L_PD, a_List] := imes[(Times @@ (Thread[{List @@Drop[L, Length[L] - (np[L] + nm[L])],a}] /. {{X[i_, j_, k_, l_], 0} :>del[i, j][Min[i, j]] del[k, l][Min[k, l]], {X[i_, j_, k_,l_], 1} :>led[l, i][Min[l, i]] led[j, k][Min[j, k]], {Y[i_, j_, k_,l_], 0} :>led[l, i][Min[l, i]] led[j, k][Min[j, k]], {Y[i_, j_, k_,l_], 1} :>del[i, j][Min[i, j]] del[k, l][Min[k, l]], {x_X, "*"} :>x, {y_Y, "*"} :> y})), (Times @@ (Take[L,Length[L] - (np[L] + nm[L])] /. {Odd[i_, j_, k_, l_] :>del[i, k][Min[i, k]] del[j, l][Min[j, l]]}))] //.rule2 //. rule3 //. ruleStarS[L_PD, s_String] := S[L, Characters[s] /. {"0" -> 0, "1" -> 1}]S produces the unenhanced state of the cube complex for L corresponding to the vertex a . MG[expr_] :=expr //. {c[m_, 0, k_] :> 1} //. {c[m_, p_, k_] :>a[p]} //. {a[i_]^_ :> a[i]}MG computes the multiple grading for an given unenhanced state. If there is no arrownumbers MG returns 1. If there are arrow numbers MG returns a product of the form a [ i ] a [ i ] . . . a [ i n ], where the i j are the distinct arrow numbers (ie i j = i k iff j = k ). Deg[expr_] := Count[expr, _v1, {0, 1}] - Count[expr, _vX, {0, 1}]V[L_PD, s_String, deg___] :=V[L, Characters[s] /. {"0" -> 0, "1" -> 1}, deg]V[L_PD, a_List] :=List @@ Expand[S[L, a] /. x_c :> ((vX @@ x) + (v1 @@ x))]V[L_PD, a_List, deg_Integer] :=Select[V[L, a], (deg == Deg[
The above subroutines provide information on the enhanced states. V replaces c[m,p,k] by vX[m,p,k]+v1[m,p,k] throughout then expands each expression. Each summandcorresponds to an enhanced (labeled by X and 1) state, which we separate into a listof enhanced states at each vertex. Deg computes ( V returns enhanced states with a given bi-degree. ((bi-degree) = (homological degree)+ ( VG[expr_] :=expr //. {vX[m_, 0, k_] :> 1} //. {v1[m_, 0, k_] :> 1} //. {vX[m_,p_, k_] :> vg[k, 1]} //. {v1[m_, p_, k_] :>vg[k, -1]} //. {vg[a_, i_]^m_ :>vg[a, i*m]} //. {vg[a_, i_] vg[a_, j_] :> vg[a, i + j]} //. {vg[a_, 0] :> 1} e need to compute the vector grading of an enhanced state. Recall that this (inf.dim.) vector is the sum over the labeling in the enhanced state where, for i >
0, wetransform vX[*,*,i] into the vector that is 1 in the i t h position and 0 elsewhere andsimilarly we transform v1[*,*,i] into the vector that is -1 in the i t h position and0 elsewhere. VG returns 1 if the vector grading is the zero vector else it returns theproduct of terms of the form v[k, n] corresponding to the k t h position in the vectorgrading having value n. d[L_PD, s_String] := d[L, Characters[s] /. {"0" -> 0, "1" -> 1}]d[L_PD, a_List] :=S[L, a] //. {(c[x__] c[y__] -> c[z__])*_. :> {v1@x v1@y -> 0,v1@x vX@y -> 0, vX@x v1@y -> 0,vX@x vX@y -> 0} /; (MG[S[L, a //. {"*" -> 0}]] =!=MG[S[L, a //. {"*" -> 1}]]), (c[z__] ->c[x__] c[y__])*_. :> {v1@z -> 0,vX@z -> 0} /; (MG[S[L, a //. {"*" -> 0}]] =!=MG[S[L, a //. {"*" -> 1}]])} //. {(c[x__] c[y__] ->c[z__])*_. :> {v1@x v1@y -> v1@z, v1@x vX@y -> vX@z,vX@x v1@y -> vX@z,vX@x vX@y ->0} /; (VG[v1@x v1@y] === VG[v1@z]) && (VG[v1@x vX@y] ===VG[vX@z]) && (VG[vX@x v1@y] === VG[vX@z]), (c[x__] c[y__] ->c[z__])*_. :> {v1@x v1@y -> 0, v1@x vX@y -> vX@z,vX@x v1@y -> vX@z,vX@x vX@y ->0} /; (VG[v1@x v1@y] =!= VG[v1@z]) && (VG[v1@x vX@y] ===VG[vX@z]) && (VG[vX@x v1@y] === VG[vX@z]), (c[x__] c[y__] ->c[z__])*_. :> {v1@x v1@y -> v1@z, v1@x vX@y -> 0,vX@x v1@y -> vX@z,vX@x vX@y ->0} /; (VG[v1@x v1@y] === VG[v1@z]) && (VG[v1@x vX@y] =!=VG[vX@z]) && (VG[vX@x v1@y] === VG[vX@z]), (c[x__] c[y__] ->c[z__])*_. :> {v1@x v1@y -> v1@z, v1@x vX@y -> vX@z,vX@x v1@y -> 0,vX@x vX@y ->0} /; (VG[v1@x v1@y] === VG[v1@z]) && (VG[v1@x vX@y] ===VG[vX@z]) && (VG[vX@x v1@y] =!= VG[vX@z]), (c[x__] c[y__] ->c[z__])*_. :> {v1@x v1@y -> 0, v1@x vX@y -> 0,vX@x v1@y -> vX@z,vX@x vX@y ->0} /; (VG[v1@x v1@y] =!= VG[v1@z]) && (VG[v1@x vX@y] =!=VG[vX@z]) && (VG[vX@x v1@y] === VG[vX@z]), (c[x__] c[y__] ->c[z__])*_. :> {v1@x v1@y -> 0, v1@x vX@y -> vX@z,vX@x v1@y -> 0,vX@x vX@y ->0} /; (VG[v1@x v1@y] =!= VG[v1@z]) && (VG[v1@x vX@y] ===VG[vX@z]) && (VG[vX@x v1@y] =!= VG[vX@z]), (c[x__] c[y__] ->c[z__])*_. :> {v1@x v1@y -> v1@z, v1@x vX@y -> 0,vX@x v1@y -> 0, X@x vX@y ->0} /; (VG[v1@x v1@y] === VG[v1@z]) && (VG[v1@x vX@y] =!=VG[vX@z]) && (VG[vX@x v1@y] =!= VG[vX@z]), (c[x__] c[y__] ->c[z__])*_. :> {v1@x v1@y -> 0, v1@x vX@y -> 0, vX@x v1@y -> 0,vX@x vX@y ->0} /; (VG[v1@x v1@y] =!= VG[v1@z]) && (VG[v1@x vX@y] =!=VG[vX@z]) && (VG[vX@x v1@y] =!= VG[vX@z]), (c[z__] ->c[x__] c[y__])*_. :> {v1@z -> v1@x vX@y + vX@x v1@y,vX@z ->vX@x vX@y} /; (VG[v1@z] === VG[v1@x vX@y]) && (VG[v1@z] ===VG[vX@x v1@y]) && (VG[vX@z] === VG[vX@x vX@y]), (c[z__] ->c[x__] c[y__])*_. :> {v1@z -> vX@x v1@y,vX@z ->vX@x vX@y} /; (VG[v1@z] =!= VG[v1@x vX@y]) && (VG[v1@z] ===VG[vX@x v1@y]) && (VG[vX@z] === VG[vX@x vX@y]), (c[z__] ->c[x__] c[y__])*_. :> {v1@z -> v1@x vX@y,vX@z ->vX@x vX@y} /; (VG[v1@z] === VG[v1@x vX@y]) && (VG[v1@z] =!=VG[vX@x v1@y]) && (VG[vX@z] === VG[vX@x vX@y]), (c[z__] ->c[x__] c[y__])*_. :> {v1@z -> v1@x vX@y + vX@x v1@y,vX@z ->0} /; (VG[v1@z] === VG[v1@x vX@y]) && (VG[v1@z] ===VG[vX@x v1@y]) && (VG[vX@z] =!= VG[vX@x vX@y]), (c[z__] ->c[x__] c[y__])*_. :> {v1@z -> 0,vX@z ->vX@x vX@y} /; (VG[v1@z] =!= VG[v1@x vX@y]) && (VG[v1@z] =!=VG[vX@x v1@y]) && (VG[vX@z] === VG[vX@x vX@y]), (c[z__] ->c[x__] c[y__])*_. :> {v1@z -> vX@x v1@y,vX@z ->0} /; (VG[v1@z] =!= VG[v1@x vX@y]) && (VG[v1@z] ===VG[vX@x v1@y]) && (VG[vX@z] =!= VG[vX@x vX@y]), (c[z__] ->c[x__] c[y__])*_. :> {v1@z -> v1@x vX@y,vX@z ->0} /; (VG[v1@z] === VG[v1@x vX@y]) && (VG[v1@z] =!=VG[vX@x v1@y]) && (VG[vX@z] =!= VG[vX@x vX@y]), (c[z__] ->c[x__] c[y__])*_. :> {v1@z -> 0,vX@z ->0} /; (VG[v1@z] =!= VG[v1@x vX@y]) && (VG[v1@z] =!=VG[vX@x v1@y]) && (VG[vX@z] =!= VG[vX@x vX@y])} //. {(c[x__] -> c[y__])*_. :> {v1@x -> 0, vX@x -> 0}}d computes the edge morphism for the edge corresponding to the label a . Here a isa list of 0’s and 1’s along with a single * where * corresponds to the crossing we areresmoothing and 0 and 1 correspond to A − and A − -smoothings at the remainingcrossings.We now have enough to construct the cube complex. The following collection of rou-tines together collect this information and constructs a graph. We then preform thepreviously mentioned graphical reduction algorithm to compute the homology. dif[L_PD, s_String] := dif[L, Characters[s] /. {"0" -> 0, "1" -> 1}] if[L_PD, a_List] :=Flatten[MapThread[ge, {V[L, a /. ("*" :> 0)],Expand[V[L, a /. ("*" :> 0)] /. d[L, a]]}] /. (ge[u___,v__ + w__] :> {ge[u, v], ge[u, w]}) /. (ge[z___, 0] :> 0)]Comp[L_PD] :=Join @@ (Join @@ {Expand[ed[((v @@ constructs the set of directed edges for the graph corresponding to an edge of thecube complex. It does the locally by applying the differential d to the tail of each edge. Comp produces the full set of vertices for the graph and connected the heads and tailsof the directed edges formed by dif . Some zero differentials remain.
Edges removesthese from the list.
KhColumn[L_PD, r_Integer] :=If[r < 0 || r > (np[L] + nm[L]), {0},Join @@ (((v @@ produces the collection of enhanced states corresponding to the enhanced statesof the complex (i.e. the nodes in the graph) by calling
KhColumn for each homologicaldegree of the planar diagram for the knot.
Perms[L_PD] :=Join @@ (Permutations[Join[{"*"}, Table[0, {(np[L] + nm[L]) - generates the lists of 0’s, 1’s and a single * corresponding to the edges of thecube complex. Height[gen___] := (gen /. {v1[a___] :> 1, vX[b___] :> 1,v[c___] :> Plus[c]});EdgeHeight[e__] := If[IntegerQ[e], -1, Height[e[[1]]]];MultGrad[e__] :=e /. {v[expr___] :> 1, vX[a_, b_, c_] :> K[b],v1[a_, b_, c_] :> K[b]} /. {K[0] :> 1} /. {K[i_]^_ :> K[i]};MultGrad takes a homology class representative and outputs its multiple grading. ddDelEdges[edges_List, e_edge] := (Off[Part::partd];listfs = Select[edges, Given an edge e , AddDelEdges looks for all local subgraphs which share the same headas e . It then computes a symmetric difference with collection of edges whose tail isthe same as e . Reduc[gens_List, edges_List,e_edge] := (Checks[(gens /. {e[[1]] :> 0, e[[2]] :> 0}),AddDelEdges[edges, e]]);Checks[gens_List, unsortededges_List] := (edges = Cases[(SortBy[unsortededges, EdgeHeight[ looks for the remaining edge whose head is in the highest homological degree.It then applies
Reduc to remove the edge and apply the symmetric difference algorithmin
AddDelEdges . HomReps[L_PD] :=Block[{$IterationLimit = Infinity, $RecursionLimit = Infinity},Checks[Cases[Gens[L], Except[0]], Cases[Edges[L], Except[0]]]];HomReps takes a planar diagram code, repeatedly applies
Checks to run the graphreduction algorithm and outputs representatives for the homology classes.
QT[gen___, L_PD] :=(r = (Height[gen] - nm[L]); (t^r)*(q^(r + Deg[gen] + np[L] - nm[L])));QT computes the associated powers of q and t for a given representative of a homologyclass.
AKh[L_PD] :=Plus @@ (QT[ lus @@ (QT[ Finally,
AKh and
ParityAKh compute the corresponding categorifications for a givenplanar diagram L.
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