Odd Khovanov homology

Jacob Rasmussen; Peter Ozsvath; Zoltan Szabo

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Odd Khovanov homology

Peter Ozsvath, Jacob Rasmussen, Zoltan Szabo

Quantum Algebra · Submitted 23 Oct 2007 · 13 references

Pith's verdict: The founding paper of odd Khovanov homology: the exterior-algebra TQFT is carried through cleanly, and the only real soft spot is the asserted four-type classification of one-handle pairs in Figure 2.

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CitationJacob Rasmussen, Peter Ozsvath, Zoltan Szabo. "Odd Khovanov homology." arXiv:0710.4300 (2007). https://pith.science/paper/Y2NVVLOJ

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This is the paper that started odd Khovanov homology, and it holds up. The idea is easy to state and not routine to carry out: swap the symmetric algebra in Khovanov's TQFT for an exterior algebra. That change makes the cobordism maps well-defined only up to sign, so the authors have to define a projective functor, show a consistent edge assignment exists on the hypercube of resolutions (Lemma 1.2, via a cocycle argument), and check the three Reidemeister moves. They do all three. The payoff is real: Kh' agrees with Khovanov homology mod 2 (Proposition 1.6) but differs over Q, and Section 5 gives the explicit 8_19 computation, rank 3 against rank 5. The reduced theory being a genuine link invariant, with no choice of component, is a structural improvement over reduced Khovanov homology. Conjecture 1.9 to Heegaard Floer is clearly labelled as a conjecture and is the right kind of question to leave open. On soft spots, the reader flagged the parity check in Lemma 2.1 as the weak link; I think that is slightly misplaced. Figure 4 labels each cube type with its (a,c,x,y) tuple, and you can read a+x and a+y even straight off the labels: (2202), (1311), (0600) and the rest all check. The genuinely asserted step is one level up, the four-type classification of one-handle pairs in Figure 2, which is what Equation (1) and hence the whole square-face analysis rests on. The text says only that the sign can be determined from the combinatorics and that the reader can easily verify the TQFT identities. That is a finite local check, not a gap in the global argument, but it is the load-bearing unproved step and deserves a couple more lines. Minor: the Section 5 tables come from an unshipped Mathematica program, so they are not independently reproducible from the paper alone, and the introduction points to Section 4 for Proposition 1.8, which is actually argued in Section 5. Neither matters much. Who should read it: anyone in link homology or on the Floer-theoretic side of knot invariants. It is the origin point for a lot of later work. Recommendation: send it out. It deserves referees and the construction deserves a careful check, but I expect it to come back clean.

Referee report

Significance

If correct, this is a substantial and genuinely new invariant: it is the natural odd (exterior-algebra) analogue of Khovanov homology, it agrees with Khovanov homology mod 2 but demonstrably differs over Q, and its reduced version is a link invariant that does not require choosing a component, unlike the ordinary reduced theory. The paper is strong on the things referees should reward. The construction is fully explicit — the four elementary cobordism maps are written down, not merely asserted to exist — and the invariance proof is carried out move by move rather than invoked. The mod 2 agreement with Khovanov homology (Proposition 1.6) is an independent external check on the construction, and the rank-3 versus rank-5 computation for 8₁₉ is a concrete, reproducible falsification of any hope for a Lee-type cancelling differential in the odd theory; the authors report this negative outcome candidly rather than burying it. Conjecture 1.9 is a sharp, falsifiable prediction tying the invariant to Heegaard Floer homology of the branched double cover, and the paper is honest that the 9₄₂ computation shows σ-thinness over Q but not over Z. The computational tables in §5 give other workers

Major objections (1)

[§1.1, Equation (1) and Figure 2] The verification that the projective functor satisfies the TQFT identities up to sign is deferred with the sentence 'The reader can easily verify that up to sign, these maps satisfy all the identities associated with a TQFT,' and the classification of pairs of one-handles into types A, C, X, Y is presented only through Figure 2, with the sign in F_{Z₂}∘F_{Z₁} = ε·F_{Z₁}∘F_{Z₂} said to be determinable 'from the combinatorics of how the two arcs Z₁ and Z₂ interact.' This is the load-bearing step for the whole paper: the square-face analysis in §1.3 that yields ∂_ε²=0 depends on exactly this classification, and Lemma 2.1's parity count is downstream of it. The maps themselves are explicit enough (merge is projection, split sends 1 to ±(a₁−a₂), death is contraction) that the four cases can be derived directly in the exterior algebra, so this is a completeness request rather than a claim of error. I would ask the authors to add a short derivation, or at minimum one fully worked example of each of the four types, so that the reader is not asked to take the classification on faith at precisely the point where the chain complex's existence is at stake.

Minor objections (9)

[Proposition 1.6] The statement reads 'the mod two reduction of Kh′(L) agrees with the mod two reduction of Khovanov's sl(2) homology Kh′(L),' which identifies the two sides by the same symbol; the displayed formula immediately below correctly uses Kh and Kh′, so this is a typographical slip in the sentence, but it should be corrected since the proposition is one of the paper's headline claims.
[Proposition 1.7 and the following paragraph] The reduced group is written with a bar (Kh̄′) in the original, but in the version I read the bar is not visible, so Proposition 1.7 as typeset reads Kh′_{m,s}(L) ≅ Kh′_{m,s−1} ⊕ Kh′_{m,s+1}(L), which is not the intended statement and is not parseable as written. The reduced and unreduced invariants must be typographically unmistakable throughout §4 and §5, including Definition 5.1, where 'Kh′(L) is said to be σ-thin' evidently refers to the reduced group.
[§1.2] The skein relation characterizing the unnormalized Jones polynomial is garbled in the display ('Ĵ(/BC ) ˙ = Ĵ (/BD ) −q · Ĵ(/C0 )'), and the notation 'f ˙ = g' for equality up to a unit is introduced without comment; both should be cleaned up.
[§5, paragraph after Proposition 5.2] The sentence 'Among the 10-crossing knots, Kh′ is σ–thin whenever which Kh is σ–thin' contains a stray 'which' and should read 'whenever Kh is σ-thin.'
[Proposition 1.8 versus §5] Proposition 1.8 is stated for non-split alternating links, but the derivation in §5 concludes 'we see that Kh′(L) ≅ Kh(L) whenever L is alternating,' dropping the hypothesis. Either the hypothesis is needed (in which case §5 should say so, and the split case should be addressed) or it is not (in which case Proposition 1.8 should be stated without it). Proposition 1.8 is also stated in the introduction with no pointer to where it is proved.
[Figure 4 and Lemma 2.1] The parity check 'We leave it to the reader to verify that in all cases, a+x and a+y are even' is in fact discharged by the labels in Figure 4, which are tuples (a,c,x,y): (2202) gives a+x=2 and a+y=4, (1311) gives 2 and 2, (0600) gives 0 and 0, and so on for every listed tuple. The caption should say explicitly that the labels record (a,c,x,y), so that this check is a glance rather than an exercise.
[§3.3, Reidemeister III] The final step, that the obstruction cocycles ψ and ψ′ in C²(P;Z/2) coincide after choosing the upward-pointing orientation at all crossings, is closed with 'this is easily seen to be the case.' Since the whole sign bookkeeping of the Reidemeister III proof rests on this, one sentence spelling out which faces are compared would make the argument checkable.
[§5, 9₄₂ discussion] The inference that σ-thinness of Kh′(9₄₂) over Z 'cannot be true' because the mod 2 reduction would then be σ-thin while Kh(9₄₂) is not, uses Proposition 1.6 together with the universal coefficient theorem; the authors should say so explicitly rather than leaving the chain of reasoning implicit.
[References] The URL in reference [12] is garbled ('http://www.geom etrie.ch/KhoHo/') and should be corrected. Reference [7] is listed as 'In preparation'; it is cited only motivationally in the paragraph following Conjecture 1.9, which is appropriate, but the authors may wish to note that no argument depends on it.

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The paper

The paper claims

Swapping the symmetric algebra in Khovanov's construction for an exterior algebra gives a new link invariant that agrees with Khovanov homology mod 2 but separates from it over the rationals.

The reading

Khovanov homology assigns to a link a bigraded abelian group whose graded Euler characteristic is the Jones polynomial; the construction builds it from a TQFT that attaches the symmetric algebra of the vector space on a collection of planar circles. This paper rebuilds the same machinery with the exterior algebra in place of the symmetric one, and shows the result is again a link invariant, $Kh'(L)$, with the same kind of bigrading and the same Jones-polynomial Euler characteristic. The obstruction is signs: the exterior-algebra maps are only well defined up to an overall sign, so the differential squares to zero only after one chooses a consistent assignment of signs to the edges of the hypercube of resolutions. The authors prove such assignments exist (of two flavours, called type X and type Y), prove the resulting homology does not depend on the diagram or on the assignment, and compute enough examples to show the invariant is genuinely new: it agrees with Khovanov homology for alternating links and mod 2 in general, but the $(3,4)$ torus knot $8_{19}$ has rank 3 in the odd theory and rank 5 in the ordinary one. The motivation is a conjectured integral lift of the known mod 2 spectral sequence from Khovanov homology to the Heegaard Floer homology of the branched double cover.

What carries the argument

The load-bearing device is the projective functor from the category of planar one-manifolds and orientable cobordisms to graded abelian groups that sends a collection $S$ of circles to $Lambda^* V(S)$, the exterior algebra on the free group generated by the components. Merges are induced by the projection $V(S_1) to V(S_1)/(a_1 - a_2)$; splits send $1$ to $pm(a_1 - a_2)$ and are well defined only up to sign; births are inclusions and deaths are contractions. Because of that sign ambiguity, the hypercube of resolutions does not by itself give a chain complex: one needs an edge assignment $epsilon : E(D) to {pm 1}$ making every face of type A and X even and every face of type C and Y odd (a type X assignment), or the analogous type Y condition. Existence is Lemma 1.2, which is proved by observing that the face-typing defines a 2-cocycle on the (contractible) hypercube with values in ${pm 1}$, hence a coboundary, hence an edge assignment; the cocycle condition is Lemma 2.1, the statement that every cube in the hypercube of resolutions contains an even number of squares of type A and X, and of type A and Y. This parity lemma is the hinge on which the whole construction turns.

Extended reading notes

Core claim

The central claim is Theorem 1.3: for a link $L$ with oriented projection $D$ and an edge assignment $e$ of type X or type Y, the bigraded homology of the complex $(C(D), partial_e)$, where $C(D) = bigoplus_{I} Lambda^* V(D(I))$ over vertices of the hypercube of resolutions, is independent of the choice of $D$ and of $e$. The differential is $partial_e(v) = sum_e epsilon(e) F_e(v)$, and $partial_e^2 = 0$ holds because the exterior-algebra maps attached to a square either anticommute or commute (types A and C) or else both composites vanish (types X and Y), with the signs chosen to match. Independence of the assignment, of the crossing orientations, and of the Reidemeister moves is checked directly, following Khovanov's original invariance argument as exposited by Bar-Natan. On top of the invariance the paper establishes: the graded Euler characteristic is the unnormalized Jones polynomial (Proposition 1.4); there is a skein exact sequence (Proposition 1.5); the mod 2 reduction coincides with that of ordinary Khovanov homology (Proposition 1.6); there is a reduced version $tilde{Kh}'(L)$ that is a link invariant in its own right, with $Kh' cong tilde{Kh}' oplus tilde{Kh}'$ shifted in $q$ (Proposition 1.7); and for non-split alternating links the odd and ordinary groups agree (Proposition 1.8).

Load-bearing premise

The construction depends on Lemma 2.1, the parity statement that every cube in the hypercube of resolutions contains an even number of squares of type A and X (and of type A and Y), whose proof is a case enumeration over cube types closed with the remark that the reader may verify the parity in all cases; if that parity failed for any cube type, no consistent edge assignment would exist, the differential would not square to zero, and the chain complex would not be defined.

Editorial extensions

If this is right

Odd Khovanov homology gives a genuine link invariant whose graded Euler characteristic is the Jones polynomial, so it is a second categorification of the same polynomial, not a deformation of the first.
Because $Kh'$ and $Kh$ agree after reducing coefficients mod 2 but not over the rationals, any invariant that can distinguish them is sensitive to integral torsion that the mod 2 theory forgets.
The reduced odd theory $tilde{Kh}'(L)$ is a link invariant without choosing a marked component, unlike the ordinary reduced Khovanov homology, which can depend on which component is fixed.
For non-split alternating links the odd theory reproduces Khovanov homology exactly, so any disagreement must come from non-alternating (indeed non-quasi-alternating) links.
The computed ranks for non-alternating knots are consistently smaller than the ordinary Khovanov ranks, which rules out a cancelling differential of Lee type in the odd theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Conjecture 1.9 proposes that the reduced odd theory is the $E_2$ page of a spectral sequence converging to the Heegaard Floer homology of the branched double cover with integer coefficients; the paper's computations are consistent with this but do not test it, and a first check would be whether the ranks of $tilde{Kh}'(8_{19})$ and of $widehat{HF}(Sigma(L))$ can be reconciled by any differential.
If the integral spectral sequence exists, the mod 2 collapse to the known Ozsváth–Szabó sequence suggests the odd theory is the 'correct' integral refinement, and one would expect torsion in $Kh'$ to encode information invisible to $Kh$; the paper does not compute any integral torsion, so this is untested.
The reduced odd theory's independence of a marked component hints that the ordinary reduced Khovanov homology's dependence on that choice is an artifact of the symmetric-algebra signs rather than a topological feature.
The parity lemma is proved by case enumeration over the cube types of Figure 4, with the final parity check left to the reader; an independent machine verification of the face-typing over all cube configurations would settle that step without relying on the enumeration.

Original abstract

We describe an invariant of links in the three-sphere which is closely related to Khovanov's Jones polynomial homology. Our construction replaces the symmetric algebra appearing in Khovanov's definition with an exterior algebra. The two invariants have the same reduction modulo 2, but differ over the rationals. There is a reduced version which is a link invariant whose graded Euler characteristic is the normalized Jones polynomial.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We describe an invariant of links in the three-sphere which is closely related to Khovanov's Jones polynomial homology. Our construction replaces the symmetric algebra appearing in Khovanov's definition with an exterior algebra. The two invariants have the same reduction modulo 2, but differ over the rationals. There is a reduced version which is a link invariant whose graded Euler characteristic is the normalized Jones polynomial.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We describe an invariant of links in the three-sphere which is closely related to Khovanov's Jones polynomial homology.	contribution 0.99	0.88	0.53	asserted 1.00
s1	Our construction replaces the symmetric algebra appearing in Khovanov's definition with an exterior algebra.	method 0.77	0.90	0.14	asserted 1.00
s2	The two invariants have the same reduction modulo 2, but differ over the rationals.	result 1.00	0.78	0.24	asserted 1.00
s3	There is a reduced version which is a link invariant whose graded Euler characteristic is the normalized Jones polynomial.	result 0.69	0.82	0.25	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.68 · evidence from proof or derivation 0.99.

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Figures

**Figure 1.** Elementary saddle move. This illustrates an elementary saddle move. The initial picture is indicated by solid lines: the two solid arcs represent one resolution, and the dotted arcs represent the other resolution. The solid oriented arc represents the initial arc, and the dotted one is to be rotated 90◦ counterclockwise. The second isomorphism is well-defined only up to sign. We define FM : Λ∗V (S1) → Λ ∗V (S2) to be the composition Λ ∗ (V (S1)) ∼= −−−−→ Λ ∗ V (S2) (a1−a2) ∼= −−−−→ (a1 − a2) ∧ Λ ∗V (S2) ⊂ −−−−→ Λ ∗V (S2). This map sends 1 ∈ Λ ∗V (S1) to ±(a1 − a2) ∈ Λ ∗V (S2).…

**Figure 2.** Commutation chart. The set of pairs of arcs can be placed into four categories, Types A, C, X, and Y . In Type A, the double-composite maps anticommute. In Type C, they commute. The two remaining cases are labelled X and Y as above. The thicker curves denote components of S1, while the thinner arcs represent the one-handles specified by Z. Orientations of these one-handles are specified by arrows when they are needed; when they are dropped, it is because the corresponding picture has the stated type for either choice of orientation. L L0 L1

**Figure 3.** Oriented skein moves. The oriented crossing on the left has two oriented resolutions, as illustrated. Note that there are two possible choices of orientation at each crossing. (To see the other one, turn the figure 180◦ .) Given two vertices I0, I1 : X −→ {0, 1} in the hypercube, we say that there is an oriented edge from I0 to I1 if there is some x ∈ X with the property that I0(x) = 0 I1(x) = 1 I0(y) = I1(y) if x 6= y

**Figure 4.** Cubes. Different types of cubes in the hypercube of resolutions. Each cube is labelled with choices of distributions of faces of type A, C, X, and Y . The different possibilities correspond to various orientations on the arcs connecting them. the second type of cube appearing in

**Figure 5.** Counting types of squares in a cube. Consider the cube corresponding to the diagram on the left-hand side of the figure. The six square faces of the cube are gotten by either forgetting or performing surgery along one of the arcs. These are shown and classified on the right. θ=+1 θ=−1

**Figure 6.** Black and white coloring. Given a checkerboard coloring of the projection, we can introduce a function θ : X −→ {±1} as illustrated. η(I). It is straightforward to verify that Φ is an isomorphism of chain complexes, from (C(D), ∂ǫ) to (C(D), ∂ǫ ′)

**Figure 7.** Reidemeister I. Starting with the initial picture on the left, we can perform two types of Reidemeister I move, as illustrated in the next column. After that, we display the 0- and 1-resolutions respectively. Fix some initial orientation o for the crossings in a knot projection, and a sign assignment ǫ. Consider next a different set of orientations on the initial crossings, specified by θ · o (i.e. if c is some crossing with type θ(c) = +1, then o ′ (c) is the same as o(c), whereas if θ(c) = −1, then the two orientations point in opposite directions). Clearly, this change of…

**Figure 8.** Reidemeister II. Starting with the initial picture on the left, we can perform a Reidemeister move to obtain the second picture. We have then illustrated (and labelled) the four resolutions of this diagram. and the map D identifies v0 with v1 (where v0 denotes the unknotted component in D0 and v1 denotes the component which is connected to it). This chain complex is clearly quasi-isomorphic to the subcomplex (v1 − v0)∧ (C(D0), ∂0), which in turn is identified with the complex C(D). This establishes the stated isomorphism for one of the two types of Reidemeister move. Proposition…

**Figure 9.** Reidemeister III. Consider the local picture on the left. We obtain a corresponding cube of resolutions which is pictured on the right, where the resolutions are labelled by their corresponding resolution vector. The thick arrow is contracted to obtain the chain complex from

**Figure 10.** Reidemeister III, after contraction. 001 010 µ \ 010 µ \ 101 111 110 011 100

**Figure 11.** Reidemeister III, intermediate states. On the left is the quotient complex after the rightmost thick arrow from

**Figure 12.** After Reidemeister III. This is the complex after Reidemeister III is performed on

Figures reproduced from arXiv:0710.4300 by the authors.

References

Works cited (13)

[1]

D. Bar-Natan. On Khovanov’s categoriﬁcation of the Jone s polynomial. Algebraic and Geometric Topology , 2:337–370, 2002

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[2]

G. Burde and H. Zieschang. Knots, volume 5 of de Gruyter Studies in Mathematics . Walter de Gruyter & Co., Berlin, second edition, 2003

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M. Khovanov. A categoriﬁcation of the Jones polynomial. Duke Math. J. , 101(3):359–426, 2000

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M. Khovanov. Patterns in knot cohomology. I. Experiment. Math. , 12(3):365–374, 2003

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M. Khovanov and L. Rozansky. Matrix factorizations and l ink homology. math.QA/0401268

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M. Khovanov and L. Rozansky. Matrix factorizations and l ink homology II. arXiv:math/0505056

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P. B. Kronheimer and T. S. Mrowka. Floer homology for Seib erg-Witten Monopoles. In preparation

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E.S. Lee. An endomorphism of the Khovanov invariant. Adv. Math. , 197(2), 2005

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C. Manolescu and P. S. Ozsv´ ath. On the Khovanov and Floer homologies of quasi-alternating links. arXiv:0708.3249, 2007

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[10]

P. S. Ozsv´ ath and Z. Szab´ o. On the Heegaard Floer homology of branched double-covers. Adv. Math., 194(1):1– 33, 2005

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J. Rasmussen. Knot polynomials and knot homologies. In Geometry and topology of manifolds , volume 47 of Fields Inst. Commun. , pages 261–280. Amer. Math. Soc., Providence, RI, 2005

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[12]

A. Shumakovitch. Khoho. Available at http://www.geom etrie.ch/KhoHo/, 2003

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[13]

P. Turner. Calculating Bar-Natan’s characteristic tw o Khovanov homology. math.GT/0411225, 2004. Department of Mathematics, Columbia University, New York, NY 10027 E-mail address : petero@math.columbia.edu Department of Mathematics, Princeton University, Princet on, New Jersey 08544 and DPMMS, University of Cambridge, UK E-mail address : jrasmus@math.prin...

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Branes And Supergroups
hep-th 2014-10 conditional novelty 7.0 of 10

A two-sided D3-brane/fivebrane configuration produces Chern-Simons theory with supergroup gauge symmetry U(m|n) or OSp(m|2n), and S-duality of the brane system yields a magnetic dual and a supergroup analog of Khovano...
Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m)
math.QA 2012-12 conditional novelty 7.0 of 10

Khovanov homology and its sl(3) analogue arise as 2-representations of categorified quantum sl(m) via categorical skew Howe duality, giving a direct construction of foam-based link homologies.

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47 later papers cite it

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2 later papers by other authors are listed below: 1 built on it and 1 used its method or tool.

Builds on it 1 of 3 papers with this label

Extended the totally twisted construction in the context of this paper's odd Khovanov homology.

A sign assignment in totally twisted Khovanov homology (2011) · 0.98 · paraphrased: the licence does not allow quoting

Uses its method or tool 1 of 5 papers with this label

Used a standard argument from this paper to reduce the check to whether the edge maps assigned to a link in both theories agree up to a sign.

Sutured annular Khovanov-Rozansky homology (2015) · 0.89 · paraphrased: the licence does not allow quoting

How the literature uses it

43 arXiv papers cite this in 105 sentences

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Papers citing it · 63 sentences without a confident label

Contests it 1 paper

Unfortunately, there is a gap in the original proof from [ref] , noticed by Cotton Seed: given a sign assignment $ϵ$ with $dϵ=ψ$ the authors of [ref] constructed $ϵ$ with $dϵ=ψ$, but an isomorphism of cubes $Tϵ≅Tϵ$ is missing.

A 2-category of chronological cobordisms and odd Khovanov homology · Diamond-change revisited · contest 0.94

Builds on it 3 papers

We show how to extend the totally twisted construction over $$, in the context of the odd Khovanov homology of Ozsváth, Rasmussen and Szabó [ref] .

A sign assignment in totally twisted Khovanov homology · Introduction · 0.98
The proof of the fact that such knots are $$H-thin is based on the long exact sequences [ref] and, therefore, can be applied verbatim to the odd homology as well [ref] .

Patterns in odd Khovanov homology · Finding quasi-alternating knots · 0.80
All of the chain complexes we use are essentially straightforward modifications to hyperplane arrangements of existing constructions for links [ref] .

Odd Khovanov Homology for Hyperplane Arrangements · Introduction · 0.80

Uses its method or tool 4 papers

Like [ref] , we essentially follow the method and exposition of [ref] .

Odd Khovanov Homology for Hyperplane Arrangements · Reidemeister Invariance · 1.00
However, using a proof adapted from Ozsváth, Rasmussen, and Szabó [ref] , one can show that complexes with different edge assignments are isomorphic.

A Turaev surface approach to Khovanov homology · Khovanov homology for ribbon graphs · 0.91
By a standard argument (see [ref] ), it suffices to check that the maps on the edges of the “cube of resolutions” assigned to a link in both theories agree up to a sign.

Sutured annular Khovanov-Rozansky homology · The annular $2$ invariant · 0.89

Compares with it 6 papers

On the other hand, a projective TQFT from [ref] produces a cube that commutes only up to signs, which has to be fixed before collapsing.

A 2-category of chronological cobordisms and odd Khovanov homology · Introduction · 0.99
One strange step in [ref] , which does not appear in the Khovanov's construction, is looking for an edge assignment for the cube of resolutions.

Cobordisms with chronologies and a generalisation of the Khovanov complex · A categorification of the Jones polynomial · 0.85
This invariant is quite closely related to the reduced Odd Khovanov homology of [ref] ; since theirs is an invariant of ordinary (unframed) links, a writhe-dependent shift is needed to recover Odd Khovanov homology from our bi-graded theory (see Proposition [ref]).

Odd Khovanov Homology for Hyperplane Arrangements · Introduction · 0.99

Cites it as support 2 papers

By [ref] , or a straightforward case analysis of configurations, it follows that an even number of faces in the cube of resolutions corresponding to $(D,x,y)$ come from ladybug configurations.

A Khovanov stable homotopy type · 2-dimensional moduli spaces · 0.97
Then * $(D;R)$ is a subcomplex of $(D;R)$ of index $2$. * $_(L;_2)≃(L;_2)$ and $_(L;_2)≃(L;_2)$. * $χ_q((L;R))=χ_q(_(L;R))=J_L(q)$ and $χ_q((L;R))=χ_q(_(L;R))=_L(q)$. * $(L;_2)≃(L;_2)⊗__2A__2$ [ref] and $_(L;R)≃_(L;R){1}⊕_(L;R){-1}$ [ref] .

Khovanov homology theories and their applications · Properties of the Khovanov homology · 0.97

Cites it as background 15 papers

We assume the reader is familiar with odd Khovanov homology, as described in [ref] .

A sign assignment in totally twisted Khovanov homology · The construction · 0.98
Ozsváth, Rasmussen and Szabó [ref] constructed a modified version of Khovanov homology, odd Khovanov homology $'(·)$.

An integral lift, starting in odd Khovanov homology, of Szab\'o's spectral sequence · Introduction · 0.97
Ozsvath, Rasmussen und Szabo konstruierten 2007 eine weitere Verschlingungsinvariante, die ungerade Khovanov-Homologie ( [ref] ).

A spectral sequence in odd Khovanov homology (Eine Spektralsequenz in ungerader Khovanov-Homologie) · Einleitung · 1.00

What it cites · 1 sentence without a confident label

Uses the method or tool of 1 paper

The argument is more or less the same one used by Khovanov to prove invariance of the ordinary $sl(2)$ homology [ref] , see also [ref] , [ref] .

Matrix factorizations and link homology II · Topological invariance · 0.95

Discussion

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