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The reduced Khovanov-Rozansky homology

Definition

Let $D$ be a marked braid diagram with nonempty closure, so its label count satisfies $m \geq 1$ and let $H (D) = ⨁_{j, k, l} H_{k, l}^{j} (D)$ be the triply graded groups of The Khovanov-Rozansky complex and trigraded braid homology, built over the polynomial ring $Q [a, x_{1}, \dots, x_{m}]$ generated by the coefficient variable $a$ and the $m$ strand labels. The reduced Khovanov-Rozansky homology $\overline{H} (D)$ is defined by repeating that construction verbatim with the label ring replaced by the ring of differences from one chosen coordinate, $Q [a, x_{1}, \dots, x_{m}] ⇝ Q [a, x_{2} - x_{1}, \dots, x_{m} - x_{1}],$ the coefficient $a$ being retained: the same marked MOY resolutions, arc and wide-edge factorizations, positive and negative crossing cones, removal of contractible summands and induced differentials, but with the $m$ strand labels replaced by the $m - 1$ differences $x_{2} - x_{1}, \dots, x_{m} - x_{1}$ ; the trivial variable of the one-mark circle is thereby dropped. The result $\overline{H} (D) = ⨁_{j, k, l} {\overline{H}}_{k, l}^{j} (D)$ is again a triply graded $Q$ -vector space, with the trigrading of that item.

By Khovanov-Rozansky II, end of section 1, one has a trigraded isomorphism $H (D) ≅ \overline{H} (D) \otimes_{Q} Q [x]$ in which $x$ is the trivial variable, and in the reduced theory the unknot has one-dimensional homology. The splitting is a statement about the two constructions, not a definition of $\overline{H} (D)$ .

Grading normalization and the unknot. In the trigrading of The Khovanov-Rozansky complex and trigraded braid homology the one-mark circle has cohomology $Q [x] {- 1, 1}$ at the resolution level (the companion computation of the matrix-factorization page, indexed there by the first and second bigrading); the reduction removes the trivial factor, leaving the one-dimensional reduced group in bidegree $(- 1, 1)$ at that level, and in the full trigraded theory the unknot class of the reduced homology sits in the raw tridegree $(- 1, 1, 0)$ in $(k, l, j)$ order, whose image under the global correction $(1, - 1, 0)$ of The Koszul-Hochschild comparison respects crossing differentials and trigradings is the class $(0, 0, 0)$ .

Caveats. $\overline{H} (D)$ is a construct of the reduced label ring and is not defined here by quotienting an arbitrary presentation of $H (D)$ ; the splitting $H (D) ≅ \overline{H} (D) \otimes_{Q} Q [x]$ is part of the source's assertion, recorded here with its grading conventions. The trivial variable $x$ is a polynomial variable carried by the one-mark circle and is not the coefficient variable $a$ . Under AC (The Axiom of Choice), invariance of $\overline{H} (D)$ up to an overall trigrading shift is inherited from Khovanov-Rozansky braid homology is an oriented link invariant up to shift and is used in HHH is an oriented-link invariant up to an overall trigrading shift; the trigrading normalization is the one of The Khovanov-Rozansky complex and trigraded braid homology, and no Wu regrading is asserted here.

The coordinate reduction and polynomial splitting are choice-free. The oriented-link-invariance claim uses AC through the indicated Markov supplier and for homogeneous basis choices in step 3.1. The zero-strand tensor unit $Q [a]$ has no chosen label $x_{1}$ and is excluded from this reduced construction.

Facts & Assumptions

Given: the nonempty marked braid diagram $D$ , its label ring $Q [a, x_{1}, \dots, x_{m}]$ , and the chosen coordinate $x_{1}$ .

[F1]

The raw braid complex is assembled from arc and wide-edge rows and crossing maps over the shared polynomial coefficient ring; its outer cohomology is $H (D)$ (The Khovanov-Rozansky complex and trigraded braid homology, The factorization of a marked MOY graph).

[F2]

Compatible homogeneous Koszul row operations are factorization isomorphisms. In particular the pair $(0, q), (a, β)$ may be replaced by $(0, q - t β), (a, β)$ for homogeneous $t$ of bidegree $(0, 2)$ (Koszul row operations and variable exclusion preserve homotopy type).

[F3]

After the indicated local row changes, the two crossing maps have forms $1 \otimes ψ (x_{4} - x_{2})$ and $1 \otimes ψ^{'} (x_{4} - x_{2})$ , with common first row $(a, x_{1} + x_{2} - x_{3} - x_{4})$ and second rows $(0, x_{2} - x_{3})$ or $(0, (x_{2} - x_{3}) (x_{4} - x_{2}))$ (The wide-edge morphisms chi-zero and chi-one).

[F4]

Under AC, a finite Markov sequence compares two ambient-isotopic closed braids by the explicit local equivalences and their stated shifts (Khovanov-Rozansky braid homology is an oriented link invariant up to shift, The Axiom of Choice).

Proof

technique · homogeneous row gauges remove the common translation coordinate; free polynomial extension preserves both cohomology operations

1.1F1F2algebra

Put $t = x_{1}$ and $u_{j} = x_{j} - x_{1}$ , $j > 1$ , with $u_{1} = 0$ . This gives the graded polynomial-ring isomorphism $Q [a, x_{1}, \dots, x_{m}] = Q [a, u_{2}, \dots, u_{m}] [t]$ , with $\deg t = (0, 2)$ . Every arc entry is a difference and is independent of $t$ . A wide-edge linear row has entry $β$ independent of $t$ , whereas its quadratic entry is $q = q_{0} + t β$ , where $q_{0}$ is the same quadratic expression in the $u$ labels. Apply [F2] to replace that second row by $(0, q_{0})$ . Thus all resolution factorizations are polynomial extensions of the corresponding reduced factorizations.

2.1F1F2F3step 1.1algebra

The same reduction respects the crossing cube. In the local forms of [F3], the entries $x_{1} + x_{2} - x_{3} - x_{4}$ , $x_{2} - x_{3}$ and $x_{4} - x_{2}$ are all unchanged by the common translation. The wide-row operation subtracting $x_{2} β = (t + u_{2}) β$ is the gauge of step 1.1 followed by subtraction of $u_{2} β$ . Both flip matrices are therefore independent of $t$ in these bases. Consequently the entire outer complex of factorizations is isomorphic to the reduced one extended by $Q [t]$ , including both crossing signs and all gradings. Taking inner and then outer cohomology commutes with this free extension: its monomial basis makes it a direct sum of shifted copies of each complex, with componentwise differentials, kernels and images. This proves $H (D) ≅ \overline{H} (D) \otimes_{Q} Q [t]$ .

2.2F1step 1.1algebra

For the one-mark circle the row is $(a, 0)$ ; setting its only label to zero leaves $Q [a] \overset{a}{\to} Q [a] {- 1, 1} \overset{0}{\to} Q [a]$ . Its odd inner cohomology is $Q {- 1, 1}$ and its even inner cohomology is zero because multiplication by $a$ is injective. The sole outer term is in degree0, giving raw $(k, l, j) = (- 1, 1, 0)$ . This verifies the stated reduced unknot value and grading.

3.1F1F4step 1.1step 2.1algebra∎

Inherit invariance at the level of graded vector spaces. Let $D_{1}, D_{2}$ have isotopic nonempty closures. Under [F4] their unreduced dimensions agree with a shift $(j_{0}, k_{0}, l_{0})$ . Write $b_{D} (j, k, l) = \dim H_{k, l}^{j} (D)$ and $r_{D} (j, k, l) = \dim {\overline{H}}_{k, l}^{j} (D)$ ; these dimensions are finite because both constructions use finite crossing cubes and finitely many polynomial variables of positive second degree. The splitting of step 2.1 gives $b_{D} (j, k, l) = \sum_{d \geq 0} r_{D} (j, k, l - 2 d)$ , a finite sum in each degree since the second gradings are bounded below. Consequently $r_{D} (j, k, l) = b_{D} (j, k, l) - b_{D} (j, k, l - 2)$ , so the same shift relates the reduced dimensions. Under AC choose bases in all homogeneous pieces; equal finite dimensions then give a trigraded vector-space isomorphism with that shift. This proves the asserted noncanonical invariance without assuming that arbitrary unreduced equivalences preserve a chosen label coordinate. Changing the chosen label only makes an invertible linear change among the differences, so the constructions of steps 1.1 and 2.1 give the same polynomial splitting. AC is used through [F4] and for the homogeneous basis choices.

Remarks

The one-strand normalization recorded above is computed on the later examples page in The trivial one-braid and the grading normalization ↗; nothing in the definition depends on that item.

Depends on

Used by

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Sources

Mikhail Khovanov and Lev Rozansky, Matrix factorizations and link homology II, arXiv:math/0505056v2; end of section 1, printed pp. 11-12 (standard reference, not scraped)
Mikhail Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules, arXiv:math/0510265v3; proof of Theorem 1, printed pp. 7-9 (standard reference, not scraped)