Matrix Factorizations and Khovanov–Rozansky Link Homology

Fix a finite set $I$, let $S=\mathbb{Q}[a,x_i:i\in I]$ be the polynomial ring of The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution in a variable $a$ and the variables $x_i$, bigraded by

$\mathrm{deg}a=(2,0),\mathrm{deg}{x}_i=(0,2),$

and for a bigraded $S$-module $M$ and $(n_1,n_2)\in {\mathbb{Z}}^2$ write $M\{n_1,n_2\}$ for the internal shift

$M\{n_1,n_2{\}}_{(k,l)}=M_{(k-n_1,l-n_2)}$

in the sense of the internal shift of Associative graded algebras, bimodules, and internal shifts (the two-parameter refinement of the one-parameter shift, applied to each bigrading separately).

Fix signs ${ϵ}_i\in \{1,-1\}$ for $i\in I$ and put $w=a{\sum }_{i\in I}{ϵ}_i{x}_i$, an element of $S$ of bidegree $(2,2)$. A bigraded matrix factorization with potential $w$ is a pair $M=(M^0,M^1,d)$ consisting of free bigraded $S$-modules $M^0,M^1$ (of arbitrary, possibly infinite, rank) together with $S$-linear maps

$d\text{\hspace{0.17em}⁣}:M^0\to M^1,d\text{\hspace{0.17em}⁣}:M^1\to M^0$

of bidegree $(1,1)$ such that

$d^2=w\cdot \mathrm{i}\mathrm{d},$

i.e. $d^2(m)=wm$ for every $m\in M^0\oplus M^1$. A morphism $f\text{\hspace{0.17em}⁣}:M\to N$ is a pair of $S$-linear maps of bidegree $(0,0)$ on the two components commuting with $d$; a homotopy between two morphisms is a pair of maps of bidegree $(-1,-1)$ satisfying the usual homotopy formula. Write ${\mathrm{m}\mathrm{f}}_w$ for the category of factorizations with potential $w$ and their bidegree-preserving morphisms, and ${\mathrm{h}\mathrm{m}\mathrm{f}}_w$ for its homotopy category, in which the morphisms are the bidegree-preserving morphisms modulo null-homotopic ones. Explicitly, $f-g=d_{N}h+h{d}_M$, where $h$ reverses inner parity. Internal shifts $M\{u,v\}$ shift both modules and retain their inner parity. Write $\mathrm{\Pi }M$ for the parity reversal: $(\mathrm{\Pi }M{)}^0=M^1$, $(\mathrm{\Pi }M{)}^1=M^0$, with the same differential. It preserves the potential, acts on morphisms by the same component maps, and satisfies ${\mathrm{\Pi }}^{2}M=M$. For $w=0$, the direct sum of the two parity cohomologies of $\mathrm{\Pi }M$ is the same bigraded vector space as that of $M$; only their parity labels change.

Caveats. Unless $w=0$, $d$ does not square to zero, so a factorization is not an ordinary complex. If $I=\mathrm{\varnothing }$ then $S=\mathbb{Q}[a]$ and a factorization with $w=0$ is a $2$-periodic complex $M^0\stackrel{d}{\to }{M}^1\stackrel{d}{\to }{M}^0$ of free bigraded $\mathbb{Q}[a]$-modules; for $w\mathrm{\ne }0$ the only failure of the complex axioms is the identity $d^2=w\cdot \mathrm{i}\mathrm{d}$. The sign vector $({ϵ}_i)$ and the bigrading are part of the data, and on this page the potential is always $a$ times a linear form with coefficients in $\{1,-1\}$. The conventions (two-variable bigrading with $\mathrm{deg}a=(2,0)$, $\mathrm{deg}{x}_i=(0,2)$; $d$ of bidegree $(1,1)$; homotopies of bidegree $(-1,-1)$; morphisms commuting with $d$) follow Khovanov-Rozansky, Matrix factorizations and link homology II, section 1, formulas (1)-(2) and the lattice picture of its Figure 3; the ungraded definitions of a duplex and a free-module factorization are Definitions 1-2 in section 2 (printed pp. 13-14) of Khovanov-Rozansky, Matrix factorizations and link homology.

In the setting of Bigraded matrix factorizations with a potential, and writing a two-term factorization $(p,q)$ for $S\stackrel{p}{\to }S\{n_1,n_2\}\stackrel{q}{\to }S$ as a Koszul row (the notation fixed for this construction in the sequel to this definition; the shifts $n_1,n_2$ are the shifts of the middle term):

(1) Oriented arc. To an oriented arc $c$ oriented from the endpoint labelled $x_2$ to the endpoint labelled $x_1$ assign the two-term factorization $C_c:=(a,x_1-x_2)=[S\stackrel{a}{\to }S\{-1,1\}\stackrel{x_1-x_2}{\to }S]$ over $S=\mathbb{Q}[a,x_1,x_2]$, with potential $w=a(x_1-x_2)$; the differentials are $a$ and $x_1-x_2$, the middle term carries the bigrading shift $\{-1,1\}$, and both maps have bidegree $(1,1)$. It is an object of ${\mathrm{h}\mathrm{m}\mathrm{f}}_w$: the square of the differential is $(x_1-x_2)\circ a=a(x_1-x_2)=w\cdot \mathrm{i}\mathrm{d}$.

(2) Wide edge. To a wide edge $t$ whose four adjacent edge labels are $x_1,x_2$ on its outgoing ends and $x_3,x_4$ on its incoming ends assign the tensor product, over $S=\mathbb{Q}[a,x_1,x_2,x_3,x_4]$, of the two Koszul rows $(a,x_1+x_2-x_3-x_4)=[S\stackrel{a}{\to }S\{-1,1\}\stackrel{x_1+x_2-x_3-x_4}{\to }S]$ and $(0,x_1{x}_2-x_3{x}_4)=[S\stackrel{0}{\to }S\{-1,3\}\stackrel{x_1{x}_2-x_3{x}_4}{\to }S],$ with potential $w=a(x_1+x_2-x_3-x_4)$. For a two-fold tensor product of rows $(a_1,b_1)\otimes (a_2,b_2)$ the total differential satisfies $d^2=(a_1{b}_1+a_2{b}_2)\cdot \mathrm{i}\mathrm{d}$ with the Koszul sign convention for the totalization, so here $d^2=a(x_1+x_2-x_3-x_4)+0\cdot (x_1{x}_2-x_3{x}_4)=w$, and the middle terms are $S\{-1,1\}\oplus S\{-1,3\}$. It is an object of ${\mathrm{h}\mathrm{m}\mathrm{f}}_w$.

Both assignments produce objects of ${\mathrm{h}\mathrm{m}\mathrm{f}}_w$ with the stated potentials. Caveats: the potential of a wide edge is linear in the $x_i$; the quadratic entry $x_1{x}_2-x_3{x}_4$ enters only through the row whose first differential is $0$; and the shifts $\{-1,1\}$ and $\{-1,3\}$ are part of the definition and must be propagated exactly (Khovanov-Rozansky II, section 1, formulas (2)-(4)). Khovanov-Rozansky I, introduction printed pp. 6-8, gives the fixed-$n$ analogue with potentials $x_i^{n+1}$ and different row entries and shifts; it is not the source of these parameter-$a$ formulas.

Let $\mathrm{\Gamma }$ be a finite planar graph in a disk whose edges are either oriented arcs between marks (including boundary points) or wide (thick) edges, each wide edge bounded by four oriented edge-ends, as in the Khovanov-Rozansky diagrams. Place finitely many marks, with at least one on every internal edge and every circle of $\mathrm{\Gamma }$, place any finite number of marks (possibly none) on each boundary edge, and label all marks and boundary points by distinct variables $x_1,\dots ,x_r$; the boundary points carry orientations ${ϵ}_p\in \{1,-1\}$. With $C_c$ and $C_t$ the arc and wide-edge factorizations of Arc and wide-edge Khovanov-Rozansky factorizations, define

$C(\mathrm{\Gamma }):={⨂}_c{C}_c\otimes {⨂}_t{C}_t,$

the tensor product taken over $S=\mathbb{Q}[a,x_1,\dots ,x_r]$ with all variables shared (the tensor product of the two-term factorizations of the local pieces, with the Koszul sign convention), and view $C(\mathrm{\Gamma })$ as a factorization over the smaller polynomial ring $R=\mathbb{Q}[a,x_p:p\text{ a boundary point}]$: the variables at internal marks are internal and are forgotten. Its potential is

$w_{\mathrm{\Gamma }}=a{\sum }_p{ϵ}_p{x}_p\mathrm{.}$

The sum is over the boundary points: every internal label occurs at exactly two edge-ends with opposite signs, whose contributions $+a{x}_i$ and $-a{x}_i$ cancel, while a boundary label occurs at exactly one edge-end, and the local potentials of the arc and wide-edge factorizations add to $w_{\mathrm{\Gamma }}$.

If $\mathrm{\Gamma }$ is closed (no boundary points), then $w_{\mathrm{\Gamma }}=0$ and $C(\mathrm{\Gamma })$ is a $2$-periodic complex $C^0(\mathrm{\Gamma })\stackrel{d}{\to }{C}^1(\mathrm{\Gamma })\stackrel{d}{\to }{C}^0(\mathrm{\Gamma })$ of bigraded $\mathbb{Q}[a]$-modules, whose cohomology is denoted $H(\mathrm{\Gamma })$. Each term is a free bigraded $\mathbb{Q}[a]$-module, generally of infinite rank. Contractible summands may be removed without changing its cohomology, but no finite-rank representative over $\mathbb{Q}[a]$ is asserted: already a one-mark circle has cohomology $\mathbb{Q}[x]\{-1,1\}$. For a nonempty closed graph, the row reduction proved below shows that $a$ acts trivially on $H(\mathrm{\Gamma })$. For the empty graph the empty tensor product is $\mathbb{Q}[a]$ in even parity, and $a$ need not act trivially. Caveats: $C(\mathrm{\Gamma })$ has infinite rank as an $R$-module whenever internal marks are present; the marks are auxiliary data, and a marking change alters $C(\mathrm{\Gamma })$ by a chain homotopy equivalence (proved later on this page); the potential vanishes exactly for closed graphs, which is why $C(\mathrm{\Gamma })$ is a genuine complex in that case.

Work over a polynomial ring $R$ and write $(a,b)={⨂}_i(a_i,b_i)$ for the Koszul factorization with rows $R\stackrel{a_i}{\to }R\stackrel{b_i}{\to }R$, so that its total differential squares to ${\sum }_i{a}_i{b}_i$. The row operations are first statements about ungraded factorizations. In the bigraded setting of Bigraded matrix factorizations with a potential, require the entries, substitutions and basis changes to be homogeneous of the degrees determined by the row shifts; only such operations give bigrading-preserving maps. Then:

(1) Row operations. For $\lambda \in R$ the replacement of two rows $(a_i,b_i),(a_j,b_j)$ by $(a_i,b_i+\lambda b_j),(a_j-\lambda a_i,b_j)$, all other rows unchanged, is an isomorphism of factorizations (it is the change of basis $\mathrm{\mid }00⟩,\mathrm{\mid }01⟩,\mathrm{\mid }10⟩,\mathrm{\mid }11⟩\mapsto \mathrm{\mid }00⟩,\mathrm{\mid }01⟩,\mathrm{\mid }10⟩+\lambda \mathrm{\mid }01⟩,\mathrm{\mid }11⟩$ on the tensor product of the two rows).

(2) Variable exclusion. Let $R=R^{\mathrm{\prime }}[y]$, let $w={\sum }_i{a}_i{b}_i\in R^{\mathrm{\prime }}$ (so $y$ is internal), and suppose one row of $(a,b)$ has the form $(0,y-\mu )$ with $\mu \in R^{\mathrm{\prime }}$. Let $(a^{\mathrm{\prime }},b^{\mathrm{\prime }})$ be the Koszul factorization over $R^{\mathrm{\prime }}$ obtained by deleting that row and substituting $y\mapsto \mu$ in all other rows, and let $(a,b{)}^{\mathrm{\prime }}$ be $(a,b)$ restricted to $R^{\mathrm{\prime }}$ (an infinite-rank factorization). Then $(a,b{)}^{\mathrm{\prime }}\simeq (a^{\mathrm{\prime }},b^{\mathrm{\prime }})$ in ${\mathrm{h}\mathrm{m}\mathrm{f}}_w(R^{\mathrm{\prime }})$: the $R^{\mathrm{\prime }}$-complex $0\to R^{\mathrm{\prime }}[y]\stackrel{y}{\to }{R}^{\mathrm{\prime }}[y]\to 0$ splits into the contractible complexes $0\to R^{\mathrm{\prime }}{y}^j\stackrel{y}{\to }{R}^{\mathrm{\prime }}{y}^{j+1}\to 0$ for $j\ge 0$ and the rank-one complex $0\to R^{\mathrm{\prime }}\to 0$.

(3) Graph factorizations. For a nonempty planar marked graph $\mathrm{\Gamma }$ with $m_1$ arcs and $m_2$ wide edges the Koszul matrix of $C(\mathrm{\Gamma })$ has $m_1+m_2$ linear rows $(a,z)$ ($z$ linear in the $x_i$) and $m_2$ quadratic rows $(0,x_i{x}_j-x_k{x}_l)$; applying the row operations of (1) with $\lambda =1$ against the first linear row turns the first row into $(a,{\sum }_p{ϵ}_p{x}_p)$ over the boundary points and all other linear rows into $(0,z)$; if $\mathrm{\Gamma }$ is closed the first row becomes $(a,0)$ and, after restricting scalars to $\mathbb{Q}$ and deleting that row with its odd parity and internal shift retained, the remaining rows are $(0,z)$ for the other linear entries and $(0,q)$ for all the quadratic entries. Their Koszul complex computes $H(\mathrm{\Gamma })$, with the parity and internal shift contributed by the removed $(a,0)$ row retained and the cyclic grading collapsed to a bigrading, and $a$ acts trivially on $H(\mathrm{\Gamma })$.

Caveats: (2) is a chain homotopy equivalence, not an isomorphism of factorizations over $R$; the substitution $y\mapsto \mu$ must be applied to every remaining row simultaneously; the collapse in (3) loses the cyclic (homological) grading because the differential has nonzero bidegree. Source: Khovanov-Rozansky II, section 2, printed pp. 12-14, and the cyclic Koszul algebra of Khovanov-Rozansky I, section 2, printed pp. 13-17. Removing $a$ here computes cohomology after restricting scalars; it does not give a free representative of that cohomology in ${\mathrm{h}\mathrm{m}\mathrm{f}}_0(\mathbb{Q}[a])$.

Let ${\mathrm{\Gamma }}_0$ be the diagram of two disjoint oriented arcs with labels $x_1,x_4$ and $x_2,x_3$ and ${\mathrm{\Gamma }}_1$ the diagram of one wide edge with the same four labels, over $R=\mathbb{Q}[a,x_1,x_2,x_3,x_4]$, and write $C({\mathrm{\Gamma }}_i)=C^0({\mathrm{\Gamma }}_i)\stackrel{P_i\mathrm{/}{Q}_i}{\to }{C}^1({\mathrm{\Gamma }}_i)\stackrel{P_i\mathrm{/}{Q}_i}{\to }{C}^0({\mathrm{\Gamma }}_i)$ in the standard product bases of Khovanov-Rozansky II: $P_0=\left(\begin{array}{cc}a& x_3-x_2\\ a& x_1-x_4\end{array}\right),P_1=\left(\begin{array}{cc}{x}_1-x_4& x_2-x_3\\ -a& a\end{array}\right),$ $Q_0=\left(\begin{array}{cc}a& x_3{x}_4-x_1{x}_2\\ 0& x_1+x_2-x_3-x_4\end{array}\right),Q_1=\left(\begin{array}{cc}{x}_1+x_2-x_3-x_4& x_1{x}_2-x_3{x}_4\\ 0& a\end{array}\right),$ with the term shifts $C^0({\mathrm{\Gamma }}_0)=R\oplus R\{-2,2\},C^1({\mathrm{\Gamma }}_0)=R\{-1,1\}\oplus R\{-1,1\},$ $C^0({\mathrm{\Gamma }}_1)=R\oplus R\{-2,4\},C^1({\mathrm{\Gamma }}_1)=R\{-1,1\}\oplus R\{-1,3\}\mathrm{.}$

Define ${\chi }_0\text{\hspace{0.17em}⁣}:C({\mathrm{\Gamma }}_0)\to C({\mathrm{\Gamma }}_1)$ by the matrices $U_0^0=\left(\begin{array}{cc}{x}_4-x_2& 0\\ 0& 1\end{array}\right),U_0^1=\left(\begin{array}{cc}{x}_4& -x_2\\ -1& 1\end{array}\right),$ and ${\chi }_1\text{\hspace{0.17em}⁣}:C({\mathrm{\Gamma }}_1)\to C({\mathrm{\Gamma }}_0)$ by $U_1^0=\left(\begin{array}{cc}1& 0\\ 0& x_4-x_2\end{array}\right),U_1^1=\left(\begin{array}{cc}1& x_2\\ 1& x_4\end{array}\right)\mathrm{.}$

Then ${\chi }_0$ is a morphism of factorizations of bidegree $(0,2)$ and ${\chi }_1$ is a morphism of bidegree $(0,0)$; and in the equivalent Koszul forms (8), (9) of the source, obtained by the row operations $[12{]}_1$ on $C({\mathrm{\Gamma }}_0)$ and $[21{]}_{-x_2}$ on $C({\mathrm{\Gamma }}_1)$, they become the flip morphisms ${\chi }_0=\mathrm{I}\mathrm{d}\otimes {\psi }^{\mathrm{\prime }}(x_4-x_2)$ and ${\chi }_1=\mathrm{I}\mathrm{d}\otimes \psi (x_4-x_2)$, where $\psi (y)$ is the morphism $(0,yz)\to (0,z)$ and ${\psi }^{\mathrm{\prime }}(y)$ its opposite. Consequently the composites ${\chi }_1{\chi }_0\text{\hspace{0.17em}⁣}:C({\mathrm{\Gamma }}_0)\to C({\mathrm{\Gamma }}_0)$ and ${\chi }_0{\chi }_1\text{\hspace{0.17em}⁣}:C({\mathrm{\Gamma }}_1)\to C({\mathrm{\Gamma }}_1)$ are nonzero endomorphisms of bidegree $(0,2)$, and on the Koszul standard forms they act as multiplication by the element $x_4-x_2\in R$.

Caveats: the maps are not inverse to each other; all signs and shifts are fixed by the printed matrices (Khovanov-Rozansky II, formulas (5)-(6); the published version writes the same two maps with half-integer cohomological degrees in formulas (12)-(13)); the bases are homogeneous for the bigrading.

With ${\chi }_0\text{\hspace{0.17em}⁣}:C({\mathrm{\Gamma }}_0)\to C({\mathrm{\Gamma }}_1)$ and ${\chi }_1\text{\hspace{0.17em}⁣}:C({\mathrm{\Gamma }}_1)\to C({\mathrm{\Gamma }}_0)$ as in The wide-edge morphisms chi-zero and chi-one, assign to a crossing $p$ of a tangle diagram the following two-term complex of matrix factorizations, using the integer grading of arXiv:math/0505056v2, Figure 6.

Positive crossing. $C_p=[0\to C({\mathrm{\Gamma }}_0)\{0,2\}\stackrel{{\chi }_0}{\to }C({\mathrm{\Gamma }}_1)\to 0],$ with $C({\mathrm{\Gamma }}_1)$ in cohomological degree $0$ (so $C({\mathrm{\Gamma }}_0)\{0,2\}$ sits in degree $-1$); the shift $\{0,2\}$ makes the differential bidegree-preserving.

Negative crossing. $C_p=[0\to C({\mathrm{\Gamma }}_1)\{0,-2\}\stackrel{{\chi }_1}{\to }C({\mathrm{\Gamma }}_0)\{0,-2\}\to 0],$ with $C({\mathrm{\Gamma }}_1)\{0,-2\}$ in cohomological degree $0$ and $C({\mathrm{\Gamma }}_0)\{0,-2\}$ in degree $1$; the overall shift $\{0,-2\}$ is the normalization required by the Reidemeister IIa move.

In both cases the differential is ${\chi }_0$ or ${\chi }_1$ and has bidegree $(0,0)$ as a map of the shifted terms.

Recorded source conflict and regrading. The arXiv prose before Figure 6 incorrectly displays the negative crossing with ${\chi }_0$ in the opposite direction. Its Figure 6, the bidegrees of the matrices (5)-(6), and the negative-crossing Euler relation in section 7 agree with the ${\chi }_1$-cone above. The published version of record corrects the direction but also changes the grading: writing $C_{p,\mathrm{v}2}^{+}$ and $C_{p,\mathrm{v}2}^{-}$ for the two complexes above, formulas (12)-(13) on printed p. 1393 give $C_{p,\mathrm{p}\mathrm{u}\mathrm{b}}^{+}=C_{p,\mathrm{v}2}^{+}\{-\frac{1}{2},-\frac{1}{2}\}[-\frac{1}{2}],C_{p,\mathrm{p}\mathrm{u}\mathrm{b}}^{-}=C_{p,\mathrm{v}2}^{-}\{\frac{1}{2},\frac{1}{2}\}[\frac{1}{2}]\mathrm{.}$ Here $C[n{]}^j=C^{j+n}$; the displayed identifications specify term degrees and maps, with the usual compatible shift signs. Both published cones have outer degrees $-\frac{1}{2},\frac{1}{2}$. Their respective term shifts are $\{-\frac{1}{2},\frac{3}{2}\},\{-\frac{1}{2},-\frac{1}{2}\}$ for the positive cone and $\{\frac{1}{2},-\frac{3}{2}\}$ on both terms of the negative cone.

Caveat: no absolute normalization is claimed; the $\{0,-2\}$ shift is fixed only by the source's IIa normalization.