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version="0.1";
category="Miscellaneous";
// summary description of the library
info="
LIBRARY: linkhomSO2N.lib Computation of link homology using web compilation
AUTHOR: Nils Carqueville, Daniel Murfet
KEYWORDS: matrix factorization, link homology
PROCEDURES:
";
LIB "linalg.lib";
LIB "matrix.lib";
LIB "ring.lib";
LIB "blow.lib";
LIB "mfweb.lib";
////////////////////////////////////////////////////////////////////
// USAGE GUIDE
//
// We expect the ambient ring to have twice the number of variables as edges
// in the link under consideration. In principle parameters should be OK, but
// this has not been checked.
//
// The references to [KR07] are to Khovanov and Rozansky "Virtual crossings,
// convolutions and a categorification of the SO(2N) Kauffman polynomial",
// arXiv:math/0701333v1.
////////////////////////////////////////////////////////////////////
// NOTES ON GRADING
//
// In [KR07, p.21] x-variables are assigned degree 2 and y-variables have
// degree 2N.
////////////////////////////////////////////////////////////////////
// NOTES ON MFs ASSIGNED TO VERTICES
//
// We follow the convention of [KR07, (3.37)] (which differs from the
// one in arXiv:math/0401268v2) and label the edges of a vertex as in
// the picture
//
// 3 4
// \ /
// \/
// /\
// / \
// 1 2
//
// where i=1,2,3,4 really means the pair of variables xi,yi.
//
// Note that the convention of [KR07] (see page 21 there) is that all
// edges of the 4-vertices considered are oriented outwards, and hence
// the associated MFs, factorise the polynomial
// W1 + W2 + W3 + W4
// (see [KR07, (3.38)]) with only plus signs. We however want to have
// a MF of
// - W1 - W2 + W3 + W4
// and hence we always replace x1,y1,x2,y2 by -x1,-y1,-x2,-y2 in the
// formulas for MFs of [KR07].
////////////////////////////////////////////////////////////////////
// w2poly
//
// Given two variables a,b, we return the polynomial w(a,b) of [KR07, (3.9)].
proc w2poly(poly a, poly b, int N)
{
poly wa = a^(2*N + 1);
poly wb = b^(2*N + 1);
poly w2 = ( wa - wb )/( a - b );
kill wa, wb;
return(w2);
}
////////////////////////////////////////////////////////////////////
// w3poly
//
// Given three variables a,b,c, we return the polynomial w(a,b,c) of [KR07, (3.9)].
proc w3poly(poly a, poly b, poly c, int N)
{
poly w1 = w2poly(a,b,N);
poly w2 = w2poly(a,c,N);
poly w3 = ( w1 - w2 )/( b - c );
kill w1, w2;
return(w3);
}
////////////////////////////////////////////////////////////////////
// w4poly
//
// Given three variables a,b,c,d, we return the polynomial w(a,b,c,d) of [KR07, (3.8)].
proc w4poly(poly a, poly b, poly c, poly d, int N)
{
poly w1 = w3poly(a,b,c,N);
poly w2 = w3poly(a,b,d,N);
poly w4 = ( w1 - w2 )/( c - d );
kill w1, w2;
return(w4);
}
////////////////////////////////////////////////////////////////////
// commonKoszul
//
// We are given four pairs of variables x1,y1,x2,y2,x3,y3,x4,y4 and
// an integer N, and we return the matrix factorisation [KR2, (3.59)]
// with x1,y1,x2,y2 replaced by -x1,-y1,-x2,-y2.
proc commonKoszul(poly x1,y1,x2,y2,x3,y3,x4,y4, int N)
{
// First define the polynomials A,B in [KR07, (3.60-61)]:
poly A = - (y3+y4)*(-y1-y2+y4) - y1*y2 + w2poly(x1,x3,N) + (-x2+x4) * w3poly(-x1,-x2,-x3,N)
- (-x2+x4)*(-x1+x4)*( w4poly(-x1,x1,-x2,x4,N) + w4poly(-x1,x1,x2,x4,N) );
poly B = x1*y1 + x2*y2 + x3*y3 - x4*y4;
matrix K1[2][2] = 0, A, -x1-x2+x3+x4, 0;
matrix K2[2][2] = 0, B, -y1-y2+y3+y4, 0;
matrix K = MFtensor(K1,K2);
kill A,B,K1,K2;
return(K);
}
////////////////////////////////////////////////////////////////////
// verticalArcs
//
// We are given four pairs of variables x1,y1,x2,y2,x3,y3,x4,y4 and
// an integer N, and we return the matrix factorisation associated to
// the "crossing"
//
// ) (
//
// (see [KR2, (3.53,58)]) with x1,y1,x2,y2 replaced by -x1,-y1,-x2,-y2.
proc verticalArcs(poly x1,y1,x2,y2,x3,y3,x4,y4, int N)
{
// First define the polynomials p1,p2,q1,q2,r1,r2,C in [KR07, (3.50-52)]:
poly p1 = -x2 + x4;
poly p2 = -y2 + y4;
poly q1 = x3 + x4;
poly q2 = y3 + y4;
poly r1 = -x1 + x4;
poly r2 = -y1 + y4;
poly C = w4poly(-x1,-x2,-x3,x4,N) + w4poly(-x1,x1,-x2,x4,N) + w4poly(-x1,x1,x2,x4,N);
// Vertical non-crossing before tensoring with commonKoszul, see [KR07, (4.2)]:
matrix Pve[2][2] = p2, p1,
q2*r2+q1*r1*C, -q2*r1-q1*r2;
matrix Qve[2][2] = q1*r2 + q2*r1, p1,
q2*r2 + q1*r1*C, -p2;
matrix K = blockmat( zeromat(2), Pve, Qve, zeromat(2) );
kill p1,p2,q1,q2,r1,r2,C,Pve,Qve;
matrix commonFactor = commonKoszul(x1,y1,x2,y2,x3,y3,x4,y4,N);
matrix mfVertical = MFtensor(commonFactor,K);
kill K,commonFactor;
return(mfVertical);
}
////////////////////////////////////////////////////////////////////
// horizontalArcs
//
// We are given four pairs of variables x1,y1,x2,y2,x3,y3,x4,y4 and
// an integer N, and we return the matrix factorisation associated to
// the "crossing"
//
// ) ( (rotated by 90 degrees)
//
// (see [KR2, (3.55,58)]) with x1,y1,x2,y2 replaced by -x1,-y1,-x2,-y2.
proc horizontalArcs(poly x1,y1,x2,y2,x3,y3,x4,y4, int N)
{
// First define the polynomials p1,p2,q1,q2,r1,r2,C in [KR07, (3.50-52)]:
poly p1 = -x2 + x4;
poly p2 = -y2 + y4;
poly q1 = x3 + x4;
poly q2 = y3 + y4;
poly r1 = -x1 + x4;
poly r2 = -y1 + y4;
poly C = w4poly(-x1,-x2,-x3,x4,N) + w4poly(-x1,x1,-x2,x4,N) + w4poly(-x1,x1,x2,x4,N);
// Horizontal non-crossing before tensoring with commonKoszul, see [KR07, (4.2)]:
matrix Pho[2][2] = q2, q1,
p2*r2+p1*r1*C, -p1*r2-p2*r1;
matrix Qho[2][2] = p1*r2+p2*r1, q1,
p2*r2+p1*r1*C, -q2;
matrix K = blockmat( zeromat(2), Pho, Qho, zeromat(2) );
kill p1,p2,q1,q2,r1,r2,C,Pho,Qho;
matrix commonFactor = commonKoszul(x1,y1,x2,y2,x3,y3,x4,y4,N);
matrix mfHorizontal = MFtensor(commonFactor,K);
kill K,commonFactor;
return(mfHorizontal);
}
////////////////////////////////////////////////////////////////////
// virtualCrossing
//
// We are given four pairs of variables x1,y1,x2,y2,x3,y3,x4,y4 and
// an integer N, and we return the matrix factorisation associated to
// the virtual crossing
//
// \ /
// \ /
// o
// / \
// / \
//
// (see [KR2, (3.54,58)]) with x1,y1,x2,y2 replaced by -x1,-y1,-x2,-y2.
proc virtualCrossing(poly x1,y1,x2,y2,x3,y3,x4,y4, int N)
{
// First define the polynomials p1,p2,q1,q2,r1,r2,C in [KR07, (3.50-52)]:
poly p1 = -x2 + x4;
poly p2 = -y2 + y4;
poly q1 = x3 + x4;
poly q2 = y3 + y4;
poly r1 = -x1 + x4;
poly r2 = -y1 + y4;
poly C = w4poly(-x1,-x2,-x3,x4,N) + w4poly(-x1,x1,-x2,x4,N) + w4poly(-x1,x1,x2,x4,N);
// Virtual crossing before tensoring with commonKoszul, see [KR07, (4.3)]:
matrix Pvi[2][2] = r2, r1,
p2*q2+p1*q1*C, -p1*q2-p2*q1;
matrix Qvi[2][2] = p1*q2+p2*q1, r1,
p2*q2+p1*q1*C, -r2;
matrix K = blockmat( zeromat(2), Pvi, Qvi, zeromat(2) );
kill p1,p2,q1,q2,r1,r2,C,Pvi,Qvi;
matrix commonFactor = commonKoszul(x1,y1,x2,y2,x3,y3,x4,y4,N);
matrix mfVirtualCrossing = MFtensor(commonFactor,K);
kill K,commonFactor;
return(mfVirtualCrossing);
}
////////////////////////////////////////////////////////////////////
// Fmap
//
// We are given four pairs of variables x1,y1,x2,y2,x3,y3,x4,y4 and
// an integer N, and we return the morphism of matrix factorisations
//
// \ /
// \ /
// ) ( --> o
// / \
// / \
//
// (see [KR2, (4.6-7)]) with x1,y1,x2,y2 replaced by -x1,-y1,-x2,-y2.
proc Fmap(poly x1,y1,x2,y2,x3,y3,x4,y4, int N)
{
// First define the polynomials p1,p2,q1,q2,r1,r2,C in [KR07, (3.50-52)]:
poly p1 = -x2 + x4;
poly p2 = -y2 + y4;
poly q1 = x3 + x4;
poly q2 = y3 + y4;
poly r1 = -x1 + x4;
poly r2 = -y1 + y4;
poly C = w4poly(-x1,-x2,-x3,x4,N) + w4poly(-x1,x1,-x2,x4,N) + w4poly(-x1,x1,x2,x4,N);
matrix F[4][4] = 0, 1, 0, 0,
q2^2 - (q1^2)*C, 0, 0, 0,
0, 0, q2, -q1,
0, 0, q1*C, -q2;
F = ZZtensor( unitmat(4), F );
kill p1,p2,q1,q2,r1,r2,C;
return(F);
}
////////////////////////////////////////////////////////////////////
// Gmap
//
// We are given four pairs of variables x1,y1,x2,y2,x3,y3,x4,y4 and
// an integer N, and we return the morphism of matrix factorisations
//
// \ /
// \ /
// o --> ) ( (rotated by 90 degrees)
// / \
// / \
//
// (see [KR2, (4.24)]) with x1,y1,x2,y2 replaced by -x1,-y1,-x2,-y2.
proc Gmap(poly x1,y1,x2,y2,x3,y3,x4,y4, int N)
{
// First define the polynomials p1,p2,q1,q2,r1,r2,C in [KR07, (3.50-52)]:
poly p1 = -x2 + x4;
poly p2 = -y2 + y4;
poly q1 = x3 + x4;
poly q2 = y3 + y4;
poly r1 = -x1 + x4;
poly r2 = -y1 + y4;
poly C = w4poly(-x1,-x2,-x3,x4,N) + w4poly(-x1,x1,-x2,x4,N) + w4poly(-x1,x1,x2,x4,N);
matrix G[4][4] = 0, 1, 0, 0,
p2^2 - (p1^2)*C, 0, 0, 0,
0, 0, p2, -p1,
0, 0, p1*C, -p2;
G = ZZtensor( unitmat(4), G );
kill p1,p2,q1,q2,r1,r2,C;
return(G);
}
////////////////////////////////////////////////////////////////////
// Xmap
//
// We are given four pairs of variables x1,y1,x2,y2,x3,y3,x4,y4 and
// an integer N, and we return the morphism of matrix factorisations
//
// ) ( --> ) ( (rotated by 90 degrees)
//
// (see [KR2, (4.27,30)]) with x1,y1,x2,y2 replaced by -x1,-y1,-x2,-y2.
proc Xmap(poly x1,y1,x2,y2,x3,y3,x4,y4, int N)
{
// First define the polynomials p1,p2,q1,q2,r1,r2,C in [KR07, (3.50-52)]:
poly p1 = -x2 + x4;
poly p2 = -y2 + y4;
poly q1 = x3 + x4;
poly q2 = y3 + y4;
poly r1 = -x1 + x4;
poly r2 = -y1 + y4;
poly C = w4poly(-x1,-x2,-x3,x4,N) + w4poly(-x1,x1,-x2,x4,N) + w4poly(-x1,x1,x2,x4,N);
matrix X[4][4] = 0, 0, 0, 0,
0, 0, p1*C, p2,
-q2, 0, 0, 0,
q1*C, 0, 0, 0;
X = ZZtensor( unitmat(4), X );
kill p1,p2,q1,q2,r1,r2,C;
return(X);
}
////////////////////////////////////////////////////////////////////
// ConvFourVertex
//
// We are given four pairs of variables x1,y1,x2,y2,x3,y3,x4,y4 and
// an integer N, and we return the matrix factorisation which is the
// convolution
//
// \ /
// \ /
// ) ( --F--> o --G--> ) ( (rotated by 90 degrees)
// / \
// / \
//
// (see [KR2, (5.57-59) & (6.10) & (6.25)]) with x1,y1,x2,y2 replaced
// by -x1,-y1,-x2,-y2.
proc ConvFourVertex(poly x1,y1,x2,y2,x3,y3,x4,y4, int N)
{
matrix ve = verticalArcs( x1,y1,x2,y2,x3,y3,x4,y4, N );
matrix ho = horizontalArcs( x1,y1,x2,y2,x3,y3,x4,y4, N );
matrix vi = virtualCrossing( x1,y1,x2,y2,x3,y3,x4,y4, N );
matrix F = Fmap( x1,y1,x2,y2,x3,y3,x4,y4, N );
matrix G = Gmap( x1,y1,x2,y2,x3,y3,x4,y4, N );
matrix X = Xmap( x1,y1,x2,y2,x3,y3,x4,y4, N );
list veBlocks = extractblockmat(ve);
list hoBlocks = extractblockmat(ho);
list viBlocks = extractblockmat(vi);
list FBlocks = extractblockmat(F);
list GBlocks = extractblockmat(G);
list XBlocks = extractblockmat(X);
matrix Pve = veBlocks[2];
matrix Pho = hoBlocks[2];
matrix Pvi = viBlocks[2];
matrix F0 = FBlocks[1];
matrix G0 = GBlocks[1];
matrix X0 = XBlocks[3];
matrix Qve = veBlocks[3];
matrix Qho = hoBlocks[3];
matrix Qvi = viBlocks[3];
matrix F1 = FBlocks[4];
matrix G1 = GBlocks[4];
matrix X1 = XBlocks[2];
matrix Pconv[24][24];
int i,j;
for( i=1; i<=8; i++ )
{
for( j=1; j<=8; j++ )
{
Pconv[i,j] = Pve[i,j];
Pconv[i+8,j] = F1[i,j];
Pconv[i+8,j+8] = -Qvi[i,j];
Pconv[i+16,j] = X1[i,j];
Pconv[i+16,j+8] = G0[i,j];
Pconv[i+16,j+16] = Pho[i,j];
}
}
matrix Qconv[24][24];
for( i=1; i<=8; i++ )
{
for( j=1; j<=8; j++ )
{
Qconv[i,j] = Qve[i,j];
Qconv[i+8,j] = F0[i,j];
Qconv[i+8,j+8] = -Pvi[i,j];
Qconv[i+16,j] = X0[i,j];
Qconv[i+16,j+8] = G1[i,j];
Qconv[i+16,j+16] = Qho[i,j];
}
}
matrix Conv = blockmat( zeromat(24), Pconv, Qconv, zeromat(24) );
// kill A,B,C,X,F,G,Fblocks,Gblocks,Fodd,Godd,AZero,CZero,FBXG;
return(Conv);
}