06-1350/Syzygies in Asymptote in Brief

Installation

See 06-1350/Syzygies in Asymptote for more detailed information.

First install Asymptote. Once installed, download syzygy.asy.

Braids

import syzygy; // Accesses the syzygy module. Braid b; // Start a new braid. b.n=3; // The braid has three strands. // The strands are numbered left to right starting at 0. b.add(bp,0); // Add a overcrossing component starting at strand 0, // the leftmost strand. b.add(bm,1); // Add an undercrossing starting at strand 1. b.add(phi,0); // Add a trivalent vertex that merges strands 0 and 1. // Strand 2 is now renumbered as strand 1. b.draw(); // Draw the resulting braid.

Relations

import syzygy; // Access the syzygy module. Braid l; // Define the left hand side of the relation. l.n=3; l.add(bp,0); l.add(bp,1); l.add(bp,0); Braid r; // Define the right hand side of the relation. r.n=3; r.add(bp,1); r.add(bp,0); r.add(bp,1); Relation r3; // Define a relation. r3.lsym="\rho_3"; // Give the relation a formula name. r3.codename="rho3"; // Give the relation a name to be used by Mathematica. r3.lhs=l; r3.rhs=r; r3.draw();

r3.toFormula() produces the formula:

$(1230)^{\star }B^{+}(1213)^{\star }B^{+}(1023)^{\star }B^{+}=(1123)^{\star }B^{+}(1203)^{\star }B^{+}(1231)^{\star }B^{+}$

r3.toLinear() produces the formula in linear form:

$\rho _{3}(x_{1},x_{2},x_{3},x_{4})=$	$b^{+}(x_{1},x_{2},x_{3})+b^{+}(x_{1}+x_{3},x_{2},x_{4})+b^{+}(x_{1},x_{3},x_{4})$
	$-b^{+}(x_{1}+x_{2},x_{3},x_{4})-b^{+}(x_{1},x_{2},x_{4})-b^{+}(x_{1}+x_{4},x_{2},x_{3})$

and r3.toCode() produces a version usable in Mathematica:

rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] + bp[x1, x3, x4]
                            - bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]

Syzygies

import syzygy; // Phi around B Braid initial; initial.n=4; initial.add(bp,2); initial.add(bp,0); initial.add(bp,1); initial.add(bp,0); initial.add(bp,2); initial.add(phi,1); Syzygy pb; pb.lsym="\Phi B"; pb.codename="PhiAroundB"; pb.initial=initial; pb.apply(r3,1,0); pb.apply(r4a,3,1); pb.swap(2,3); pb.apply(r4b,0,1); pb.apply(-r3,1,0); pb.apply(-r4a,0,0); pb.swap(2,3); pb.apply(-r4b,3,0); pb.apply(r3,1,1); pb.draw();

Again, like relations, we can use pb.toLinear()

$\Phi B(x_{1},x_{2},x_{3},x_{4},x_{5})=$	$\rho _{3}(x_{1},x_{2},x_{3},x_{5})+\rho _{4a}(x_{1}+x_{5},x_{2},x_{3},x_{4})+\rho _{4b}(x_{1}+x_{2},x_{3},x_{4},x_{5})$
	$-\rho _{3}(x_{1},x_{2},x_{3}+x_{4},x_{5})-\rho _{4a}(x_{1},x_{2},x_{3},x_{4})$
	$-\rho _{4b}(x_{1},x_{3},x_{4},x_{5})+\rho _{3}(x_{1}+x_{3},x_{2},x_{4},x_{5}).$

and pb.toCode()

PhiAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + rho4a[x1 + x5, x2, x3, x4]
 + rho4b[x1 + x2, x3, x4, x5] - rho3[x1, x2, x3 + x4, x5] - rho4a[x1, x2, x3, x4]
 - rho4b[x1, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5]

to produce formulas.