L11n194

L11n193

L11n195

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Link Presentations

[edit Notes on L11n194's Link Presentations]

Planar diagram presentation	X8192 X16,6,17,5 X9,20,10,21 X21,10,22,11 X18,16,19,15 X14,20,15,19 X2,11,3,12 X12,3,13,4 X4758 X22,14,7,13 X6,18,1,17
Gauss code	{1, -7, 8, -9, 2, -11}, {9, -1, -3, 4, 7, -8, 10, -6, 5, -2, 11, -5, 6, 3, -4, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{3 t(2)^2 t(1)^2-4 t(2) t(1)^2+t(1)^2-4 t(2)^2 t(1)+9 t(2) t(1)-4 t(1)+t(2)^2-4 t(2)+3}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{5/2}-5 q^{3/2}+7 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^3+a^5 z-a^3 z^5-a^3 z^3+a^3 z+a^3 z^{-1} -2 a z^5-6 a z^3+2 z^3 a^{-1} -6 a z-a z^{-1} +3 z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^3 z^9-a z^9-3 a^4 z^8-5 a^2 z^8-2 z^8-4 a^5 z^7-5 a^3 z^7-2 a z^7-z^7 a^{-1} -3 a^6 z^6+a^4 z^6+6 a^2 z^6+2 z^6-a^7 z^5+6 a^5 z^5+8 a^3 z^5-2 a z^5-3 z^5 a^{-1} +6 a^6 z^4+4 a^4 z^4-4 a^2 z^4-3 z^4 a^{-2} -5 z^4+2 a^7 z^3-a^5 z^3+11 a z^3+8 z^3 a^{-1} -3 a^6 z^2-a^4 z^2+3 a^2 z^2+4 z^2 a^{-2} +5 z^2-a^7 z-3 a^3 z-9 a z-5 z a^{-1} -a^2+a^3 z^{-1} +a z^{-1} }[/math] (db)

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                   2 -2 4                 3   3 2               4 2   -2 0             7 3     4 -2           5 5       0 -4         6 6         0 -6       4 6           2 -8     2 5             -3 -10   1 4               3 -12   2                 -2 -14 1                   1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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