L11n414

L11n413

L11n415

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Link Presentations

[edit Notes on L11n414's Link Presentations]

Planar diagram presentation	X8192 X5,15,6,14 X10,3,11,4 X13,5,14,4 X2738 X6,9,1,10 X18,12,19,11 X12,18,7,17 X15,20,16,21 X19,22,20,13 X21,16,22,17
Gauss code	{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, 7, -8}, {-4, 2, -9, 11, 8, -7, -10, 9, -11, 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{-t(1) t(3)^3+t(1) t(2) t(3)^3-t(2) t(3)^3+t(1)^2 t(3)^2+t(2)^2 t(3)^2-t(1)^2 t(3)-t(2)^2 t(3)+t(1) t(2)^2+t(1)^2 t(2)-t(1) t(2)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^3-q^2+2 q-1+2 q^{-1} + q^{-3} + q^{-4} - q^{-5} + q^{-6} - q^{-7} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 \left(-z^2\right)-a^6 z^{-2} -2 a^6+a^4 z^4+5 a^4 z^2+4 a^4 z^{-2} +8 a^4-a^2 z^4-5 a^2 z^2-5 a^2 z^{-2} +z^2 a^{-2} -8 a^2+2 a^{-2} -z^4-3 z^2+2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^7-6 a^7 z^5+10 a^7 z^3-5 a^7 z+a^7 z^{-1} +a^6 z^8-6 a^6 z^6+10 a^6 z^4-7 a^6 z^2-a^6 z^{-2} +4 a^6+2 a^5 z^7-14 a^5 z^5+28 a^5 z^3-21 a^5 z+5 a^5 z^{-1} +a^4 z^8-8 a^4 z^6+20 a^4 z^4-25 a^4 z^2-4 a^4 z^{-2} +17 a^4+2 a^3 z^7-16 a^3 z^5+37 a^3 z^3-33 a^3 z+9 a^3 z^{-1} +a^2 z^8-8 a^2 z^6+z^6 a^{-2} +22 a^2 z^4-5 z^4 a^{-2} -31 a^2 z^2+6 z^2 a^{-2} -5 a^2 z^{-2} +20 a^2-2 a^{-2} +2 a z^7+z^7 a^{-1} -12 a z^5-4 z^5 a^{-1} +21 a z^3+2 z^3 a^{-1} -16 a z+z a^{-1} +5 a z^{-1} +z^8-5 z^6+7 z^4-7 z^2-2 z^{-2} +6 }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     1 1 0 3                   1     1 1               1 1 1     1 -1             1 3 1       1 -3           1   1         2 -5         1 4 2           1 -7       1   1             2 -9       1 1               0 -11   1 1                   0 -13                         0 -15 1                       -1

Integral Khovanov Homology (db, data source)

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