L11n235

L11n234

L11n236

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Link Presentations

[edit Notes on L11n235's Link Presentations]

Planar diagram presentation	X10,1,11,2 X5,20,6,21 X14,3,15,4 X4,15,5,16 X19,22,20,9 X16,7,17,8 X18,12,19,11 X12,18,13,17 X2,9,3,10 X8,13,1,14 X21,6,22,7
Gauss code	{1, -9, 3, -4, -2, 11, 6, -10}, {9, -1, 7, -8, 10, -3, 4, -6, 8, -7, -5, 2, -11, 5}

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(2)^3 t(1)^3-2 t(2)^2 t(1)^3-t(2)^3 t(1)^2+t(2)^2 t(1)^2-t(1)^2-t(2)^3 t(1)+t(2) t(1)-t(1)-2 t(2)+1}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{3}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{3}{q^{5/2}}+\frac{2}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{2}{q^{11/2}}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ 2 z a^9+a^9 z^{-1} -z^5 a^7-5 z^3 a^7-5 z a^7-a^7 z^{-1} +z^7 a^5+5 z^5 a^5+6 z^3 a^5+2 z a^5-z^5 a^3-4 z^3 a^3-3 z a^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}+5 z^3 a^{11}-4 z a^{11}+z^4 a^{10}-z^2 a^{10}+z^5 a^9-2 z^3 a^9+3 z a^9-a^9 z^{-1} -z^8 a^8+5 z^6 a^8-4 z^4 a^8-z^2 a^8+a^8-z^9 a^7+4 z^7 a^7-7 z^3 a^7+5 z a^7-a^7 z^{-1} -3 z^8 a^6+15 z^6 a^6-18 z^4 a^6+5 z^2 a^6-z^9 a^5+3 z^7 a^5+3 z^5 a^5-7 z^3 a^5+z a^5-2 z^8 a^4+10 z^6 a^4-13 z^4 a^4+5 z^2 a^4-z^7 a^3+5 z^5 a^3-7 z^3 a^3+3 z a^3 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                     1 1 -2                   1   -1 -4                 2 1   1 -6               2 2     0 -8           1 2 1       0 -10           1 2         1 -12       1 3 2           0 -14         1             1 -16     1 2               -1 -18 1                     1 -20 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}_2 }[/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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