L11n100

L11n99

L11n101

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Link Presentations

[edit Notes on L11n100's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X20,13,21,14 X7,17,8,16 X19,9,20,8 X9,19,10,18 X17,11,18,10 X22,15,5,16 X14,21,15,22 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, -6, 7, 11, -2, 3, -9, 8, 4, -7, 6, -5, -3, 9, -8}

A Braid Representative	{{{braid_table}}}
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{2 u v^3-3 u v^2+u v-u-v^3+v^2-3 v+2}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{3}{q^{9/2}}-\frac{4}{q^{7/2}}-q^{5/2}+\frac{4}{q^{5/2}}+2 q^{3/2}-\frac{5}{q^{3/2}}-\frac{2}{q^{11/2}}-3 \sqrt{q}+\frac{4}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^{-1} -a^5 z^3-3 a^5 z-2 a^5 z^{-1} +a^3 z^5+3 a^3 z^3+2 a^3 z+2 a^3 z^{-1} +a z^5+3 a z^3-z^3 a^{-1} +a z-a z^{-1} -2 z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^3 z^9-a z^9-a^4 z^8-3 a^2 z^8-2 z^8+4 a^3 z^7+3 a z^7-z^7 a^{-1} +2 a^4 z^6+12 a^2 z^6+10 z^6-4 a^5 z^5-9 a^3 z^5+5 z^5 a^{-1} -a^6 z^4+a^4 z^4-12 a^2 z^4-14 z^4+11 a^5 z^3+17 a^3 z^3-a z^3-7 z^3 a^{-1} -a^6 z^2-2 a^4 z^2+4 a^2 z^2+5 z^2-3 a^7 z-9 a^5 z-10 a^3 z-2 a z+2 z a^{-1} +a^4+a^7 z^{-1} +2 a^5 z^{-1} +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   \ -4 -3 -2 -1 0 1 2 3 4 χ 6                 1 1 4               1   -1 2             2 1   1 0           2 1     -1 -2         3 2       1 -4       2 3         1 -6     2 2           0 -8   1 2             1 -10 1 2               -1 -12 2                 2

Integral Khovanov Homology (db, data source)

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