L11n112

L11n111

L11n113

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Link Presentations

[edit Notes on L11n112's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X7,18,8,19 X19,22,20,5 X9,21,10,20 X21,9,22,8 X11,17,12,16 X17,15,18,14 X15,11,16,10 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 6, -5, 9, -7, -2, 11, 8, -9, 7, -8, 3, -4, 5, -6, 4}

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-1) (v-1)}{\sqrt{u} \sqrt{v}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -4 q^{9/2}+3 q^{7/2}-3 q^{5/2}-\frac{1}{q^{5/2}}+2 q^{3/2}+\frac{1}{q^{3/2}}+q^{15/2}-2 q^{13/2}+3 q^{11/2}-\sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^{-7} + a^{-7} z^{-1} -2 z^3 a^{-5} -5 z a^{-5} -4 a^{-5} z^{-1} +z^5 a^{-3} +5 z^3 a^{-3} +10 z a^{-3} +6 a^{-3} z^{-1} -z^5 a^{-1} +a z^3-6 z^3 a^{-1} +3 a z-9 z a^{-1} +2 a z^{-1} -5 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{-2} -z^8 a^{-4} -z^8 a^{-6} -z^8-a z^7-2 z^7 a^{-1} -3 z^7 a^{-3} -4 z^7 a^{-5} -2 z^7 a^{-7} +7 z^6 a^{-2} +3 z^6 a^{-4} +z^6 a^{-6} -z^6 a^{-8} +6 z^6+6 a z^5+16 z^5 a^{-1} +20 z^5 a^{-3} +18 z^5 a^{-5} +8 z^5 a^{-7} -10 z^4 a^{-2} +2 z^4 a^{-4} +8 z^4 a^{-6} +4 z^4 a^{-8} -8 z^4-10 a z^3-33 z^3 a^{-1} -39 z^3 a^{-3} -24 z^3 a^{-5} -8 z^3 a^{-7} -7 z^2 a^{-4} -9 z^2 a^{-6} -4 z^2 a^{-8} +2 z^2+7 a z+22 z a^{-1} +27 z a^{-3} +16 z a^{-5} +4 z a^{-7} + a^{-2} +3 a^{-4} +3 a^{-6} + a^{-8} +1-2 a z^{-1} -5 a^{-1} z^{-1} -6 a^{-3} z^{-1} -4 a^{-5} z^{-1} - a^{-7} 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 5 6 7 χ 16                       1 -1 14                     1   1 12                   2 1   -1 10                 2 1     1 8             1 2 2       1 6             2 2         0 4         1 2 2           1 2       1 2 2             -1 0       1 3               2 -2   1 1                   0 -4                         0 -6 1                       1

Integral Khovanov Homology (db, data source)

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