L11n119

L11n118

L11n120

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Link Presentations

[edit Notes on L11n119's Link Presentations]

Planar diagram presentation	X6172 X3,15,4,14 X16,10,17,9 X11,21,12,20 X21,9,22,8 X7,19,8,18 X19,13,20,12 X10,16,11,15 X17,5,18,22 X2536 X13,1,14,4
Gauss code	{1, -10, -2, 11}, {10, -1, -6, 5, 3, -8, -4, 7, -11, 2, 8, -3, -9, 6, -7, 4, -5, 9}

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{(u-1) (v-1) \left(2 v^2-v+2\right)}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 7 q^{9/2}-7 q^{7/2}+4 q^{5/2}-3 q^{3/2}+2 q^{17/2}-4 q^{15/2}+6 q^{13/2}-7 q^{11/2} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -2 z^5 a^{-5} +3 z^3 a^{-3} -8 z^3 a^{-5} +2 z^3 a^{-7} +7 z a^{-3} -11 z a^{-5} +4 z a^{-7} +3 a^{-3} z^{-1} -5 a^{-5} z^{-1} +2 a^{-7} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 z^4 a^{-10} -5 z^2 a^{-10} + a^{-10} +z^7 a^{-9} +z^5 a^{-9} -4 z^3 a^{-9} +z^8 a^{-8} +z^6 a^{-8} -3 z^4 a^{-8} +5 z^7 a^{-7} -11 z^5 a^{-7} +12 z^3 a^{-7} -7 z a^{-7} +2 a^{-7} z^{-1} +z^8 a^{-6} +4 z^6 a^{-6} -12 z^4 a^{-6} +14 z^2 a^{-6} -5 a^{-6} +4 z^7 a^{-5} -12 z^5 a^{-5} +22 z^3 a^{-5} -17 z a^{-5} +5 a^{-5} z^{-1} +3 z^6 a^{-4} -6 z^4 a^{-4} +9 z^2 a^{-4} -5 a^{-4} +6 z^3 a^{-3} -10 z a^{-3} +3 a^{-3} 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   \ 0 1 2 3 4 5 6 7 χ 18               2 -2 16             2   2 14           4 2   -2 12         3 2     1 10       4 4       0 8     3 3         0 6   1 4           3 4 2 3             -1 2 3               3

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=4 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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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