L11n171

L11n170

L11n172

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Link Presentations

[edit Notes on L11n171's Link Presentations]

Planar diagram presentation	X8192 X11,21,12,20 X10,4,11,3 X2,17,3,18 X14,5,15,6 X6718 X16,10,17,9 X13,19,14,18 X22,16,7,15 X19,13,20,12 X4,22,5,21
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, -2, 10, -8, -5, 9, -7, 4, 8, -10, 2, 11, -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{t(1)^2 t(2)^4-t(1) t(2)^4-3 t(1)^2 t(2)^3+6 t(1) t(2)^3-t(2)^3+3 t(1)^2 t(2)^2-9 t(1) t(2)^2+3 t(2)^2-t(1)^2 t(2)+6 t(1) t(2)-3 t(2)-t(1)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{11/2}-5 q^{9/2}+9 q^{7/2}-12 q^{5/2}+13 q^{3/2}-14 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{1}{q^{7/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^{-5} + a^{-5} z^{-1} +z^5 a^{-3} +z^3 a^{-3} -2 z a^{-3} -2 a^{-3} z^{-1} -z^7 a^{-1} +a z^5-4 z^5 a^{-1} +2 a z^3-5 z^3 a^{-1} +a z-z a^{-1} +a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^9 a^{-1} -2 z^9 a^{-3} -9 z^8 a^{-2} -3 z^8 a^{-4} -6 z^8-7 a z^7-6 z^7 a^{-1} -z^7 a^{-5} -4 a^2 z^6+20 z^6 a^{-2} +6 z^6 a^{-4} +10 z^6-a^3 z^5+14 a z^5+18 z^5 a^{-1} -3 z^5 a^{-5} +6 a^2 z^4-22 z^4 a^{-2} -14 z^4 a^{-4} -3 z^4 a^{-6} -5 z^4+a^3 z^3-7 a z^3-12 z^3 a^{-1} +z^3 a^{-3} +5 z^3 a^{-5} +12 z^2 a^{-2} +15 z^2 a^{-4} +5 z^2 a^{-6} +2 z^2+2 a z-3 z a^{-3} -z a^{-5} -3 a^{-2} -5 a^{-4} -2 a^{-6} +1-a z^{-1} +2 a^{-3} z^{-1} + a^{-5} 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 χ 12                   2 -2 10                 3   3 8               6 2   -4 6             6 3     3 4           7 6       -1 2         7 6         1 0       4 8           4 -2     4 6             -2 -4   1 5               4 -6   3                 -3 -8 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{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/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}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/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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