L11a301

L11a300

L11a302

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Link Presentations

[edit Notes on L11a301's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X16,6,17,5 X20,14,21,13 X22,16,9,15 X14,22,15,21 X18,8,19,7 X6,18,7,17 X8,20,1,19
Gauss code	{1, -2, 3, -4, 5, -10, 9, -11}, {4, -1, 2, -3, 6, -8, 7, -5, 10, -9, 11, -6, 8, -7}

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(1)-1) (t(2)-1) (t(1) t(2)+1)^2 \left(t(2)^2+1\right)}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 9 q^{9/2}-10 q^{7/2}+9 q^{5/2}-\frac{1}{q^{5/2}}-9 q^{3/2}+\frac{2}{q^{3/2}}+q^{17/2}-2 q^{15/2}+4 q^{13/2}-7 q^{11/2}+6 \sqrt{q}-\frac{4}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^9 a^{-3} +z^7 a^{-1} -8 z^7 a^{-3} +z^7 a^{-5} +6 z^5 a^{-1} -24 z^5 a^{-3} +6 z^5 a^{-5} +12 z^3 a^{-1} -32 z^3 a^{-3} +12 z^3 a^{-5} +9 z a^{-1} -17 z a^{-3} +8 z a^{-5} +2 a^{-1} z^{-1} -3 a^{-3} z^{-1} + a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^{10} a^{-2} -z^{10} a^{-4} -2 z^9 a^{-1} -5 z^9 a^{-3} -3 z^9 a^{-5} +z^8 a^{-2} -z^8 a^{-4} -4 z^8 a^{-6} -2 z^8-a z^7+8 z^7 a^{-1} +23 z^7 a^{-3} +10 z^7 a^{-5} -4 z^7 a^{-7} +6 z^6 a^{-2} +12 z^6 a^{-4} +12 z^6 a^{-6} -3 z^6 a^{-8} +9 z^6+5 a z^5-11 z^5 a^{-1} -46 z^5 a^{-3} -18 z^5 a^{-5} +10 z^5 a^{-7} -2 z^5 a^{-9} -10 z^4 a^{-2} -22 z^4 a^{-4} -17 z^4 a^{-6} +5 z^4 a^{-8} -z^4 a^{-10} -11 z^4-7 a z^3+12 z^3 a^{-1} +51 z^3 a^{-3} +17 z^3 a^{-5} -12 z^3 a^{-7} +3 z^3 a^{-9} +9 z^2 a^{-2} +17 z^2 a^{-4} +6 z^2 a^{-6} -3 z^2 a^{-8} +2 z^2 a^{-10} +3 z^2+2 a z-9 z a^{-1} -21 z a^{-3} -8 z a^{-5} +2 z a^{-7} -3 a^{-2} -3 a^{-4} - a^{-6} +2 a^{-1} z^{-1} +3 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 6 7 χ 18                       1 -1 16                     1   1 14                   3 1   -2 12                 4 1     3 10               5 3       -2 8             5 4         1 6           4 5           1 4         5 5             0 2       3 6               3 0     1 3                 -2 -2   1 3                   2 -4   1                     -1 -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=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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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