L11a148

L11a147

L11a149

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Link Presentations

[edit Notes on L11a148's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X20,11,21,12 X12,6,13,5 X4,20,5,19 X14,18,15,17 X16,8,17,7 X22,16,7,15 X18,14,19,13 X6,21,1,22
Gauss code	{1, -2, 3, -6, 5, -11}, {8, -1, 2, -3, 4, -5, 10, -7, 9, -8, 7, -10, 6, -4, 11, -9}

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{2 u^2 v^4-5 u^2 v^3+5 u^2 v^2-2 u^2 v-2 u v^4+6 u v^3-9 u v^2+6 u v-2 u-2 v^3+5 v^2-5 v+2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 4 q^{9/2}-\frac{3}{q^{9/2}}-7 q^{7/2}+\frac{6}{q^{7/2}}+11 q^{5/2}-\frac{11}{q^{5/2}}-15 q^{3/2}+\frac{14}{q^{3/2}}-q^{11/2}+\frac{1}{q^{11/2}}+16 \sqrt{q}-\frac{17}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^5-z^5 a^{-3} -3 a^3 z^3-2 z^3 a^{-3} -2 a^3 z+z a^{-3} + a^{-3} z^{-1} +a z^7+z^7 a^{-1} +4 a z^5+3 z^5 a^{-1} +6 a z^3+5 a z-5 z a^{-1} +2 a z^{-1} -3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^4-a^6 z^2+z^7 a^{-5} +3 a^5 z^5-3 z^5 a^{-5} -3 a^5 z^3+2 z^3 a^{-5} +a^5 z+4 z^8 a^{-4} +5 a^4 z^6-15 z^6 a^{-4} -4 a^4 z^4+15 z^4 a^{-4} +a^4 z^2-2 z^2 a^{-4} - a^{-4} +5 z^9 a^{-3} +7 a^3 z^7-17 z^7 a^{-3} -9 a^3 z^5+15 z^5 a^{-3} +6 a^3 z^3-3 z^3 a^{-3} -a^3 z-z a^{-3} + a^{-3} z^{-1} +2 z^{10} a^{-2} +7 a^2 z^8+3 z^8 a^{-2} -9 a^2 z^6-26 z^6 a^{-2} +2 a^2 z^4+25 z^4 a^{-2} +2 a^2 z^2-2 z^2 a^{-2} -3 a^{-2} +5 a z^9+10 z^9 a^{-1} -3 a z^7-28 z^7 a^{-1} -12 a z^5+18 z^5 a^{-1} +15 a z^3+z^3 a^{-1} -8 a z-7 z a^{-1} +2 a z^{-1} +3 a^{-1} z^{-1} +2 z^{10}+6 z^8-25 z^6+17 z^4-3 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                     3   -3 8                   4 1   3 6                 7 3     -4 4               8 4       4 2             8 7         -1 0           9 8           1 -2         6 9             3 -4       5 8               -3 -6     2 7                 5 -8   1 4                   -3 -10   2                     2 -12 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=0 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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}_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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