L11a176

L11a175

L11a177

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Link Presentations

[edit Notes on L11a176's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X20,13,21,14 X10,4,11,3 X14,6,15,5 X16,8,17,7 X22,16,7,15 X4,12,5,11 X12,19,13,20 X6,21,1,22 X2,18,3,17
Gauss code	{1, -11, 4, -8, 5, -10}, {6, -1, 2, -4, 8, -9, 3, -5, 7, -6, 11, -2, 9, -3, 10, -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)^2 t(2)^6-t(1) t(2)^6-3 t(1)^2 t(2)^5+4 t(1) t(2)^5-t(2)^5+5 t(1)^2 t(2)^4-8 t(1) t(2)^4+3 t(2)^4-5 t(1)^2 t(2)^3+11 t(1) t(2)^3-5 t(2)^3+3 t(1)^2 t(2)^2-8 t(1) t(2)^2+5 t(2)^2-t(1)^2 t(2)+4 t(1) t(2)-3 t(2)-t(1)+1}{t(1) t(2)^3} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{13/2}+4 q^{11/2}-9 q^{9/2}+15 q^{7/2}-21 q^{5/2}+23 q^{3/2}-24 \sqrt{q}+\frac{20}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-3} -4 z^5 a^{-3} -5 z^3 a^{-3} -2 z a^{-3} - a^{-3} z^{-1} +z^9 a^{-1} -a z^7+6 z^7 a^{-1} -4 a z^5+13 z^5 a^{-1} -5 a z^3+12 z^3 a^{-1} -2 a z+5 z a^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -5 z^4 a^{-6} +z^2 a^{-6} +8 z^7 a^{-5} -12 z^5 a^{-5} +6 z^3 a^{-5} -z a^{-5} +10 z^8 a^{-4} +a^4 z^6-15 z^6 a^{-4} -2 a^4 z^4+8 z^4 a^{-4} +a^4 z^2-z^2 a^{-4} +8 z^9 a^{-3} +4 a^3 z^7-8 z^7 a^{-3} -9 a^3 z^5+z^5 a^{-3} +5 a^3 z^3+2 z a^{-3} - a^{-3} z^{-1} +3 z^{10} a^{-2} +7 a^2 z^8+10 z^8 a^{-2} -15 a^2 z^6-27 z^6 a^{-2} +8 a^2 z^4+19 z^4 a^{-2} -a^2 z^2-4 z^2 a^{-2} + a^{-2} +7 a z^9+15 z^9 a^{-1} -12 a z^7-32 z^7 a^{-1} +6 a z^5+29 z^5 a^{-1} -6 a z^3-18 z^3 a^{-1} +3 a z+6 z a^{-1} - a^{-1} z^{-1} +3 z^{10}+7 z^8-24 z^6+16 z^4-4 z^2 }[/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 14                       1 1 12                     3   -3 10                   6 1   5 8                 9 3     -6 6               12 6       6 4             12 10         -2 2           12 11           1 0         9 13             4 -2       6 11               -5 -4     3 9                 6 -6   1 6                   -5 -8   3                     3 -10 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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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