L11a308

L11a307

L11a309

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Link Presentations

[edit Notes on L11a308's Link Presentations]

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

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)+t(2)-1) (t(1) t(2)+1) (t(2) t(1)-t(1)-t(2)) \left(t(2)^2-t(2)+1\right)}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{15}{q^{9/2}}+\frac{17}{q^{11/2}}-\frac{17}{q^{13/2}}+\frac{15}{q^{15/2}}-\frac{12}{q^{17/2}}+\frac{7}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{1}{q^{23/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 \left(-z^5\right)-3 a^9 z^3-3 a^9 z-a^9 z^{-1} +a^7 z^7+4 a^7 z^5+7 a^7 z^3+8 a^7 z+3 a^7 z^{-1} +a^5 z^7+3 a^5 z^5-5 a^5 z-2 a^5 z^{-1} -a^3 z^5-3 a^3 z^3-2 a^3 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^4-a^{14} z^2+3 a^{13} z^5-2 a^{13} z^3+6 a^{12} z^6-6 a^{12} z^4+3 a^{12} z^2+9 a^{11} z^7-15 a^{11} z^5+13 a^{11} z^3-3 a^{11} z+9 a^{10} z^8-16 a^{10} z^6+12 a^{10} z^4-4 a^{10} z^2+a^{10}+6 a^9 z^9-7 a^9 z^7-3 a^9 z^5+4 a^9 z^3-a^9 z^{-1} +2 a^8 z^{10}+7 a^8 z^8-29 a^8 z^6+29 a^8 z^4-17 a^8 z^2+3 a^8+10 a^7 z^9-30 a^7 z^7+31 a^7 z^5-23 a^7 z^3+12 a^7 z-3 a^7 z^{-1} +2 a^6 z^{10}+a^6 z^8-18 a^6 z^6+21 a^6 z^4-11 a^6 z^2+3 a^6+4 a^5 z^9-13 a^5 z^7+12 a^5 z^5-7 a^5 z^3+7 a^5 z-2 a^5 z^{-1} +3 a^4 z^8-11 a^4 z^6+11 a^4 z^4-2 a^4 z^2+a^3 z^7-4 a^3 z^5+5 a^3 z^3-2 a^3 z }[/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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                       1 1 -2                     2   -2 -4                   5 1   4 -6                 6 3     -3 -8               9 4       5 -10             8 6         -2 -12           9 9           0 -14         7 9             2 -16       5 8               -3 -18     2 7                 5 -20   1 5                   -4 -22   2                     2 -24 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/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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