L11a167

L11a166

L11a168

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Link Presentations

[edit Notes on L11a167's Link Presentations]

Planar diagram presentation	X8192 X16,9,17,10 X6718 X20,13,21,14 X10,4,11,3 X14,6,15,5 X4,12,5,11 X22,17,7,18 X18,21,19,22 X12,19,13,20 X2,16,3,15
Gauss code	{1, -11, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -10, 4, -6, 11, -2, 8, -9, 10, -4, 9, -8}

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(2) t(1)-t(1)+1) (t(1) t(2)-t(2)+1) \left(2 t(2)^2-3 t(2)+2\right)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{13}{q^{9/2}}-q^{7/2}+\frac{18}{q^{7/2}}+3 q^{5/2}-\frac{21}{q^{5/2}}-7 q^{3/2}+\frac{19}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{9}{q^{11/2}}+12 \sqrt{q}-\frac{18}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^5 z^5-2 a^5 z^3-a^5 z-a^5 z^{-1} +a^3 z^7+3 a^3 z^5+2 a^3 z^3+2 a^3 z^{-1} +a z^7+4 a z^5-z^5 a^{-1} +7 a z^3-3 z^3 a^{-1} +5 a z-3 z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^4 z^{10}-2 a^2 z^{10}-6 a^5 z^9-11 a^3 z^9-5 a z^9-7 a^6 z^8-9 a^4 z^8-8 a^2 z^8-6 z^8-4 a^7 z^7+10 a^5 z^7+22 a^3 z^7+3 a z^7-5 z^7 a^{-1} -a^8 z^6+17 a^6 z^6+30 a^4 z^6+22 a^2 z^6-3 z^6 a^{-2} +7 z^6+9 a^7 z^5-2 a^5 z^5-15 a^3 z^5+4 a z^5+7 z^5 a^{-1} -z^5 a^{-3} +2 a^8 z^4-11 a^6 z^4-26 a^4 z^4-20 a^2 z^4+5 z^4 a^{-2} -2 z^4-4 a^7 z^3+2 a^5 z^3+3 a^3 z^3-9 a z^3-4 z^3 a^{-1} +2 z^3 a^{-3} -a^8 z^2+4 a^6 z^2+12 a^4 z^2+7 a^2 z^2-2 z^2 a^{-2} -2 z^2-a^5 z+5 a z+3 z a^{-1} -z a^{-3} -2 a^6-5 a^4-3 a^2+1+a^5 z^{-1} +2 a^3 z^{-1} - a^{-1} 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     2   -2 4                   5 1   4 2                 8 3     -5 0               10 4       6 -2             10 9         -1 -4           11 9           2 -6         7 10             3 -8       6 11               -5 -10     3 7                 4 -12   1 6                   -5 -14   3                     3 -16 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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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