L11a268

L11a267

L11a269

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Link Presentations

[edit Notes on L11a268's Link Presentations]

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

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{u^3 v^3-3 u^3 v^2+3 u^3 v-u^3-3 u^2 v^3+9 u^2 v^2-10 u^2 v+3 u^2+3 u v^3-10 u v^2+9 u v-3 u-v^3+3 v^2-3 v+1}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{14}{q^{9/2}}-q^{7/2}+\frac{18}{q^{7/2}}+4 q^{5/2}-\frac{21}{q^{5/2}}-8 q^{3/2}+\frac{21}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{8}{q^{11/2}}+13 \sqrt{q}-\frac{19}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^7+3 z^3 a^5+4 z a^5+a^5 z^{-1} -3 z^5 a^3-8 z^3 a^3-7 z a^3-a^3 z^{-1} +z^7 a+4 z^5 a+7 z^3 a+3 z a-z^5 a^{-1} -2 z^3 a^{-1} -z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^4 z^{10}-a^2 z^{10}-4 a^5 z^9-8 a^3 z^9-4 a z^9-6 a^6 z^8-15 a^4 z^8-16 a^2 z^8-7 z^8-4 a^7 z^7-4 a^5 z^7-3 a^3 z^7-10 a z^7-7 z^7 a^{-1} -a^8 z^6+12 a^6 z^6+36 a^4 z^6+32 a^2 z^6-4 z^6 a^{-2} +5 z^6+10 a^7 z^5+29 a^5 z^5+40 a^3 z^5+33 a z^5+11 z^5 a^{-1} -z^5 a^{-3} +2 a^8 z^4-5 a^6 z^4-22 a^4 z^4-17 a^2 z^4+6 z^4 a^{-2} +4 z^4-8 a^7 z^3-29 a^5 z^3-42 a^3 z^3-29 a z^3-7 z^3 a^{-1} +z^3 a^{-3} -a^8 z^2+3 a^4 z^2+a^2 z^2-2 z^2 a^{-2} -3 z^2+2 a^7 z+11 a^5 z+14 a^3 z+7 a z+2 z a^{-1} +a^4-a^5 z^{-1} -a^3 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                     3   -3 4                   5 1   4 2                 8 3     -5 0               11 5       6 -2             11 9         -2 -4           10 10           0 -6         8 11             3 -8       6 10               -4 -10     3 9                 6 -12   1 5                   -4 -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}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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