L11a246

L11a245

L11a247

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Link Presentations

[edit Notes on L11a246's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X14,6,15,5 X16,7,17,8 X20,15,21,16 X18,14,19,13 X6,22,7,21 X22,18,9,17 X4,19,5,20 X2,9,3,10 X8,11,1,12
Gauss code	{1, -10, 2, -9, 3, -7, 4, -11}, {10, -1, 11, -2, 6, -3, 5, -4, 8, -6, 9, -5, 7, -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{u^3 v^4-3 u^3 v^3+3 u^3 v^2-u^3 v+u^2 v^5-5 u^2 v^4+9 u^2 v^3-10 u^2 v^2+5 u^2 v-u^2-u v^5+5 u v^4-10 u v^3+9 u v^2-5 u v+u-v^4+3 v^3-3 v^2+v}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{15}{q^{9/2}}-q^{7/2}+\frac{21}{q^{7/2}}+5 q^{5/2}-\frac{25}{q^{5/2}}-11 q^{3/2}+\frac{25}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{8}{q^{11/2}}+17 \sqrt{q}-\frac{23}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^7+a z^7-a^5 z^5+3 a^3 z^5+2 a z^5-z^5 a^{-1} -2 a^5 z^3+3 a^3 z^3-a z^3-z^3 a^{-1} -3 a z+z a^{-1} +a^5 z^{-1} -a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^6-2 a^8 z^4+4 a^7 z^7-10 a^7 z^5+6 a^7 z^3+7 a^6 z^8-17 a^6 z^6+13 a^6 z^4-3 a^6 z^2+7 a^5 z^9-12 a^5 z^7+3 a^5 z^5+6 a^5 z^3-4 a^5 z+a^5 z^{-1} +3 a^4 z^{10}+9 a^4 z^8-32 a^4 z^6+29 a^4 z^4-7 a^4 z^2-a^4+16 a^3 z^9-30 a^3 z^7+17 a^3 z^5+z^5 a^{-3} -a^3 z^3-3 a^3 z+a^3 z^{-1} +3 a^2 z^{10}+15 a^2 z^8-36 a^2 z^6+5 z^6 a^{-2} +21 a^2 z^4-4 z^4 a^{-2} -4 a^2 z^2+9 a z^9-3 a z^7+11 z^7 a^{-1} -12 a z^5-15 z^5 a^{-1} +3 a z^3+4 z^3 a^{-1} +2 a z+z a^{-1} +13 z^8-17 z^6+3 z^4 }[/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                     4   -4 4                   7 1   6 2                 10 4     -6 0               13 7       6 -2             13 11         -2 -4           12 12           0 -6         9 13             4 -8       6 12               -6 -10     3 10                 7 -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}^{10}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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