L11a248

L11a247

L11a249

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Link Presentations

[edit Notes on L11a248's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X22,12,9,11 X2,9,3,10 X4,22,5,21 X14,5,15,6 X20,13,21,14 X6,19,7,20 X16,8,17,7 X18,16,19,15 X8,18,1,17
Gauss code	{1, -4, 2, -5, 6, -8, 9, -11}, {4, -1, 3, -2, 7, -6, 10, -9, 11, -10, 8, -7, 5, -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-1) (v-1) \left(u^2 v^2-2 u^2 v+u^2-2 u v^2+6 u v-2 u+v^2-2 v+1\right)}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{13/2}+5 q^{11/2}-10 q^{9/2}+15 q^{7/2}-21 q^{5/2}+23 q^{3/2}-23 \sqrt{q}+\frac{19}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{-5} +2 z^5 a^{-3} -a^3 z^3+4 z^3 a^{-3} -a^3 z+3 z a^{-3} -z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +5 a z^3-9 z^3 a^{-1} +5 a z-7 z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^{10} a^{-2} -z^{10}-4 a z^9-9 z^9 a^{-1} -5 z^9 a^{-3} -6 a^2 z^8-21 z^8 a^{-2} -10 z^8 a^{-4} -17 z^8-4 a^3 z^7-5 a z^7-5 z^7 a^{-1} -14 z^7 a^{-3} -10 z^7 a^{-5} -a^4 z^6+12 a^2 z^6+41 z^6 a^{-2} +9 z^6 a^{-4} -5 z^6 a^{-6} +40 z^6+10 a^3 z^5+30 a z^5+47 z^5 a^{-1} +43 z^5 a^{-3} +15 z^5 a^{-5} -z^5 a^{-7} +2 a^4 z^4-6 a^2 z^4-19 z^4 a^{-2} +4 z^4 a^{-4} +5 z^4 a^{-6} -26 z^4-8 a^3 z^3-30 a z^3-46 z^3 a^{-1} -30 z^3 a^{-3} -6 z^3 a^{-5} -a^4 z^2-3 z^2 a^{-4} +4 z^2+2 a^3 z+10 a z+14 z a^{-1} +6 z a^{-3} +1-a 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     4   -4 10                   6 1   5 8                 9 4     -5 6               12 6       6 4             11 9         -2 2           12 12           0 0         9 13             4 -2       5 10               -5 -4     3 9                 6 -6   1 5                   -4 -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}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\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^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} }[/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}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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