L11a198

L11a197

L11a199

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Link Presentations

[edit Notes on L11a198's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X20,15,21,16 X6,14,1,13 X18,11,19,12 X16,6,17,5 X12,17,13,18 X4,20,5,19 X14,21,15,22
Gauss code	{1, -4, 2, -10, 8, -6}, {4, -1, 3, -2, 7, -9, 6, -11, 5, -8, 9, -7, 10, -5, 11, -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{t(1)^2 t(2)^4-t(1) t(2)^4-5 t(1)^2 t(2)^3+8 t(1) t(2)^3-4 t(2)^3+8 t(1)^2 t(2)^2-15 t(1) t(2)^2+8 t(2)^2-4 t(1)^2 t(2)+8 t(1) t(2)-5 t(2)-t(1)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{11/2}-4 q^{9/2}+9 q^{7/2}-15 q^{5/2}+19 q^{3/2}-23 \sqrt{q}+\frac{21}{\sqrt{q}}-\frac{19}{q^{3/2}}+\frac{14}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{1}{q^{11/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+8 a z^3-8 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-4 a^3 z+8 a z-6 z a^{-1} +2 z a^{-3} -a^3 z^{-1} +3 a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} +a^5 z^7-3 a^5 z^5+4 z^5 a^{-5} +3 a^5 z^3-z^3 a^{-5} -a^5 z+4 a^4 z^8-14 a^4 z^6+9 z^6 a^{-4} +16 a^4 z^4-7 z^4 a^{-4} -5 a^4 z^2+2 z^2 a^{-4} -a^4+5 a^3 z^9-12 a^3 z^7+14 z^7 a^{-3} +2 a^3 z^5-20 z^5 a^{-3} +9 a^3 z^3+13 z^3 a^{-3} -4 a^3 z-4 z a^{-3} +a^3 z^{-1} +2 a^2 z^{10}+10 a^2 z^8+14 z^8 a^{-2} -46 a^2 z^6-20 z^6 a^{-2} +45 a^2 z^4+6 z^4 a^{-2} -9 a^2 z^2-3 a^2+13 a z^9+8 z^9 a^{-1} -24 a z^7+3 z^7 a^{-1} -10 a z^5-39 z^5 a^{-1} +27 a z^3+35 z^3 a^{-1} -11 a z-12 z a^{-1} +3 a z^{-1} +2 a^{-1} z^{-1} +2 z^{10}+20 z^8-61 z^6+43 z^4-6 z^2-3 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   6 1   -5 6                 9 3     6 4               11 7       -4 2             12 8         4 0           10 12           2 -2         9 11             -2 -4       5 10               5 -6     3 9                 -6 -8   1 5                   4 -10   3                     -3 -12 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/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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