L11a267

L11a266

L11a268

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Link Presentations

[edit Notes on L11a267'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 X20,18,21,17 X18,7,19,8 X6,19,7,20 X8,22,1,21
Gauss code	{1, -4, 2, -7, 6, -10, 9, -11}, {4, -1, 3, -2, 5, -6, 7, -5, 8, -9, 10, -8, 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{u^3 v^3-3 u^3 v^2+3 u^3 v-u^3-3 u^2 v^3+7 u^2 v^2-6 u^2 v+3 u^2+3 u v^3-6 u v^2+7 u v-3 u-v^3+3 v^2-3 v+1}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{11/2}-4 q^{9/2}+7 q^{7/2}-11 q^{5/2}+16 q^{3/2}-17 \sqrt{q}+\frac{16}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z+a^5 z^{-1} +z^5 a^{-3} -3 a^3 z^3+2 z^3 a^{-3} -6 a^3 z-a^3 z^{-1} -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +9 a z^3-6 z^3 a^{-1} +6 a z-3 z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} +a^5 z^7-4 a^5 z^5+4 z^5 a^{-5} +6 a^5 z^3-3 z^3 a^{-5} -4 a^5 z+a^5 z^{-1} +3 a^4 z^8-11 a^4 z^6+7 z^6 a^{-4} +12 a^4 z^4-7 z^4 a^{-4} -3 a^4 z^2+3 z^2 a^{-4} -a^4+3 a^3 z^9-5 a^3 z^7+8 z^7 a^{-3} -10 a^3 z^5-6 z^5 a^{-3} +21 a^3 z^3-z^3 a^{-3} -10 a^3 z-z a^{-3} +a^3 z^{-1} +a^2 z^{10}+8 a^2 z^8+7 z^8 a^{-2} -33 a^2 z^6-2 z^6 a^{-2} +30 a^2 z^4-14 z^4 a^{-2} -6 a^2 z^2+11 z^2 a^{-2} +7 a z^9+4 z^9 a^{-1} -8 a z^7+6 z^7 a^{-1} -24 a z^5-28 z^5 a^{-1} +36 a z^3+23 z^3 a^{-1} -13 a z-8 z a^{-1} +z^{10}+12 z^8-31 z^6+12 z^4+5 z^2 }[/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                   4 1   -3 6                 7 3     4 4               9 4       -5 2             8 7         1 0           9 10           1 -2         6 7             -1 -4       4 9               5 -6     3 6                 -3 -8   1 5                   4 -10   2                     -2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/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}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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