L11a187

L11a186

L11a188

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Link Presentations

[edit Notes on L11a187's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X20,14,21,13 X16,5,17,6 X4,15,5,16 X18,12,19,11 X22,18,7,17 X14,20,15,19 X12,22,13,21 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 6, -9, 3, -8, 5, -4, 7, -6, 8, -3, 9, -7}

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{2 t(1) t(2)^4+2 t(1)^2 t(2)^3-7 t(1) t(2)^3+4 t(2)^3-5 t(1)^2 t(2)^2+11 t(1) t(2)^2-5 t(2)^2+4 t(1)^2 t(2)-7 t(1) t(2)+2 t(2)+2 t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{7}{q^{9/2}}-3 q^{7/2}+\frac{10}{q^{7/2}}+6 q^{5/2}-\frac{15}{q^{5/2}}-10 q^{3/2}+\frac{16}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{3}{q^{11/2}}+14 \sqrt{q}-\frac{16}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^3+a^5 z+a^5 z^{-1} -a^3 z^5-a^3 z^3+z^3 a^{-3} -a^3 z+z a^{-3} -2 a z^5-z^5 a^{-1} -4 a z^3-z^3 a^{-1} -4 a z-2 a z^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^2 z^{10}-z^{10}-4 a^3 z^9-7 a z^9-3 z^9 a^{-1} -6 a^4 z^8-9 a^2 z^8-4 z^8 a^{-2} -7 z^8-6 a^5 z^7+4 a^3 z^7+16 a z^7+3 z^7 a^{-1} -3 z^7 a^{-3} -3 a^6 z^6+12 a^4 z^6+29 a^2 z^6+10 z^6 a^{-2} -z^6 a^{-4} +25 z^6-a^7 z^5+15 a^5 z^5+a^3 z^5-20 a z^5+4 z^5 a^{-1} +9 z^5 a^{-3} +5 a^6 z^4-9 a^4 z^4-38 a^2 z^4-7 z^4 a^{-2} +3 z^4 a^{-4} -34 z^4+2 a^7 z^3-15 a^5 z^3-5 a^3 z^3+17 a z^3-2 z^3 a^{-1} -7 z^3 a^{-3} +19 a^2 z^2+4 z^2 a^{-2} -2 z^2 a^{-4} +25 z^2+7 a^5 z-9 a z-z a^{-1} +z a^{-3} +a^4-3 a^2-2 a^{-2} -5-a^5 z^{-1} +2 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     2   2 6                   4 1   -3 4                 6 2     4 2               8 4       -4 0             8 6         2 -2           9 9           0 -4         6 7             -1 -6       4 9               5 -8     3 6                 -3 -10     4                   4 -12 1 3                     -2 -14 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/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}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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