L11a382

L11a381

L11a383

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Link Presentations

[edit Notes on L11a382's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X14,3,15,4 X16,7,17,8 X18,10,19,9 X20,6,21,5 X6,22,7,21 X22,18,11,17 X10,20,1,19 X8,11,9,12 X4,15,5,16
Gauss code	{1, -2, 3, -11, 6, -7, 4, -10, 5, -9}, {10, -1, 2, -3, 11, -4, 8, -5, 9, -6, 7, -8}

A Braid Representative	{{{braid_table}}}
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{\left(t(1)^2 t(2)^2-t(1) t(2)^2+2 t(1) t(2)-t(2)-t(1)+1\right) \left(t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+2 t(2) t(1)-t(1)+1\right)}{t(1)^2 t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-3 q^{7/2}+6 q^{5/2}-10 q^{3/2}+13 \sqrt{q}-\frac{16}{\sqrt{q}}+\frac{15}{q^{3/2}}-\frac{14}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a z^9+a^3 z^7-7 a z^7+z^7 a^{-1} +5 a^3 z^5-18 a z^5+5 z^5 a^{-1} +8 a^3 z^3-20 a z^3+8 z^3 a^{-1} +4 a^3 z-9 a z+4 z a^{-1} +a^3 z^{-1} -a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-2 a^7 z^3+a^7 z+3 a^6 z^6-6 a^6 z^4+3 a^6 z^2+4 a^5 z^7-5 a^5 z^5-a^5 z^3+a^5 z+4 a^4 z^8-3 a^4 z^6+z^6 a^{-4} -3 a^4 z^4-3 z^4 a^{-4} +a^4 z^2+z^2 a^{-4} +4 a^3 z^9-8 a^3 z^7+3 z^7 a^{-3} +14 a^3 z^5-9 z^5 a^{-3} -14 a^3 z^3+5 z^3 a^{-3} +5 a^3 z-z a^{-3} -a^3 z^{-1} +2 a^2 z^{10}+5 z^8 a^{-2} -6 a^2 z^6-16 z^6 a^{-2} +12 a^2 z^4+14 z^4 a^{-2} -6 a^2 z^2-5 z^2 a^{-2} +a^2+9 a z^9+5 z^9 a^{-1} -31 a z^7-16 z^7 a^{-1} +47 a z^5+18 z^5 a^{-1} -30 a z^3-10 z^3 a^{-1} +9 a z+3 z a^{-1} -a z^{-1} +2 z^{10}+z^8-17 z^6+26 z^4-10 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 10                       1 -1 8                     2   2 6                   4 1   -3 4                 6 2     4 2               7 4       -3 0             9 6         3 -2           7 8           1 -4         7 8             -1 -6       4 8               4 -8     2 6                 -4 -10   1 4                   3 -12   2                     -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{ 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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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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