L11a359

L11a358

L11a360

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Link Presentations

[edit Notes on L11a359's Link Presentations]

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

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