L11a511

L11a510

L11a512

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Link Presentations

[edit Notes on L11a511's Link Presentations]

Planar diagram presentation	X8192 X14,3,15,4 X20,12,21,11 X18,10,19,9 X22,16,13,15 X12,20,7,19 X10,22,11,21 X16,6,17,5 X2738 X4,13,5,14 X6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 4, -7, 3, -6}, {10, -2, 5, -8, 11, -4, 6, -3, 7, -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{-t(1)^2 t(3)^3+t(1) t(3)^3+t(1)^2 t(2) t(3)^3-2 t(1) t(2) t(3)^3+t(2) t(3)^3+2 t(1)^2 t(3)^2+t(1)^2 t(2)^2 t(3)^2-2 t(1) t(2)^2 t(3)^2+t(2)^2 t(3)^2-2 t(1) t(3)^2-3 t(1)^2 t(2) t(3)^2+5 t(1) t(2) t(3)^2-3 t(2) t(3)^2+t(3)^2-t(1)^2 t(3)-t(1)^2 t(2)^2 t(3)+2 t(1) t(2)^2 t(3)-2 t(2)^2 t(3)+2 t(1) t(3)+3 t(1)^2 t(2) t(3)-5 t(1) t(2) t(3)+3 t(2) t(3)-t(3)-t(1) t(2)^2+t(2)^2-t(1)^2 t(2)+2 t(1) t(2)-t(2)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+3 q^7-6 q^6+11 q^5-14 q^4+17 q^3-16 q^2+15 q-10+7 q^{-1} -3 q^{-2} + q^{-3} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^{-6} -2 z^2 a^{-6} - a^{-6} +z^6 a^{-4} +3 z^4 a^{-4} +5 z^2 a^{-4} + a^{-4} z^{-2} +4 a^{-4} +z^6 a^{-2} +z^4 a^{-2} +a^2 z^2-3 z^2 a^{-2} -2 a^{-2} z^{-2} +a^2-5 a^{-2} -2 z^4-3 z^2+ z^{-2} +1 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-9} -2 z^3 a^{-9} +3 z^6 a^{-8} -6 z^4 a^{-8} +2 z^2 a^{-8} +5 z^7 a^{-7} -10 z^5 a^{-7} +6 z^3 a^{-7} -z a^{-7} +6 z^8 a^{-6} -14 z^6 a^{-6} +17 z^4 a^{-6} -9 z^2 a^{-6} +2 a^{-6} +4 z^9 a^{-5} -4 z^7 a^{-5} -2 z^5 a^{-5} +9 z^3 a^{-5} -3 z a^{-5} +z^{10} a^{-4} +10 z^8 a^{-4} -35 z^6 a^{-4} +53 z^4 a^{-4} -35 z^2 a^{-4} - a^{-4} z^{-2} +10 a^{-4} +7 z^9 a^{-3} -13 z^7 a^{-3} +7 z^5 a^{-3} +5 z^3 a^{-3} -8 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +8 z^8 a^{-2} +a^2 z^6-27 z^6 a^{-2} -3 a^2 z^4+35 z^4 a^{-2} +3 a^2 z^2-29 z^2 a^{-2} -2 a^{-2} z^{-2} -a^2+12 a^{-2} +3 z^9 a^{-1} +3 a z^7-z^7 a^{-1} -8 a z^5-10 z^5 a^{-1} +5 a z^3+9 z^3 a^{-1} -6 z a^{-1} +2 a^{-1} z^{-1} +4 z^8-8 z^6+2 z^4-2 z^2- z^{-2} +4 }[/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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     2   2 13                   4 1   -3 11                 7 2     5 9               8 5       -3 7             9 6         3 5           8 9           1 3         7 8             -1 1       4 9               5 -1     3 6                 -3 -3   1 5                   4 -5   2                     -2 -7 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\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^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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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