L10a161

L10a160

L10a162

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Link Presentations

[edit Notes on L10a161's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X10,20,11,19 X18,10,19,9 X20,12,13,11 X12,14,7,13 X16,6,17,5 X2738 X4,16,5,15 X6,18,1,17
Gauss code	{1, -8, 2, -9, 7, -10}, {8, -1, 4, -3, 5, -6}, {6, -2, 9, -7, 10, -4, 3, -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(2)^2 t(3)^3-t(1) t(2)^2 t(3)^3-t(1)^2 t(2) t(3)^3-t(1)^2 t(3)^2+t(1) t(2)^2 t(3)^2-t(2)^2 t(3)^2+t(1)^2 t(2) t(3)^2-t(1) t(2) t(3)^2+t(1)^2 t(3)+t(2)^2 t(3)-t(1) t(3)+t(1) t(2) t(3)-t(2) t(3)+t(1)+t(2)-1}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{11}-2 q^{10}+3 q^9-4 q^8+5 q^7-4 q^6+5 q^5-3 q^4+3 q^3-q^2+q }[/math] (db)
Signature	6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-8} +5 z^4 a^{-8} +7 z^2 a^{-8} + a^{-8} z^{-2} +3 a^{-8} -z^8 a^{-6} -7 z^6 a^{-6} -17 z^4 a^{-6} -18 z^2 a^{-6} -2 a^{-6} z^{-2} -9 a^{-6} +z^6 a^{-4} +6 z^4 a^{-4} +11 z^2 a^{-4} + a^{-4} z^{-2} +6 a^{-4} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +4 z^8 a^{-6} +3 z^8 a^{-8} -5 z^7 a^{-5} -z^7 a^{-7} +4 z^7 a^{-9} -7 z^6 a^{-4} -23 z^6 a^{-6} -11 z^6 a^{-8} +5 z^6 a^{-10} +5 z^5 a^{-5} -11 z^5 a^{-7} -12 z^5 a^{-9} +4 z^5 a^{-11} +17 z^4 a^{-4} +43 z^4 a^{-6} +9 z^4 a^{-8} -14 z^4 a^{-10} +3 z^4 a^{-12} +6 z^3 a^{-5} +20 z^3 a^{-7} +6 z^3 a^{-9} -6 z^3 a^{-11} +2 z^3 a^{-13} -17 z^2 a^{-4} -32 z^2 a^{-6} -3 z^2 a^{-8} +9 z^2 a^{-10} -2 z^2 a^{-12} +z^2 a^{-14} -9 z a^{-5} -9 z a^{-7} +7 a^{-4} +11 a^{-6} +3 a^{-8} -2 a^{-10} +2 a^{-5} z^{-1} +2 a^{-7} z^{-1} - a^{-4} z^{-2} -2 a^{-6} z^{-2} - a^{-8} 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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 23                     1 1 21                   2 1 -1 19                 1     1 17               3 2     -1 15             2 1       1 13           2 3         1 11         3 2           1 9       2 4             2 7     1 1               0 5   1 3                 2 3                       0 1 1                     1

Integral Khovanov Homology (db, data source)

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