L11a197

L11a196

L11a198

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Link Presentations

[edit Notes on L11a197's Link Presentations]

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

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) t(2)^4-t(2)^4+4 t(1)^2 t(2)^3-10 t(1) t(2)^3+5 t(2)^3-8 t(1)^2 t(2)^2+17 t(1) t(2)^2-8 t(2)^2+5 t(1)^2 t(2)-10 t(1) t(2)+4 t(2)-t(1)^2+t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{21/2}-4 q^{19/2}+9 q^{17/2}-15 q^{15/2}+21 q^{13/2}-24 q^{11/2}+24 q^{9/2}-22 q^{7/2}+15 q^{5/2}-10 q^{3/2}+4 \sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-3} -3 z^5 a^{-5} -z^5 a^{-7} +z^3 a^{-1} +2 z^3 a^{-3} -6 z^3 a^{-5} +z^3 a^{-7} +z^3 a^{-9} +5 z a^{-3} -6 z a^{-5} +2 z a^{-7} +2 a^{-3} z^{-1} -3 a^{-5} z^{-1} + a^{-7} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-12} -2 z^4 a^{-12} +z^2 a^{-12} +4 z^7 a^{-11} -9 z^5 a^{-11} +6 z^3 a^{-11} -z a^{-11} +7 z^8 a^{-10} -15 z^6 a^{-10} +10 z^4 a^{-10} -3 z^2 a^{-10} +6 z^9 a^{-9} -4 z^7 a^{-9} -12 z^5 a^{-9} +11 z^3 a^{-9} -3 z a^{-9} +2 z^{10} a^{-8} +15 z^8 a^{-8} -41 z^6 a^{-8} +29 z^4 a^{-8} -6 z^2 a^{-8} - a^{-8} +13 z^9 a^{-7} -13 z^7 a^{-7} -18 z^5 a^{-7} +25 z^3 a^{-7} -8 z a^{-7} + a^{-7} z^{-1} +2 z^{10} a^{-6} +19 z^8 a^{-6} -46 z^6 a^{-6} +31 z^4 a^{-6} -2 z^2 a^{-6} -3 a^{-6} +7 z^9 a^{-5} +4 z^7 a^{-5} -31 z^5 a^{-5} +33 z^3 a^{-5} -13 z a^{-5} +3 a^{-5} z^{-1} +11 z^8 a^{-4} -17 z^6 a^{-4} +10 z^4 a^{-4} -3 a^{-4} +9 z^7 a^{-3} -15 z^5 a^{-3} +12 z^3 a^{-3} -7 z a^{-3} +2 a^{-3} z^{-1} +4 z^6 a^{-2} -4 z^4 a^{-2} +z^5 a^{-1} -z^3 a^{-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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 22                       1 -1 20                     3   3 18                   6 1   -5 16                 9 3     6 14               12 6       -6 12             12 9         3 10           12 12           0 8         10 12             -2 6       6 13               7 4     4 9                 -5 2   1 7                   6 0   3                     -3 -2 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=4 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=9 }[/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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