L11a141

L11a140

L11a142

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Link Presentations

[edit Notes on L11a141's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X10,4,11,3 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X20,14,21,13 X22,16,7,15 X4,20,5,19 X14,22,15,21
Gauss code	{1, -4, 3, -10, 5, -6}, {6, -1, 7, -3, 2, -5, 8, -11, 9, -7, 4, -2, 10, -8, 11, -9}

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)^6-t(1) t(2)^6-3 t(1)^2 t(2)^5+4 t(1) t(2)^5-t(2)^5+4 t(1)^2 t(2)^4-8 t(1) t(2)^4+3 t(2)^4-4 t(1)^2 t(2)^3+9 t(1) t(2)^3-4 t(2)^3+3 t(1)^2 t(2)^2-8 t(1) t(2)^2+4 t(2)^2-t(1)^2 t(2)+4 t(1) t(2)-3 t(2)-t(1)+1}{t(1) t(2)^3} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{13/2}+4 q^{11/2}-8 q^{9/2}+14 q^{7/2}-19 q^{5/2}+21 q^{3/2}-22 \sqrt{q}+\frac{18}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^9 a^{-1} -a z^7+6 z^7 a^{-1} -z^7 a^{-3} -4 a z^5+12 z^5 a^{-1} -4 z^5 a^{-3} -4 a z^3+7 z^3 a^{-1} -4 z^3 a^{-3} +a z-3 z a^{-1} +z a^{-3} +2 a z^{-1} -3 a^{-1} z^{-1} + a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 z^{10} a^{-2} -3 z^{10}-7 a z^9-14 z^9 a^{-1} -7 z^9 a^{-3} -7 a^2 z^8-5 z^8 a^{-2} -8 z^8 a^{-4} -4 z^8-4 a^3 z^7+16 a z^7+36 z^7 a^{-1} +9 z^7 a^{-3} -7 z^7 a^{-5} -a^4 z^6+18 a^2 z^6+16 z^6 a^{-2} +9 z^6 a^{-4} -4 z^6 a^{-6} +22 z^6+10 a^3 z^5-11 a z^5-38 z^5 a^{-1} -6 z^5 a^{-3} +10 z^5 a^{-5} -z^5 a^{-7} +2 a^4 z^4-12 a^2 z^4-10 z^4 a^{-2} +2 z^4 a^{-4} +6 z^4 a^{-6} -20 z^4-5 a^3 z^3+2 a z^3+14 z^3 a^{-1} +3 z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} +2 a^2 z^2-3 z^2 a^{-2} -5 z^2 a^{-4} -2 z^2 a^{-6} +2 z^2+3 a z+4 z a^{-1} +z a^{-3} +3 a^{-2} + a^{-4} +3-2 a z^{-1} -3 a^{-1} z^{-1} - a^{-3} z^{-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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   5 1   4 8                 9 3     -6 6               10 5       5 4             11 9         -2 2           11 10           1 0         8 12             4 -2       6 10               -4 -4     3 9                 6 -6   1 5                   -4 -8   3                     3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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 }[/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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