L11a533

L11a532

L11a534

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Link Presentations

[edit Notes on L11a533's Link Presentations]

Planar diagram presentation	X6172 X2536 X18,12,19,11 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X22,16,13,15 X20,18,21,17 X16,22,17,21 X12,20,9,19
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, {7, -6, 8, -10, 9, -3, 11, -9, 10, -8}

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