L11a432

L11a431

L11a433

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Link Presentations

[edit Notes on L11a432's Link Presentations]

Planar diagram presentation	X6172 X14,6,15,5 X8493 X2,16,3,15 X16,7,17,8 X18,14,19,13 X20,9,21,10 X12,19,5,20 X22,11,13,12 X10,21,11,22 X4,17,1,18
Gauss code	{1, -4, 3, -11}, {2, -1, 5, -3, 7, -10, 9, -8}, {6, -2, 4, -5, 11, -6, 8, -7, 10, -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)-1) (t(2)-1) (t(3)-1) (t(3) t(2)-t(2)+1) (t(2) t(3)-t(3)+1)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-8} +5 q^{-7} -10 q^{-6} +16 q^{-5} -20 q^{-4} +q^3+24 q^{-3} -4 q^2-22 q^{-2} +8 q+20 q^{-1} -13 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^2 z^8+2 a^4 z^6-5 a^2 z^6+z^6-a^6 z^4+6 a^4 z^4-9 a^2 z^4+3 z^4-a^6 z^2+3 a^4 z^2-4 a^2 z^2+2 z^2+a^6-4 a^4+3 a^2+a^6 z^{-2} -2 a^4 z^{-2} +a^2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^5+5 a^8 z^6-5 a^8 z^4+a^8+10 a^7 z^7-14 a^7 z^5+3 a^7 z^3+11 a^6 z^8-14 a^6 z^6+3 a^6 z^4-2 a^6 z^2+a^6 z^{-2} -a^6+7 a^5 z^9+a^5 z^7-16 a^5 z^5+5 a^5 z^3+4 a^5 z-2 a^5 z^{-1} +2 a^4 z^{10}+17 a^4 z^8-44 a^4 z^6+36 a^4 z^4-9 a^4 z^2+2 a^4 z^{-2} -4 a^4+13 a^3 z^9-22 a^3 z^7+7 a^3 z^5+3 a^3 z^3+4 a^3 z-2 a^3 z^{-1} +2 a^2 z^{10}+13 a^2 z^8-45 a^2 z^6+z^6 a^{-2} +47 a^2 z^4-2 z^4 a^{-2} -13 a^2 z^2+a^2 z^{-2} -3 a^2+6 a z^9-9 a z^7+4 z^7 a^{-1} -2 a z^5-10 z^5 a^{-1} +6 a z^3+5 z^3 a^{-1} +7 z^8-19 z^6+17 z^4-6 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     3   -3 3                   5 1   4 1                 8 3     -5 -1               12 5       7 -3             12 10         -2 -5           12 10           2 -7         8 12             4 -9       8 12               -4 -11     4 10                 6 -13   1 6                   -5 -15   4                     4 -17 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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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