L11a452

L11a451

L11a453

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Link Presentations

[edit Notes on L11a452's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
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(3)^2 t(2)^3-t(3)^2 t(2)^3-t(1) t(3) t(2)^3+2 t(3) t(2)^3-t(2)^3+t(1) t(3)^3 t(2)^2-t(3)^3 t(2)^2-2 t(1) t(3)^2 t(2)^2+2 t(3)^2 t(2)^2-t(1) t(2)^2+2 t(1) t(3) t(2)^2-2 t(3) t(2)^2+2 t(2)^2-2 t(1) t(3)^3 t(2)+t(3)^3 t(2)+2 t(1) t(3)^2 t(2)-2 t(3)^2 t(2)+t(1) t(2)-2 t(1) t(3) t(2)+2 t(3) t(2)-t(2)+t(1) t(3)^3-2 t(1) t(3)^2+t(3)^2+t(1) t(3)-t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^6+3 q^5-6 q^4+9 q^3-10 q^2+12 q-11+10 q^{-1} -6 q^{-2} +5 q^{-3} -2 q^{-4} + q^{-5} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^{-4} +a^4 z^2-2 z^2 a^{-4} +a^4 z^{-2} +2 a^4- a^{-4} +z^6 a^{-2} -2 a^2 z^4+3 z^4 a^{-2} -6 a^2 z^2+3 z^2 a^{-2} -2 a^2 z^{-2} -5 a^2+2 a^{-2} +z^6+3 z^4+3 z^2+ z^{-2} +2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^2 z^{10}+z^{10}+2 a^3 z^9+6 a z^9+4 z^9 a^{-1} +a^4 z^8+a^2 z^8+7 z^8 a^{-2} +7 z^8-10 a^3 z^7-24 a z^7-5 z^7 a^{-1} +9 z^7 a^{-3} -6 a^4 z^6-24 a^2 z^6-13 z^6 a^{-2} +9 z^6 a^{-4} -40 z^6+14 a^3 z^5+24 a z^5-12 z^5 a^{-1} -16 z^5 a^{-3} +6 z^5 a^{-5} +13 a^4 z^4+52 a^2 z^4-14 z^4 a^{-4} +3 z^4 a^{-6} +56 z^4-2 a^3 z^3+a z^3+12 z^3 a^{-1} +4 z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} -13 a^4 z^2-39 a^2 z^2+z^2 a^{-2} +7 z^2 a^{-4} -32 z^2-5 a^3 z-8 a z-3 z a^{-1} +z a^{-3} +z a^{-5} +6 a^4+13 a^2- a^{-4} +9+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 13                       1 -1 11                     2   2 9                   4 1   -3 7                 5 2     3 5               5 4       -1 3             7 5         2 1           7 8           1 -1         3 4             -1 -3       3 7               4 -5     2 3                 -1 -7   1 4                   3 -9   1                     -1 -11 1                       1

Integral Khovanov Homology (db, data source)

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