L11n265

L11n264

L11n266

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Link Presentations

[edit Notes on L11n265's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,19,12,18 X15,21,16,20 X17,9,18,22 X21,17,22,16 X19,13,20,12 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, -5, 9, -4, 3, -6, 8, -7, 5, -9, 6, -8, 7}

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{-2 u v w^4+3 u v w^3-3 u v w^2+2 u v w-u v+u w^4-2 u w^3+2 u w^2-u w+v w^4-2 v w^3+2 v w^2-v w+w^5-2 w^4+3 w^3-3 w^2+2 w}{\sqrt{u} \sqrt{v} w^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{11}+2 q^{10}-6 q^9+9 q^8-11 q^7+12 q^6-10 q^5+10 q^4-4 q^3+3 q^2 }[/math] (db)
Signature	4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -2 z^6 a^{-6} +3 z^4 a^{-4} -10 z^4 a^{-6} +3 z^4 a^{-8} +10 z^2 a^{-4} -21 z^2 a^{-6} +11 z^2 a^{-8} -z^2 a^{-10} +10 a^{-4} -22 a^{-6} +15 a^{-8} -3 a^{-10} +3 a^{-4} z^{-2} -8 a^{-6} z^{-2} +7 a^{-8} z^{-2} -2 a^{-10} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^9 a^{-7} +z^9 a^{-9} +4 z^8 a^{-6} +7 z^8 a^{-8} +3 z^8 a^{-10} +3 z^7 a^{-5} +7 z^7 a^{-7} +7 z^7 a^{-9} +3 z^7 a^{-11} -13 z^6 a^{-6} -18 z^6 a^{-8} -3 z^6 a^{-10} +2 z^6 a^{-12} -9 z^5 a^{-5} -26 z^5 a^{-7} -21 z^5 a^{-9} -3 z^5 a^{-11} +z^5 a^{-13} +6 z^4 a^{-4} +31 z^4 a^{-6} +32 z^4 a^{-8} +4 z^4 a^{-10} -3 z^4 a^{-12} +21 z^3 a^{-5} +48 z^3 a^{-7} +28 z^3 a^{-9} -2 z^3 a^{-11} -3 z^3 a^{-13} -16 z^2 a^{-4} -41 z^2 a^{-6} -35 z^2 a^{-8} -10 z^2 a^{-10} -24 z a^{-5} -45 z a^{-7} -21 z a^{-9} +3 z a^{-11} +3 z a^{-13} +13 a^{-4} +28 a^{-6} +22 a^{-8} +7 a^{-10} + a^{-12} +8 a^{-5} z^{-1} +15 a^{-7} z^{-1} +7 a^{-9} z^{-1} - a^{-11} z^{-1} - a^{-13} z^{-1} -3 a^{-4} z^{-2} -8 a^{-6} z^{-2} -7 a^{-8} z^{-2} -2 a^{-10} 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   \ 0 1 2 3 4 5 6 7 8 9 χ 23                   1 -1 21                 1   1 19               5 1   -4 17             4 1     3 15           7 5       -2 13         5 4         1 11       6 8           2 9     4 4             0 7     6               6 5 3 4                 -1 3 3                   3

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=5 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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