L11n307

L11n306

L11n308

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Link Presentations

[edit Notes on L11n307's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X7,17,8,16 X9,21,10,20 X15,9,16,8 X19,5,20,10 X13,19,14,18 X17,11,18,22 X21,15,22,14 X2536 X11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 5, -4, 6}, {-11, 2, -7, 9, -5, 3, -8, 7, -6, 4, -9, 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 u v^2 w^2+3 u v^2 w-u v^2-u v w^3+3 u v w^2-2 u v w+u w^3-u w^2+v^2 w^2-v^2 w+2 v w^3-3 v w^2+v w+w^4-3 w^3+3 w^2}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{10}+3 q^9-6 q^8+8 q^7-9 q^6+11 q^5-8 q^4+8 q^3-4 q^2+2 q }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^{-4} -3 z^4 a^{-6} +2 z^2 a^{-2} +2 z^2 a^{-4} -9 z^2 a^{-6} +4 z^2 a^{-8} + a^{-2} +5 a^{-4} -12 a^{-6} +7 a^{-8} - a^{-10} +2 a^{-4} z^{-2} -5 a^{-6} z^{-2} +4 a^{-8} z^{-2} - a^{-10} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^7 a^{-11} -4 z^5 a^{-11} +6 z^3 a^{-11} -4 z a^{-11} + a^{-11} z^{-1} +3 z^8 a^{-10} -12 z^6 a^{-10} +14 z^4 a^{-10} -6 z^2 a^{-10} - a^{-10} z^{-2} +3 a^{-10} +2 z^9 a^{-9} -z^7 a^{-9} -21 z^5 a^{-9} +36 z^3 a^{-9} -19 z a^{-9} +5 a^{-9} z^{-1} +10 z^8 a^{-8} -39 z^6 a^{-8} +48 z^4 a^{-8} -32 z^2 a^{-8} -4 a^{-8} z^{-2} +15 a^{-8} +2 z^9 a^{-7} +5 z^7 a^{-7} -38 z^5 a^{-7} +52 z^3 a^{-7} -33 z a^{-7} +9 a^{-7} z^{-1} +7 z^8 a^{-6} -24 z^6 a^{-6} +35 z^4 a^{-6} -37 z^2 a^{-6} -5 a^{-6} z^{-2} +20 a^{-6} +7 z^7 a^{-5} -20 z^5 a^{-5} +25 z^3 a^{-5} -18 z a^{-5} +5 a^{-5} z^{-1} +3 z^6 a^{-4} +z^4 a^{-4} -8 z^2 a^{-4} -2 a^{-4} z^{-2} +8 a^{-4} +z^5 a^{-3} +3 z^3 a^{-3} +3 z^2 a^{-2} - a^{-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 21                   1 -1 19                 2   2 17               4 1   -3 15             4 2     2 13           6 5       -1 11         5 3         2 9       3 6           3 7     5 5             0 5     4               4 3 2 4                 -2 1 2                   2

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