L11n319

L11n318

L11n320

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Link Presentations

[edit Notes on L11n319's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,5,15,6 X11,21,12,20 X17,11,18,22 X21,17,22,16 X10,13,5,14 X8,20,9,19 X18,8,19,7 X2,9,3,10 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {3, -1, 9, -8, 10, -7}, {-4, -2, 7, -3, 11, 6, -5, -9, 8, 4, -6, 5}

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{u v^2 w^4-u v^2 w^3-u v w^4+3 u v w^3-2 u v w^2+2 u v w-u v-u w^3+u w^2-u w+u-v^2 w^4+v^2 w^3-v^2 w^2+v^2 w+v w^4-2 v w^3+2 v w^2-3 v w+v+w-1}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q^6+4 q^5-7 q^4+10 q^3-9 q^2+11 q-7+6 q^{-1} -3 q^{-2} + q^{-3} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^8 a^{-2} -6 z^6 a^{-2} +z^6 a^{-4} +z^6-13 z^4 a^{-2} +5 z^4 a^{-4} +4 z^4-14 z^2 a^{-2} +9 z^2 a^{-4} -z^2 a^{-6} +5 z^2-10 a^{-2} +8 a^{-4} -2 a^{-6} +4-5 a^{-2} z^{-2} +4 a^{-4} z^{-2} - a^{-6} z^{-2} +2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 z^9 a^{-1} +2 z^9 a^{-3} +9 z^8 a^{-2} +5 z^8 a^{-4} +4 z^8+3 a z^7-z^7 a^{-1} +4 z^7 a^{-5} +a^2 z^6-34 z^6 a^{-2} -19 z^6 a^{-4} +z^6 a^{-6} -13 z^6-9 a z^5-11 z^5 a^{-1} -15 z^5 a^{-3} -13 z^5 a^{-5} -3 a^2 z^4+50 z^4 a^{-2} +35 z^4 a^{-4} +2 z^4 a^{-6} +14 z^4+5 a z^3+14 z^3 a^{-1} +31 z^3 a^{-3} +25 z^3 a^{-5} +3 z^3 a^{-7} +2 a^2 z^2-37 z^2 a^{-2} -26 z^2 a^{-4} -4 z^2 a^{-6} -13 z^2-11 z a^{-1} -24 z a^{-3} -18 z a^{-5} -5 z a^{-7} +15 a^{-2} +12 a^{-4} +3 a^{-6} +7+5 a^{-1} z^{-1} +9 a^{-3} z^{-1} +5 a^{-5} z^{-1} + a^{-7} z^{-1} -5 a^{-2} z^{-2} -4 a^{-4} z^{-2} - a^{-6} z^{-2} -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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 13                   2 -2 11                 2   2 9               5 2   -3 7             5 2     3 5           5 6       1 3         6 4         2 1       3 7           4 -1     3 4             -1 -3   1 4               3 -5   2                 -2 -7 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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