L11n349

L11n348

L11n350

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Link Presentations

[edit Notes on L11n349's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X20,12,21,11 X18,8,19,7 X22,18,13,17 X9,17,10,16 X15,11,16,10 X12,20,5,19 X8,22,9,21 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, -6, 7, 3, -8}, {11, -2, -7, 6, 5, -4, 8, -3, 9, -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^3-2 u v^2 w^2+2 u v^2 w-u v^2-u v w^3+2 u v w^2-3 u v w+2 u v+u w+v^3 \left(-w^2\right)-2 v^2 w^3+3 v^2 w^2-2 v^2 w+v^2+v w^3-2 v w^2+2 v w-v}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^6+5 q^5-7 q^4+10 q^3+ q^{-3} -10 q^2-2 q^{-2} +10 q+6 q^{-1} -8 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-2} +2 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2-2 z^2 a^{-2} -4 z^2+2 a^2-5 a^{-2} +3 a^{-4} -2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{-7} +5 z^4 a^{-6} -z^2 a^{-6} -2 a^{-6} +2 z^7 a^{-5} +z^3 a^{-5} +z a^{-5} +3 z^8 a^{-4} -5 z^6 a^{-4} +8 z^4 a^{-4} -6 z^2 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +z^9 a^{-3} +5 z^7 a^{-3} -15 z^5 a^{-3} +15 z^3 a^{-3} -8 z a^{-3} +2 a^{-3} z^{-1} +6 z^8 a^{-2} +a^2 z^6-14 z^6 a^{-2} -4 a^2 z^4+14 z^4 a^{-2} +5 a^2 z^2-16 z^2 a^{-2} -2 a^{-2} z^{-2} -2 a^2+9 a^{-2} +z^9 a^{-1} +2 a z^7+5 z^7 a^{-1} -5 a z^5-20 z^5 a^{-1} +2 a z^3+17 z^3 a^{-1} +a z-8 z a^{-1} +2 a^{-1} z^{-1} +3 z^8-8 z^6+7 z^4-6 z^2- z^{-2} +3 }[/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                   1 -1 11                 4   4 9               5 3   -2 7             5 2     3 5           5 5       0 3         5 5         0 1       4 6           2 -1     2 4             -2 -3     4               4 -5 1 2                 -1 -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{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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