L11n309

L11n308

L11n310

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Link Presentations

[edit Notes on L11n309's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,17,8,16 X9,21,10,20 X15,9,16,8 X19,5,20,10 X18,13,19,14 X22,17,11,18 X21,15,22,14 X2536 X4,11,1,12
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{u v^2 w-u v^2+u v w^2-4 u v w+3 u v-u w^2+3 u w-u+v^2 w^2-3 v^2 w+v^2-3 v w^2+4 v w-v+w^2-w}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^3+5 q^2-6 q+10-10 q^{-1} +10 q^{-2} -8 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6-3 z^2 a^4-2 a^4+2 z^4 a^2+3 z^2 a^2+a^2 z^{-2} +3 a^2+z^4-2 z^2-2 z^{-2} -4-z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+3 a^5 z^7-9 a^5 z^5+7 a^5 z^3-2 a^5 z+3 a^4 z^8-5 a^4 z^6-5 a^4 z^4+6 a^4 z^2-a^4+a^3 z^9+6 a^3 z^7-24 a^3 z^5+20 a^3 z^3+z^3 a^{-3} -6 a^3 z+6 a^2 z^8-12 a^2 z^6+a^2 z^4+5 z^4 a^{-2} +6 a^2 z^2-2 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -3 a^2-4 a^{-2} +a z^9+5 a z^7+2 z^7 a^{-1} -16 a z^5-z^5 a^{-1} +13 a z^3+z^3 a^{-1} +4 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +3 z^8-6 z^6+8 z^4+z^2+2 z^{-2} -6 }[/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 χ 7                   1 -1 5                 4   4 3               4 3   -1 1             6 2     4 -1           5 5       0 -3         5 5         0 -5       3 5           2 -7     3 5             -2 -9   1 4               3 -11   2                 -2 -13 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=1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/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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