L10n113

L10n112

L11a1

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Link Presentations

[edit Notes on L10n113's Link Presentations]

Planar diagram presentation	X6172 X2536 X11,19,12,18 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X20,15,17,16 X16,19,13,20 X17,9,18,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 10}, {-7, 6, 8, -9}, {-10, 3, 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 w+u v x y-u v y+u v+u w y-u x y+v w-v x-w x y+w x-w+x y}{\sqrt{u} \sqrt{v} \sqrt{w} \sqrt{x} \sqrt{y}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^6-q^5+5 q^4-q^3+ q^{-3} +5 q^2+ q^{-1} +5 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{-6} z^{-4} +2 a^{-6} z^{-2} + a^{-6} -4 a^{-4} z^{-4} -3 z^2 a^{-4} -9 a^{-4} z^{-2} -8 a^{-4} +2 z^4 a^{-2} +a^2 z^{-4} +6 a^{-2} z^{-4} +a^2 z^2+9 z^2 a^{-2} +3 a^2 z^{-2} +15 a^{-2} z^{-2} +3 a^2+16 a^{-2} -z^4-4 z^{-4} -7 z^2-11 z^{-2} -12 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-6} -5 z^4 a^{-6} - a^{-6} z^{-4} +10 z^2 a^{-6} +5 a^{-6} z^{-2} -10 a^{-6} +z^7 a^{-5} -z^5 a^{-5} -10 z^3 a^{-5} +4 a^{-5} z^{-3} +20 z a^{-5} -15 a^{-5} z^{-1} +6 z^6 a^{-4} -25 z^4 a^{-4} -4 a^{-4} z^{-4} +30 z^2 a^{-4} +14 a^{-4} z^{-2} -25 a^{-4} +z^7 a^{-3} +3 z^5 a^{-3} -30 z^3 a^{-3} +12 a^{-3} z^{-3} +55 z a^{-3} -41 a^{-3} z^{-1} +a^2 z^6+5 z^6 a^{-2} -6 a^2 z^4-25 z^4 a^{-2} -a^2 z^{-4} -6 a^{-2} z^{-4} +10 a^2 z^2+40 z^2 a^{-2} +5 a^2 z^{-2} +18 a^{-2} z^{-2} -10 a^2-31 a^{-2} +a z^5+5 z^5 a^{-1} -10 a z^3-30 z^3 a^{-1} +4 a z^{-3} +12 a^{-1} z^{-3} +20 a z+55 z a^{-1} -15 a z^{-1} -41 a^{-1} z^{-1} +z^6-11 z^4-4 z^{-4} +30 z^2+14 z^{-2} -25 }[/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 6 χ 13                     1 1 11                       0 9                 5 1   4 7               1 5     4 5             4         4 3         4   1         5 1       1 10 4           5 -1         6             6 -3     1                 1 -5 1                     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=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{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/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.

L10n112 L11a1