L11n340

L11n339

L11n341

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Link Presentations

[edit Notes on L11n340's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,12,19,11 X7,14,8,15 X13,8,14,9 X22,20,13,19 X20,16,21,15 X16,22,17,21 X12,18,5,17 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, 3, -9}, {-5, 4, 7, -8, 9, -3, 6, -7, 8, -6}

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{(v-1) (w-1) \left(-u v w^2+2 u v w-u v-u w+v^2 w+v w^2-2 v w+v\right)}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+3 q^4-7 q^3+11 q^2-13 q+14-12 q^{-1} +11 q^{-2} -5 q^{-3} +3 q^{-4} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^{-4} +2 a^4 z^{-2} +2 a^4- a^{-4} +a^2 z^4+2 z^4 a^{-2} -2 a^2 z^2+3 z^2 a^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -7 a^2+ a^{-2} -z^6-2 z^4+4 z^{-2} +5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-5} -2 z^3 a^{-5} +z a^{-5} +3 z^6 a^{-4} +6 a^4 z^4-5 z^4 a^{-4} -13 a^4 z^2+3 z^2 a^{-4} -2 a^4 z^{-2} +9 a^4- a^{-4} +3 a^3 z^7+5 z^7 a^{-3} -6 a^3 z^5-8 z^5 a^{-3} +14 a^3 z^3+5 z^3 a^{-3} -16 a^3 z-3 z a^{-3} +5 a^3 z^{-1} + a^{-3} z^{-1} +4 a^2 z^8+4 z^8 a^{-2} -8 a^2 z^6+18 a^2 z^4-10 z^4 a^{-2} -26 a^2 z^2+6 z^2 a^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} +18 a^2+a z^9+z^9 a^{-1} +11 a z^7+13 z^7 a^{-1} -32 a z^5-35 z^5 a^{-1} +43 a z^3+36 z^3 a^{-1} -32 a z-20 z a^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} +8 z^8-11 z^6+7 z^4-10 z^2-4 z^{-2} +11 }[/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 11                   1 -1 9                 2   2 7               5 1   -4 5             6 2     4 3           7 5       -2 1         7 6         1 -1       7 9           2 -3     4 5             -1 -5   1 7               6 -7 2 4                 -2 -9 3                   3

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=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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