L11n439

L11n438

L11n440

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Link Presentations

[edit Notes on L11n439'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 X22,19,13,20 X20,15,21,16 X16,21,17,22 X17,9,18,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 11}, {-7, 6, 9, -10, -11, 3, 8, -9, 10, -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^3+u v x^2-u v x+u v-u w x^2+u w x+u x^3+v w+v x^2-v x+w x^3-w x^2+w x-w-x^3}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{11/2}+q^{9/2}-5 q^{7/2}+2 q^{5/2}-5 q^{3/2}+\sqrt{q}-\frac{3}{\sqrt{q}}+\frac{1}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^5-2 z^5 a^{-1} -a^3 z^3+7 a z^3-11 z^3 a^{-1} +3 z^3 a^{-3} -3 a^3 z+15 a z-22 z a^{-1} +11 z a^{-3} -z a^{-5} -3 a^3 z^{-1} +14 a z^{-1} -22 a^{-1} z^{-1} +14 a^{-3} z^{-1} -3 a^{-5} z^{-1} -a^3 z^{-3} +5 a z^{-3} -9 a^{-1} z^{-3} +7 a^{-3} z^{-3} -2 a^{-5} z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{-2} -z^8 a^{-4} -a^3 z^7-a z^7-5 z^7 a^{-1} -6 z^7 a^{-3} -z^7 a^{-5} -a^4 z^6-2 a^2 z^6-3 z^6 a^{-2} +2 z^6 a^{-4} -6 z^6+5 a^3 z^5+9 a z^5+28 z^5 a^{-1} +30 z^5 a^{-3} +6 z^5 a^{-5} +5 a^4 z^4+17 a^2 z^4+37 z^4 a^{-2} +9 z^4 a^{-4} +40 z^4-4 a^3 z^3-18 a z^3-46 z^3 a^{-1} -46 z^3 a^{-3} -14 z^3 a^{-5} -6 a^4 z^2-34 a^2 z^2-73 z^2 a^{-2} -26 z^2 a^{-4} -75 z^2+2 a^3 z+16 a z+37 z a^{-1} +39 z a^{-3} +16 z a^{-5} +4 a^4+24 a^2+60 a^{-2} +23 a^{-4} +58-2 a^3 z^{-1} -12 a z^{-1} -24 a^{-1} z^{-1} -23 a^{-3} z^{-1} -9 a^{-5} z^{-1} -a^4 z^{-2} -7 a^2 z^{-2} -19 a^{-2} z^{-2} -7 a^{-4} z^{-2} -18 z^{-2} +a^3 z^{-3} +5 a z^{-3} +9 a^{-1} z^{-3} +7 a^{-3} z^{-3} +2 a^{-5} z^{-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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                         0 8                   5 1   4 6                 1 4     3 4               4 1       3 2           4 1 1         4 0         1 7 4           2 -2       1   1             2 -4       1 3               -2 -6   1 1                   0 -8                         0 -10 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/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}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/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.

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