L11n446

L11n445

L11n447

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Link Presentations

[edit Notes on L11n446's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X18,12,19,11 X22,19,17,20 X16,21,9,22 X20,15,21,16 X12,18,13,17 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {9, -5, 6, -8, 7, -6}, {-11, 2, 5, -9, -4, 3, 8, -7}

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{(w-1) (x-1) (u-x) (x-v)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}+\frac{1}{q^{9/2}}-3 q^{7/2}-\frac{2}{q^{7/2}}+q^{5/2}+\frac{2}{q^{5/2}}-2 q^{3/2}-\frac{2}{q^{3/2}}-q^{11/2}-2 \sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ - a^{-5} z^{-3} -z a^{-5} -2 a^{-5} z^{-1} -a^3 z^3+2 z^3 a^{-3} +3 a^{-3} z^{-3} -2 a^3 z+7 z a^{-3} -a^3 z^{-1} +8 a^{-3} z^{-1} +a z^5-z^5 a^{-1} +5 a z^3-6 z^3 a^{-1} +a z^{-3} -3 a^{-1} z^{-3} +8 a z-12 z a^{-1} +6 a z^{-1} -11 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^2 z^8-z^8 a^{-2} -z^8 a^{-4} -z^8-2 a^3 z^7-3 a z^7-4 z^7 a^{-1} -4 z^7 a^{-3} -z^7 a^{-5} -a^4 z^6+3 a^2 z^6+4 z^6 a^{-2} +4 z^6 a^{-4} +4 z^6+9 a^3 z^5+20 a z^5+29 z^5 a^{-1} +24 z^5 a^{-3} +6 z^5 a^{-5} +4 a^4 z^4+6 a^2 z^4+6 z^4 a^{-2} +8 z^4-9 a^3 z^3-36 a z^3-57 z^3 a^{-1} -42 z^3 a^{-3} -12 z^3 a^{-5} -3 a^4 z^2-14 a^2 z^2-24 z^2 a^{-2} -9 z^2 a^{-4} -26 z^2+6 a^3 z+27 a z+44 z a^{-1} +34 z a^{-3} +11 z a^{-5} +a^4+6 a^2+21 a^{-2} +9 a^{-4} +18-2 a^3 z^{-1} -11 a z^{-1} -18 a^{-1} z^{-1} -14 a^{-3} z^{-1} -5 a^{-5} z^{-1} -6 a^{-2} z^{-2} -3 a^{-4} z^{-2} -3 z^{-2} +a z^{-3} +3 a^{-1} z^{-3} +3 a^{-3} z^{-3} + 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                   3 1   2 6               1 1 2     2 4               2 1       1 2           5 2 1         4 0         2 7 2           3 -2       1 1 3             3 -4     1 2 1               0 -6   1 1                   0 -8   1                     1 -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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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