L11n447

L11n446

L11n448

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Link Presentations

[edit Notes on L11n447's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X19,17,20,22 X21,9,22,16 X15,21,16,20 X17,12,18,13 X2536 X4,9,1,10
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) \left(-u v+u x^2-u x+u+v x^2-v x+v-x^2\right)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{12}{q^{9/2}}+\frac{9}{q^{7/2}}-\frac{11}{q^{5/2}}+\frac{8}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{6}{q^{13/2}}+\frac{7}{q^{11/2}}+2 \sqrt{q}-\frac{6}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-1} +a^9 z^{-3} -4 z a^7-7 a^7 z^{-1} -3 a^7 z^{-3} +5 z^3 a^5+13 z a^5+12 a^5 z^{-1} +3 a^5 z^{-3} -2 z^5 a^3-7 z^3 a^3-11 z a^3-7 a^3 z^{-1} -a^3 z^{-3} +2 z^3 a+2 z a+a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^7-5 a^9 z^5+10 a^9 z^3-a^9 z^{-3} -10 a^9 z+5 a^9 z^{-1} +2 a^8 z^8-7 a^8 z^6+5 a^8 z^4+6 a^8 z^2+3 a^8 z^{-2} -9 a^8+a^7 z^9+5 a^7 z^7-34 a^7 z^5+55 a^7 z^3-3 a^7 z^{-3} -38 a^7 z+14 a^7 z^{-1} +8 a^6 z^8-22 a^6 z^6+a^6 z^4+27 a^6 z^2+6 a^6 z^{-2} -21 a^6+a^5 z^9+15 a^5 z^7-65 a^5 z^5+83 a^5 z^3-3 a^5 z^{-3} -54 a^5 z+18 a^5 z^{-1} +6 a^4 z^8-8 a^4 z^6-18 a^4 z^4+33 a^4 z^2+3 a^4 z^{-2} -18 a^4+11 a^3 z^7-35 a^3 z^5+45 a^3 z^3-a^3 z^{-3} -31 a^3 z+11 a^3 z^{-1} +7 a^2 z^6-14 a^2 z^4+15 a^2 z^2-6 a^2+a z^5+7 a z^3-5 a z+2 a z^{-1} +3 z^2-1 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 2                   2 -2 0                 4   4 -2               6 4   -2 -4             5 3 1   3 -6           4 6       2 -8         8 5         3 -10       5 10           5 -12     1 2             -1 -14   1 5               4 -16   1                 -1 -18 1                   1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/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}^{10}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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