L11n89

L11n88

L11n90

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Link Presentations

[edit Notes on L11n89's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,16,8,17 X17,22,18,5 X9,15,10,14 X19,10,20,11 X21,9,22,8 X13,18,14,19 X15,21,16,20 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 7, -5, 6, 11, -2, -8, 5, -9, 3, -4, 8, -6, 9, -7, 4}

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{t(1) t(2)^5-3 t(1) t(2)^4-t(2)^4+2 t(1) t(2)^3+2 t(2)^3+2 t(1) t(2)^2+2 t(2)^2-t(1) t(2)-3 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{3/2}-3 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{2}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{2}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-1} -4 a^7 z-3 a^7 z^{-1} +a^5 z^5+6 a^5 z^3+8 a^5 z+4 a^5 z^{-1} -a^3 z^7-5 a^3 z^5-7 a^3 z^3-6 a^3 z-2 a^3 z^{-1} +a z^5+3 a z^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^7 a^9+5 z^5 a^9-6 z^3 a^9+4 z a^9-a^9 z^{-1} -2 z^8 a^8+11 z^6 a^8-14 z^4 a^8+6 z^2 a^8-a^8-z^9 a^7+4 z^7 a^7+6 z^5 a^7-24 z^3 a^7+16 z a^7-3 a^7 z^{-1} -3 z^8 a^6+21 z^6 a^6-36 z^4 a^6+18 z^2 a^6-3 a^6-z^9 a^5+3 z^7 a^5+14 z^5 a^5-40 z^3 a^5+25 z a^5-4 a^5 z^{-1} -3 z^8 a^4+18 z^6 a^4-28 z^4 a^4+15 z^2 a^4-2 a^4-5 z^7 a^3+25 z^5 a^3-32 z^3 a^3+15 z a^3-2 a^3 z^{-1} -2 z^8 a^2+7 z^6 a^2-3 z^4 a^2+2 z^2 a^2-a^2-3 z^7 a+12 z^5 a-10 z^3 a+2 z a-z^6+3 z^4-z^2 }[/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 2 3 χ 4                       1 -1 2                     2   2 0                   1 1   0 -2               1 4 2     1 -4             1 2 2       1 -6           1 3 3         -1 -8         2 3 2           1 -10       1 2 2             -1 -12     1 2 3               2 -14   1 1                   0 -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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/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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