L11n91

L11n90

L11n92

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Link Presentations

[edit Notes on L11n91's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X7,16,8,17 X11,20,12,21 X17,22,18,5 X21,18,22,19 X19,10,20,11 X9,14,10,15 X15,8,16,9 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, -8, 7, -4, -2, 11, 8, -9, 3, -5, 6, -7, 4, -6, 5}

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)-1) (t(2)-1) \left(t(2)^2+1\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\sqrt{q}+\frac{2}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{17/2}}+\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 (-z)-a^9 z^{-1} +a^7 z^3+3 a^7 z+2 a^7 z^{-1} -a^5 z^3-2 a^5 z-a^5 z^{-1} +a^3 z^5+3 a^3 z^3+2 a^3 z+a^3 z^{-1} -a z^3-2 a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{10}+7 z^6 a^{10}-15 z^4 a^{10}+12 z^2 a^{10}-3 a^{10}-z^9 a^9+7 z^7 a^9-14 z^5 a^9+10 z^3 a^9-3 z a^9+a^9 z^{-1} -2 z^8 a^8+16 z^6 a^8-36 z^4 a^8+27 z^2 a^8-7 a^8-z^9 a^7+8 z^7 a^7-18 z^5 a^7+14 z^3 a^7-8 z a^7+2 a^7 z^{-1} -z^8 a^6+8 z^6 a^6-20 z^4 a^6+15 z^2 a^6-4 a^6-4 z^5 a^5+10 z^3 a^5-7 z a^5+a^5 z^{-1} -3 z^6 a^4+6 z^4 a^4-z^2 a^4-z^7 a^3-z^5 a^3+9 z^3 a^3-5 z a^3+a^3 z^{-1} -2 z^6 a^2+5 z^4 a^2-z^2 a^2-a^2-z^5 a+3 z^3 a-3 z a+a z^{-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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     1   -1 -2                   3 1   2 -4                 2 3     1 -6             1 3 1       1 -8             1 2         1 -10         1 3 3           -1 -12       1   1             2 -14       1 2               -1 -16   1 1                   0 -18                         0 -20 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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/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} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/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}^{3} }[/math] [math]\displaystyle{ r=1 }[/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 }[/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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