L11n55

L11n54

L11n56

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Link Presentations

[edit Notes on L11n55's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,11,17,12 X7,15,8,14 X15,9,16,8 X20,13,21,14 X22,17,5,18 X18,21,19,22 X12,19,13,20 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, 3, -9, 6, 4, -5, -3, 7, -8, 9, -6, 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{2 t(1) t(2)^3-3 t(2)^3-6 t(1) t(2)^2+6 t(2)^2+6 t(1) t(2)-6 t(2)-3 t(1)+2}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{11}{q^{9/2}}-\frac{12}{q^{7/2}}+\frac{9}{q^{5/2}}-\frac{6}{q^{3/2}}+\frac{2}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{8}{q^{13/2}}-\frac{11}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^9 z^{-1} -a^7 z^3+2 a^7 z+3 a^7 z^{-1} +a^5 z^5-3 a^5 z-2 a^5 z^{-1} +a^3 z^5+a^3 z^3-a z^3-a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 z^4 a^{10}+4 z^2 a^{10}-a^{10}-z^7 a^9-3 z^5 a^9+6 z^3 a^9-3 z a^9+a^9 z^{-1} -2 z^8 a^8+z^6 a^8-4 z^4 a^8+8 z^2 a^8-3 a^8-z^9 a^7-3 z^7 a^7+z^5 a^7+7 z^3 a^7-8 z a^7+3 a^7 z^{-1} -5 z^8 a^6+4 z^6 a^6+3 z^2 a^6-3 a^6-z^9 a^5-6 z^7 a^5+11 z^5 a^5-3 z^3 a^5-3 z a^5+2 a^5 z^{-1} -3 z^8 a^4+7 z^4 a^4-4 z^2 a^4-4 z^7 a^3+6 z^5 a^3-2 z^3 a^3+z a^3-3 z^6 a^2+6 z^4 a^2-3 z^2 a^2-z^5 a+2 z^3 a-z a }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                   1 1 0                 2   -2 -2               4 1   3 -4             6 3     -3 -6           6 3       3 -8         5 6         1 -10       6 6           0 -12     3 6             3 -14   2 5               -3 -16   3                 3 -18 2                   -2

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{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=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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