L10n38

L10n37

L10n39

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Link Presentations

[edit Notes on L10n38's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X16,13,17,14 X14,9,15,10 X10,15,11,16 X17,5,18,20 X7,19,8,18 X19,9,20,8 X2536 X4,11,1,12
Gauss code	{1, -9, 2, -10}, {9, -1, -7, 8, 4, -5, 10, -2, 3, -4, 5, -3, -6, 7, -8, 6}

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)) \left(t(2)^2-t(2)+1\right)^2}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{6}{q^{9/2}}+\frac{5}{q^{7/2}}-\frac{7}{q^{5/2}}+\frac{5}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{4}{q^{11/2}}+2 \sqrt{q}-\frac{4}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^7-2 a^7 z^{-1} +3 z^3 a^5+9 z a^5+7 a^5 z^{-1} -2 z^5 a^3-9 z^3 a^3-14 z a^3-7 a^3 z^{-1} +2 z^3 a+4 z a+2 a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^8+4 z^4 a^8-5 z^2 a^8+2 a^8-2 z^7 a^7+7 z^5 a^7-7 z^3 a^7+4 z a^7-2 a^7 z^{-1} -z^8 a^6-2 z^6 a^6+17 z^4 a^6-20 z^2 a^6+8 a^6-6 z^7 a^5+20 z^5 a^5-22 z^3 a^5+17 z a^5-7 a^5 z^{-1} -z^8 a^4-5 z^6 a^4+24 z^4 a^4-29 z^2 a^4+13 a^4-4 z^7 a^3+12 z^5 a^3-18 z^3 a^3+16 z a^3-7 a^3 z^{-1} -4 z^6 a^2+11 z^4 a^2-17 z^2 a^2+8 a^2-z^5 a-3 z^3 a+3 z a-2 a z^{-1} -3 z^2+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   \ -7 -6 -5 -4 -3 -2 -1 0 1 χ 2                 2 -2 0               2   2 -2             4 3   -1 -4           3 1     2 -6         2 4       2 -8       4 3         1 -10     1 3           2 -12   1 3             -2 -14   1               1 -16 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{ 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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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