L10a35

L10a34

L10a36

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Link Presentations

[edit Notes on L10a35's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,12,15,11 X20,15,5,16 X16,7,17,8 X18,9,19,10 X8,17,9,18 X10,19,11,20 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 5, -7, 6, -8, 3, -2, 10, -3, 4, -5, 7, -6, 8, -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)-1) (t(2)-1) \left(t(2)^4-2 t(2)^3+t(2)^2-2 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{3/2}-3 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{10}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{7}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^7-3 z a^7-2 a^7 z^{-1} +2 z^5 a^5+8 z^3 a^5+10 z a^5+5 a^5 z^{-1} -z^7 a^3-5 z^5 a^3-9 z^3 a^3-8 z a^3-3 a^3 z^{-1} +z^5 a+3 z^3 a+z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^4-2 a^{10} z^2+a^{10}+2 a^9 z^5-2 a^9 z^3+3 a^8 z^6-3 a^8 z^4+a^8 z^2+3 a^7 z^7-3 a^7 z^5+4 a^7 z^3-5 a^7 z+2 a^7 z^{-1} +2 a^6 z^8+a^6 z^6-7 a^6 z^4+10 a^6 z^2-5 a^6+a^5 z^9+3 a^5 z^7-12 a^5 z^5+18 a^5 z^3-15 a^5 z+5 a^5 z^{-1} +5 a^4 z^8-12 a^4 z^6+5 a^4 z^4+7 a^4 z^2-5 a^4+a^3 z^9+3 a^3 z^7-18 a^3 z^5+22 a^3 z^3-12 a^3 z+3 a^3 z^{-1} +3 a^2 z^8-9 a^2 z^6+5 a^2 z^4+a^2 z^2+3 a z^7-11 a z^5+10 a z^3-2 a z+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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                     1 -1 2                   2   2 0                 2 1   -1 -2               5 2     3 -4             5 4       -1 -6           5 3         2 -8         3 5           2 -10       4 5             -1 -12     1 3               2 -14   1 4                 -3 -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{ 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}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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