L10a99

L10a98

L10a100

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Link Presentations

[edit Notes on L10a99's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X20,12,9,11 X2,9,3,10 X18,14,19,13 X14,7,15,8 X16,5,17,6 X6,15,7,16 X4,17,5,18 X8,20,1,19
Gauss code	{1, -4, 2, -9, 7, -8, 6, -10}, {4, -1, 3, -2, 5, -6, 8, -7, 9, -5, 10, -3}

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(2)^2 t(1)^3-2 t(2) t(1)^3+t(1)^3+t(2)^3 t(1)^2-3 t(2)^2 t(1)^2+3 t(2) t(1)^2-2 t(1)^2-2 t(2)^3 t(1)+3 t(2)^2 t(1)-3 t(2) t(1)+t(1)+t(2)^3-2 t(2)^2+t(2)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{9/2}+3 q^{7/2}-5 q^{5/2}+7 q^{3/2}-8 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z+a^5 z^{-1} -2 a^3 z^3-z^3 a^{-3} -4 a^3 z-a^3 z^{-1} -z a^{-3} +a z^5+z^5 a^{-1} +2 a z^3+2 z^3 a^{-1} +a z+z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^7-5 a^5 z^5+8 a^5 z^3+z^3 a^{-5} -5 a^5 z+a^5 z^{-1} +2 a^4 z^8-9 a^4 z^6+11 a^4 z^4+3 z^4 a^{-4} -2 a^4 z^2-z^2 a^{-4} -a^4+a^3 z^9-13 a^3 z^5+5 z^5 a^{-3} +21 a^3 z^3-4 z^3 a^{-3} -9 a^3 z+z a^{-3} +a^3 z^{-1} +5 a^2 z^8-17 a^2 z^6+6 z^6 a^{-2} +14 a^2 z^4-8 z^4 a^{-2} -2 a^2 z^2+3 z^2 a^{-2} +a z^9+4 a z^7+5 z^7 a^{-1} -20 a z^5-7 z^5 a^{-1} +19 a z^3+z^3 a^{-1} -6 a z-z a^{-1} +3 z^8-2 z^6-8 z^4+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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 10                     1 1 8                   2   -2 6                 3 1   2 4               4 2     -2 2             4 3       1 0           5 5         0 -2         3 3           0 -4       2 5             3 -6     2 3               -1 -8   1 3                 2 -10   1                   -1 -12 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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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