L11n232

L11n231

L11n233

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Link Presentations

[edit Notes on L11n232's Link Presentations]

Planar diagram presentation	X10,1,11,2 X18,11,19,12 X8,9,1,10 X22,19,9,20 X20,6,21,5 X4,22,5,21 X7,15,8,14 X12,4,13,3 X13,16,14,17 X15,7,16,6 X2,18,3,17
Gauss code	{1, -11, 8, -6, 5, 10, -7, -3}, {3, -1, 2, -8, -9, 7, -10, 9, 11, -2, 4, -5, 6, -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{u^3 \left(-v^2\right)+2 u^3 v-u^3+u^2 v^2-u^2 v+u^2+u v^3-u v^2+u v-v^3+2 v^2-v}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{9/2}-4 q^{7/2}+4 q^{5/2}-\frac{1}{q^{5/2}}-5 q^{3/2}+\frac{3}{q^{3/2}}-q^{11/2}+4 \sqrt{q}-\frac{4}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-1} +a z^3-3 z^3 a^{-1} +2 z^3 a^{-3} +a z-2 z a^{-1} +4 z a^{-3} -z a^{-5} + a^{-3} z^{-1} - a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^7 a^{-5} -5 z^5 a^{-5} +8 z^3 a^{-5} -5 z a^{-5} + a^{-5} z^{-1} +2 z^8 a^{-4} -9 z^6 a^{-4} +10 z^4 a^{-4} -2 z^2 a^{-4} - a^{-4} +z^9 a^{-3} -z^7 a^{-3} -11 z^5 a^{-3} +18 z^3 a^{-3} -8 z a^{-3} + a^{-3} z^{-1} +4 z^8 a^{-2} -17 z^6 a^{-2} +17 z^4 a^{-2} +a^2 z^2-4 z^2 a^{-2} +z^9 a^{-1} +a z^7-z^7 a^{-1} -4 a z^5-10 z^5 a^{-1} +5 a z^3+15 z^3 a^{-1} -2 a z-5 z a^{-1} +2 z^8-8 z^6+7 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   \ -2 -1 0 1 2 3 4 5 6 χ 12                 1 1 10               1   -1 8             3 1   2 6           2 2     0 4         3 2       1 2     1 2 2         1 0     3 3           0 -2   1 2             1 -4   2               -2 -6 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{ i=2 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/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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