L11n245

L11n244

L11n246

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Link Presentations

[edit Notes on L11n245's Link Presentations]

Planar diagram presentation	X12,1,13,2 X10,11,1,12 X14,5,15,6 X9,19,10,18 X17,3,18,2 X16,8,17,7 X3849 X20,16,21,15 X22,13,11,14 X4,20,5,19 X6,21,7,22
Gauss code	{1, 5, -7, -10, 3, -11, 6, 7, -4, -2}, {2, -1, 9, -3, 8, -6, -5, 4, 10, -8, 11, -9}

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-1) (v-1) \left(2 u^2 v-u^2+u v^2-u v+u-v^2+2 v\right)}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{2}{q^{9/2}}-3 q^{7/2}+\frac{6}{q^{7/2}}+6 q^{5/2}-\frac{9}{q^{5/2}}-10 q^{3/2}+\frac{11}{q^{3/2}}+11 \sqrt{q}-\frac{13}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -2 a z^5-z^5 a^{-1} +2 a^3 z^3-5 a z^3-z^3 a^{-1} +z^3 a^{-3} +2 a^3 z-3 a z+z a^{-3} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a z^9-2 z^9 a^{-1} -5 a^2 z^8-4 z^8 a^{-2} -9 z^8-4 a^3 z^7-3 a z^7-2 z^7 a^{-1} -3 z^7 a^{-3} -a^4 z^6+10 a^2 z^6+10 z^6 a^{-2} -z^6 a^{-4} +22 z^6+3 a^3 z^5+10 a z^5+16 z^5 a^{-1} +9 z^5 a^{-3} -6 a^4 z^4-14 a^2 z^4-5 z^4 a^{-2} +3 z^4 a^{-4} -16 z^4-3 a^5 z^3-2 a^3 z^3-4 a z^3-13 z^3 a^{-1} -8 z^3 a^{-3} +5 a^4 z^2+8 a^2 z^2-2 z^2 a^{-4} +5 z^2+a^5 z+4 z a^{-1} +3 z a^{-3} +1-a z^{-1} - a^{-1} z^{-1} }[/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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               4 1   -3 4             6 2     4 2           5 4       -1 0         8 6         2 -2       5 7           2 -4     4 6             -2 -6   2 5               3 -8   4                 -4 -10 2                   2

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=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/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}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/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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