L11a168

L11a167

L11a169

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Link Presentations

[edit Notes on L11a168's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X10,4,11,3 X2,17,3,18 X14,5,15,6 X6718 X16,10,17,9 X20,13,21,14 X12,19,13,20 X22,16,7,15 X4,22,5,21
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, 2, -9, 8, -5, 10, -7, 4, -2, 9, -8, 11, -10}

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^2 v^6-3 u^2 v^5+4 u^2 v^4-4 u^2 v^3+3 u^2 v^2-u^2 v-u v^6+4 u v^5-7 u v^4+7 u v^3-7 u v^2+4 u v-u-v^5+3 v^4-4 v^3+4 v^2-3 v+1}{u v^3} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{5/2}+4 q^{3/2}-8 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{18}{q^{3/2}}+\frac{19}{q^{5/2}}-\frac{20}{q^{7/2}}+\frac{18}{q^{9/2}}-\frac{13}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{1}{q^{17/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^5 z^7-4 a^5 z^5-4 a^5 z^3+a^3 z^9+6 a^3 z^7+12 a^3 z^5+8 a^3 z^3+a^3 z^{-1} -a z^7-4 a z^5-4 a z^3-a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^4+4 a^9 z^5-2 a^9 z^3+8 a^8 z^6-7 a^8 z^4+2 a^8 z^2+11 a^7 z^7-13 a^7 z^5+4 a^7 z^3+11 a^6 z^8-14 a^6 z^6+4 a^6 z^2+8 a^5 z^9-9 a^5 z^7-4 a^5 z^5+2 a^5 z^3+3 a^4 z^{10}+7 a^4 z^8-30 a^4 z^6+18 a^4 z^4+14 a^3 z^9-42 a^3 z^7+38 a^3 z^5-14 a^3 z^3+a^3 z-a^3 z^{-1} +3 a^2 z^{10}-22 a^2 z^6+23 a^2 z^4-4 a^2 z^2+a^2+6 a z^9-21 a z^7+z^7 a^{-1} +22 a z^5-3 z^5 a^{-1} -8 a z^3+2 z^3 a^{-1} +a z-a z^{-1} +4 z^8-14 z^6+13 z^4-2 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 4 χ 6                       1 1 4                     3   -3 2                   5 1   4 0                 7 3     -4 -2               11 5       6 -4             9 8         -1 -6           11 10           1 -8         8 10             2 -10       5 10               -5 -12     3 8                 5 -14   1 5                   -4 -16   3                     3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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