L11n133

L11n132

L11n134

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Link Presentations

[edit Notes on L11n133's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X3,10,4,11 X2,17,3,18 X12,5,13,6 X6718 X9,16,10,17 X13,20,14,21 X15,22,16,7 X19,4,20,5 X21,14,22,15
Gauss code	{1, -4, -3, 10, 5, -6}, {6, -1, -7, 3, 2, -5, -8, 11, -9, 7, 4, -2, -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^2 v^6+u v^4-u v^3+u v^2+1}{u v^3} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{9/2}}+\frac{1}{q^{25/2}}-\frac{1}{q^{23/2}}+\frac{1}{q^{21/2}}-\frac{1}{q^{19/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{13}-3 z a^{13}-2 a^{13} z^{-1} +z^7 a^{11}+8 z^5 a^{11}+20 z^3 a^{11}+18 z a^{11}+5 a^{11} z^{-1} -z^9 a^9-9 z^7 a^9-28 z^5 a^9-36 z^3 a^9-18 z a^9-3 a^9 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{16} z^2-a^{16}+a^{15} z^3-a^{15} z+a^{14} z^2-a^{13} z^3+4 a^{13} z-2 a^{13} z^{-1} +a^{12} z^8-8 a^{12} z^6+20 a^{12} z^4-18 a^{12} z^2+5 a^{12}+a^{11} z^9-9 a^{11} z^7+28 a^{11} z^5-38 a^{11} z^3+23 a^{11} z-5 a^{11} z^{-1} +a^{10} z^8-8 a^{10} z^6+20 a^{10} z^4-18 a^{10} z^2+5 a^{10}+a^9 z^9-9 a^9 z^7+28 a^9 z^5-36 a^9 z^3+18 a^9 z-3 a^9 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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -8                   1 1 -10                   1 1 -12               1     1 -14           1         1 -16           1 1       0 -18       2 1           1 -20       1 1           0 -22   1 1               0 -24                     0 -26 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-10 }[/math] [math]\displaystyle{ i=-8 }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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