L11n137

L11n136

L11n138

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Link Presentations

[edit Notes on L11n137's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X10,4,11,3 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X13,20,14,21 X15,22,16,7 X4,20,5,19 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^4-3 u^2 v^3+3 u^2 v^2-u^2 v-u v^4+5 u v^3-7 u v^2+5 u v-u-v^3+3 v^2-3 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{12}{q^{9/2}}-\frac{11}{q^{7/2}}+\frac{7}{q^{5/2}}-\frac{5}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{10}{q^{13/2}}-\frac{12}{q^{11/2}}+\frac{1}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^5+2 a^7 z^3-a^7 z^{-1} -a^5 z^7-4 a^5 z^5-4 a^5 z^3+2 a^5 z+3 a^5 z^{-1} +a^3 z^5+a^3 z^3-3 a^3 z-2 a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^5-a^{11} z^3+4 a^{10} z^6-7 a^{10} z^4+2 a^{10} z^2+6 a^9 z^7-11 a^9 z^5+4 a^9 z^3+4 a^8 z^8-a^8 z^6-10 a^8 z^4+7 a^8 z^2-a^8+a^7 z^9+8 a^7 z^7-16 a^7 z^5+7 a^7 z^3-a^7 z+a^7 z^{-1} +5 a^6 z^8-2 a^6 z^6-8 a^6 z^4+9 a^6 z^2-3 a^6+a^5 z^9+2 a^5 z^7+a^5 z^5-2 a^5 z^3-4 a^5 z+3 a^5 z^{-1} +a^4 z^8+3 a^4 z^6-4 a^4 z^4+4 a^4 z^2-3 a^4+5 a^3 z^5-4 a^3 z^3-3 a^3 z+2 a^3 z^{-1} +a^2 z^4 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 0                   1 -1 -2                 4   4 -4               4 2   -2 -6             7 3     4 -8           6 5       -1 -10         6 6         0 -12       4 6           2 -14     3 6             -3 -16   1 4               3 -18   3                 -3 -20 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/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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