L10a74

L10a73

L10a75

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Link Presentations

[edit Notes on L10a74's Link Presentations]

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

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{2 u^2 v^4-2 u^2 v^3+2 u^2 v^2-u^2 v-u v^4+3 u v^3-3 u v^2+3 u v-u-v^3+2 v^2-2 v+2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}-\frac{1}{q^{27/2}}+\frac{3}{q^{25/2}}-\frac{5}{q^{23/2}}+\frac{7}{q^{21/2}}-\frac{8}{q^{19/2}}+\frac{8}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{5}{q^{13/2}}-\frac{4}{q^{11/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{11} z^5+3 a^{11} z^3+a^{11} z-a^{11} z^{-1} -a^9 z^7-4 a^9 z^5-2 a^9 z^3+5 a^9 z+3 a^9 z^{-1} -a^7 z^7-6 a^7 z^5-12 a^7 z^3-9 a^7 z-2 a^7 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{17} z^3+3 a^{16} z^4-a^{16} z^2+5 a^{15} z^5-4 a^{15} z^3+a^{15} z+6 a^{14} z^6-8 a^{14} z^4+3 a^{14} z^2+5 a^{13} z^7-7 a^{13} z^5+a^{13} z+3 a^{12} z^8-3 a^{12} z^6-5 a^{12} z^4+3 a^{12} z^2-a^{12}+a^{11} z^9+2 a^{11} z^7-10 a^{11} z^5+4 a^{11} z^3+a^{11} z^{-1} +4 a^{10} z^8-12 a^{10} z^6+5 a^{10} z^4+6 a^{10} z^2-3 a^{10}+a^9 z^9-2 a^9 z^7-4 a^9 z^5+11 a^9 z^3-9 a^9 z+3 a^9 z^{-1} +a^8 z^8-3 a^8 z^6-a^8 z^4+7 a^8 z^2-3 a^8+a^7 z^7-6 a^7 z^5+12 a^7 z^3-9 a^7 z+2 a^7 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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                     1 1 -8                   1 1 0 -10                 3     3 -12               2 1     -1 -14             5 3       2 -16           3 2         -1 -18         5 5           0 -20       3 4             1 -22     2 4               -2 -24   1 3                 2 -26   2                   -2 -28 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-8 }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/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}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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