L10a121

L10a120

L10a122

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Link Presentations

[edit Notes on L10a121's Link Presentations]

Planar diagram presentation	X12,1,13,2 X20,5,11,6 X16,3,17,4 X10,15,1,16 X8,19,9,20 X18,9,19,10 X4,17,5,18 X2,11,3,12 X6,13,7,14 X14,7,15,8
Gauss code	{1, -8, 3, -7, 2, -9, 10, -5, 6, -4}, {8, -1, 9, -10, 4, -3, 7, -6, 5, -2}

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^4 \left(-v^2\right)-3 u^3 v^3+5 u^3 v^2-3 u^3 v-u^2 v^4+5 u^2 v^3-9 u^2 v^2+5 u^2 v-u^2-3 u v^3+5 u v^2-3 u v-v^2}{u^2 v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{11}{q^{11/2}}-\frac{14}{q^{13/2}}+\frac{16}{q^{15/2}}-\frac{14}{q^{17/2}}+\frac{11}{q^{19/2}}-\frac{8}{q^{21/2}}+\frac{3}{q^{23/2}}-\frac{1}{q^{25/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{11} z^3+a^{11} z+a^{11} z^{-1} -a^9 z^5+2 a^9 z-a^9 z^{-1} -3 a^7 z^5-9 a^7 z^3-7 a^7 z-a^5 z^5-2 a^5 z^3-a^5 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{15}+2 z^3 a^{15}-z a^{15}-3 z^6 a^{14}+4 z^4 a^{14}-z^2 a^{14}-6 z^7 a^{13}+12 z^5 a^{13}-12 z^3 a^{13}+7 z a^{13}-5 z^8 a^{12}+3 z^6 a^{12}+6 z^4 a^{12}-5 z^2 a^{12}-2 z^9 a^{11}-8 z^7 a^{11}+18 z^5 a^{11}-9 z^3 a^{11}+z a^{11}+a^{11} z^{-1} -10 z^8 a^{10}+12 z^6 a^{10}+4 z^4 a^{10}-8 z^2 a^{10}-a^{10}-2 z^9 a^9-8 z^7 a^9+18 z^5 a^9-9 z^3 a^9+z a^9+a^9 z^{-1} -5 z^8 a^8+3 z^6 a^8+6 z^4 a^8-5 z^2 a^8-6 z^7 a^7+12 z^5 a^7-12 z^3 a^7+7 z a^7-3 z^6 a^6+4 z^4 a^6-z^2 a^6-z^5 a^5+2 z^3 a^5-z a^5 }[/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 -4                     1 1 -6                   3 1 -2 -8                 5     5 -10               6 3     -3 -12             8 5       3 -14           8 6         -2 -16         6 8           -2 -18       5 8             3 -20     3 6               -3 -22     5                 5 -24 1 3                   -2 -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=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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