L10a114

L10a113

L10a115

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Link Presentations

[edit Notes on L10a114's Link Presentations]

Planar diagram presentation	X12,1,13,2 X20,9,11,10 X14,3,15,4 X16,5,17,6 X18,7,19,8 X4,15,5,16 X6,17,7,18 X8,19,9,20 X2,11,3,12 X10,13,1,14
Gauss code	{1, -9, 3, -6, 4, -7, 5, -8, 2, -10}, {9, -1, 10, -3, 6, -4, 7, -5, 8, -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{-t(2)^3 t(1)^4-t(2)^4 t(1)^3+t(2)^3 t(1)^3-t(2)^2 t(1)^3-t(2)^3 t(1)^2+t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+t(2) t(1)-t(1)-t(2)}{t(1)^2 t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}-\frac{1}{q^{27/2}}+\frac{1}{q^{25/2}}-\frac{2}{q^{23/2}}+\frac{3}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{2}{q^{11/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{11}+5 z^3 a^{11}+6 z a^{11}+a^{11} z^{-1} -z^7 a^9-6 z^5 a^9-11 z^3 a^9-7 z a^9-a^9 z^{-1} -z^7 a^7-6 z^5 a^7-10 z^3 a^7-4 z a^7 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^3 a^{17}+2 z a^{17}-z^4 a^{16}+z^2 a^{16}-z^5 a^{15}+z^3 a^{15}-z a^{15}-z^6 a^{14}+2 z^4 a^{14}-3 z^2 a^{14}-z^7 a^{13}+3 z^5 a^{13}-4 z^3 a^{13}+z a^{13}-z^8 a^{12}+4 z^6 a^{12}-6 z^4 a^{12}+4 z^2 a^{12}-z^9 a^{11}+6 z^7 a^{11}-15 z^5 a^{11}+19 z^3 a^{11}-8 z a^{11}+a^{11} z^{-1} -2 z^8 a^{10}+10 z^6 a^{10}-15 z^4 a^{10}+9 z^2 a^{10}-a^{10}-z^9 a^9+6 z^7 a^9-13 z^5 a^9+15 z^3 a^9-8 z a^9+a^9 z^{-1} -z^8 a^8+5 z^6 a^8-6 z^4 a^8+z^2 a^8-z^7 a^7+6 z^5 a^7-10 z^3 a^7+4 z a^7 }[/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                 1     1 -12               1 1     0 -14             2 1       1 -16           1 1         0 -18         2 2           0 -20       1 1             0 -22     1 2               -1 -24     1                 1 -26 1 1                   0 -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{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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