L10a115

L10a114

L10a116

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Link Presentations

[edit Notes on L10a115's Link Presentations]

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