L11a270

L11a269

L11a271

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Link Presentations

[edit Notes on L11a270's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X14,5,15,6 X18,8,19,7 X20,15,21,16 X22,17,9,18 X16,21,17,22 X6,20,7,19 X8,13,1,14
Gauss code	{1, -2, 3, -4, 5, -10, 6, -11}, {4, -1, 2, -3, 11, -5, 7, -9, 8, -6, 10, -7, 9, -8}

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^3 v^4-u^3 v^3+u^3 v^2-u^3 v+u^2 v^5-4 u^2 v^4+5 u^2 v^3-5 u^2 v^2+3 u^2 v-u^2-u v^5+3 u v^4-5 u v^3+5 u v^2-4 u v+u-v^4+v^3-v^2+v}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{12}{q^{9/2}}+\frac{14}{q^{11/2}}-\frac{15}{q^{13/2}}+\frac{13}{q^{15/2}}-\frac{11}{q^{17/2}}+\frac{7}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{1}{q^{23/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 \left(-z^5\right)-3 a^9 z^3-3 a^9 z-a^9 z^{-1} +a^7 z^7+4 a^7 z^5+6 a^7 z^3+6 a^7 z+3 a^7 z^{-1} +a^5 z^7+4 a^5 z^5+4 a^5 z^3-a^5 z-2 a^5 z^{-1} -a^3 z^5-4 a^3 z^3-4 a^3 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^4-a^{14} z^2+3 a^{13} z^5-2 a^{13} z^3+6 a^{12} z^6-7 a^{12} z^4+4 a^{12} z^2+8 a^{11} z^7-13 a^{11} z^5+11 a^{11} z^3-3 a^{11} z+7 a^{10} z^8-11 a^{10} z^6+8 a^{10} z^4-3 a^{10} z^2+a^{10}+4 a^9 z^9-3 a^9 z^7-a^9 z^5-3 a^9 z^3+2 a^9 z-a^9 z^{-1} +a^8 z^{10}+7 a^8 z^8-22 a^8 z^6+21 a^8 z^4-14 a^8 z^2+3 a^8+6 a^7 z^9-17 a^7 z^7+21 a^7 z^5-24 a^7 z^3+14 a^7 z-3 a^7 z^{-1} +a^6 z^{10}+2 a^6 z^8-13 a^6 z^6+14 a^6 z^4-8 a^6 z^2+3 a^6+2 a^5 z^9-5 a^5 z^7+a^5 z^5+5 a^5 z-2 a^5 z^{-1} +2 a^4 z^8-8 a^4 z^6+9 a^4 z^4-2 a^4 z^2+a^3 z^7-5 a^3 z^5+8 a^3 z^3-4 a^3 z }[/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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                       1 1 -2                     1   -1 -4                   4 1   3 -6                 5 2     -3 -8               7 3       4 -10             7 5         -2 -12           8 7           1 -14         6 8             2 -16       5 7               -2 -18     2 6                 4 -20   1 5                   -4 -22   2                     2 -24 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=-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}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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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