L11a147

L11a146

L11a148

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Link Presentations

[edit Notes on L11a147's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X18,12,19,11 X12,6,13,5 X4,19,5,20 X16,7,17,8 X20,13,21,14 X22,15,7,16 X14,21,15,22 X6,18,1,17
Gauss code	{1, -2, 3, -6, 5, -11}, {7, -1, 2, -3, 4, -5, 8, -10, 9, -7, 11, -4, 6, -8, 10, -9}

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^2 v^6-2 u^2 v^5+2 u^2 v^4-2 u^2 v^3+2 u^2 v^2-u^2 v-u v^6+3 u v^5-3 u v^4+3 u v^3-3 u v^2+3 u v-u-v^5+2 v^4-2 v^3+2 v^2-2 v+1}{u v^3} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{11}{q^{9/2}}+\frac{9}{q^{7/2}}-\frac{8}{q^{5/2}}+\frac{4}{q^{3/2}}-\frac{1}{q^{21/2}}+\frac{3}{q^{19/2}}-\frac{5}{q^{17/2}}+\frac{8}{q^{15/2}}-\frac{10}{q^{13/2}}+\frac{11}{q^{11/2}}+\sqrt{q}-\frac{3}{\sqrt{q}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^5 z^9+a^7 z^7-7 a^5 z^7+a^3 z^7+5 a^7 z^5-17 a^5 z^5+5 a^3 z^5+7 a^7 z^3-16 a^5 z^3+6 a^3 z^3+2 a^7 z-2 a^5 z-a^3 z-a^7 z^{-1} +3 a^5 z^{-1} -2 a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^3 a^{13}-3 z^4 a^{12}+z^2 a^{12}-5 z^5 a^{11}+3 z^3 a^{11}-7 z^6 a^{10}+9 z^4 a^{10}-2 z^2 a^{10}-8 z^7 a^9+16 z^5 a^9-7 z^3 a^9+z a^9-7 z^8 a^8+16 z^6 a^8-4 z^4 a^8-3 z^2 a^8+a^8-5 z^9 a^7+13 z^7 a^7-4 z^5 a^7-z^3 a^7-a^7 z^{-1} -2 z^{10} a^6+z^8 a^6+17 z^6 a^6-20 z^4 a^6+z^2 a^6+3 a^6-8 z^9 a^5+38 z^7 a^5-56 z^5 a^5+29 z^3 a^5-z a^5-3 a^5 z^{-1} -2 z^{10} a^4+7 z^8 a^4-z^6 a^4-11 z^4 a^4+3 z^2 a^4+3 a^4-3 z^9 a^3+17 z^7 a^3-31 z^5 a^3+19 z^3 a^3-2 a^3 z^{-1} -z^8 a^2+5 z^6 a^2-7 z^4 a^2+2 z^2 a^2 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 2                       1 -1 0                     2   2 -2                   2 1   -1 -4                 6 2     4 -6               4 3       -1 -8             7 5         2 -10           5 5           0 -12         5 6             -1 -14       3 5               2 -16     2 5                 -3 -18   1 3                   2 -20   2                     -2 -22 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=-8 }[/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}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/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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