L11a162

L11a161

L11a163

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Link Presentations

[edit Notes on L11a162's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X16,11,17,12 X14,6,15,5 X4,16,5,15 X20,13,21,14 X22,17,7,18 X18,21,19,22 X12,19,13,20
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -11, 8, -6, 7, -5, 9, -10, 11, -8, 10, -9}

A Braid Representative	{{{braid_table}}}
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{2 t(1)^2 t(2)^4-2 t(1) t(2)^4-4 t(1)^2 t(2)^3+4 t(1) t(2)^3-2 t(2)^3+4 t(1)^2 t(2)^2-5 t(1) t(2)^2+4 t(2)^2-2 t(1)^2 t(2)+4 t(1) t(2)-4 t(2)-2 t(1)+2}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{11}{q^{9/2}}+\frac{7}{q^{7/2}}-\frac{5}{q^{5/2}}+\frac{2}{q^{3/2}}+\frac{1}{q^{23/2}}-\frac{3}{q^{21/2}}+\frac{6}{q^{19/2}}-\frac{9}{q^{17/2}}+\frac{12}{q^{15/2}}-\frac{13}{q^{13/2}}+\frac{12}{q^{11/2}}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^9-3 z^3 a^9-2 z a^9-a^9 z^{-1} +z^7 a^7+4 z^5 a^7+5 z^3 a^7+4 z a^7+2 a^7 z^{-1} +z^7 a^5+4 z^5 a^5+4 z^3 a^5+z a^5-z^5 a^3-4 z^3 a^3-4 z a^3-a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^4-a^{14} z^2+3 a^{13} z^5-3 a^{13} z^3+5 a^{12} z^6-6 a^{12} z^4+2 a^{12} z^2+6 a^{11} z^7-9 a^{11} z^5+6 a^{11} z^3-a^{11} z+6 a^{10} z^8-13 a^{10} z^6+15 a^{10} z^4-8 a^{10} z^2+2 a^{10}+4 a^9 z^9-8 a^9 z^7+8 a^9 z^5-6 a^9 z^3+3 a^9 z-a^9 z^{-1} +a^8 z^{10}+6 a^8 z^8-27 a^8 z^6+36 a^8 z^4-22 a^8 z^2+5 a^8+6 a^7 z^9-20 a^7 z^7+23 a^7 z^5-16 a^7 z^3+7 a^7 z-2 a^7 z^{-1} +a^6 z^{10}+2 a^6 z^8-17 a^6 z^6+22 a^6 z^4-12 a^6 z^2+3 a^6+2 a^5 z^9-5 a^5 z^7-2 a^5 z^5+7 a^5 z^3-2 a^5 z+2 a^4 z^8-8 a^4 z^6+8 a^4 z^4-a^4 z^2-a^4+a^3 z^7-5 a^3 z^5+8 a^3 z^3-5 a^3 z+a^3 z^{-1} }[/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                 4 2     -2 -8               7 3       4 -10             6 5         -1 -12           7 6           1 -14         5 6             1 -16       4 7               -3 -18     2 5                 3 -20   1 4                   -3 -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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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