L11a205

L11a204

L11a206

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Link Presentations

[edit Notes on L11a205's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X18,15,19,16 X16,6,17,5 X4,18,5,17 X20,11,21,12 X22,13,7,14 X12,21,13,22 X14,19,15,20
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 8, -10, 9, -11, 5, -6, 7, -5, 11, -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{2 t(1)^2 t(2)^4-2 t(1) t(2)^4-3 t(1)^2 t(2)^3+4 t(1) t(2)^3-2 t(2)^3+3 t(1)^2 t(2)^2-5 t(1) t(2)^2+3 t(2)^2-2 t(1)^2 t(2)+4 t(1) t(2)-3 t(2)-2 t(1)+2}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{9}{q^{9/2}}+\frac{6}{q^{7/2}}-\frac{4}{q^{5/2}}+\frac{1}{q^{3/2}}+\frac{1}{q^{23/2}}-\frac{3}{q^{21/2}}+\frac{6}{q^{19/2}}-\frac{9}{q^{17/2}}+\frac{11}{q^{15/2}}-\frac{12}{q^{13/2}}+\frac{11}{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+z^7 a^7+4 z^5 a^7+4 z^3 a^7-a^7 z^{-1} +z^7 a^5+5 z^5 a^5+9 z^3 a^5+8 z a^5+3 a^5 z^{-1} -z^5 a^3-5 z^3 a^3-7 z a^3-2 a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^4 a^{14}+z^2 a^{14}-3 z^5 a^{13}+3 z^3 a^{13}-5 z^6 a^{12}+6 z^4 a^{12}-2 z^2 a^{12}-6 z^7 a^{11}+10 z^5 a^{11}-8 z^3 a^{11}+z a^{11}-5 z^8 a^{10}+9 z^6 a^{10}-9 z^4 a^{10}+3 z^2 a^{10}-3 z^9 a^9+5 z^7 a^9-7 z^5 a^9+8 z^3 a^9-3 z a^9-z^{10} a^8-z^8 a^8+3 z^6 a^8+2 z^4 a^8-2 z^2 a^8+a^8-4 z^9 a^7+13 z^7 a^7-18 z^5 a^7+15 z^3 a^7-2 z a^7-a^7 z^{-1} -z^{10} a^6+3 z^8 a^6-8 z^6 a^6+19 z^4 a^6-15 z^2 a^6+3 a^6-z^9 a^5+z^7 a^5+8 z^5 a^5-16 z^3 a^5+11 z a^5-3 a^5 z^{-1} -z^8 a^4+3 z^6 a^4+z^4 a^4-7 z^2 a^4+3 a^4-z^7 a^3+6 z^5 a^3-12 z^3 a^3+9 z a^3-2 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                         0 -4                   4 1   3 -6                 3 1     -2 -8               6 3       3 -10             6 4         -2 -12           6 5           1 -14         5 6             1 -16       4 6               -2 -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}^{6}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/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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