L11a379

L11a378

L11a380

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Link Presentations

[edit Notes on L11a379's Link Presentations]

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

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{u^4 v^2-u^4 v-u^3 v^4+4 u^3 v^3-6 u^3 v^2+4 u^3 v-u^3+2 u^2 v^4-7 u^2 v^3+9 u^2 v^2-7 u^2 v+2 u^2-u v^4+4 u v^3-6 u v^2+4 u v-u-v^3+v^2}{u^2 v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{5/2}+3 q^{3/2}-7 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{17}{q^{3/2}}+\frac{19}{q^{5/2}}-\frac{21}{q^{7/2}}+\frac{18}{q^{9/2}}-\frac{14}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{1}{q^{17/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 \left(-z^3\right)-a^7 z+2 a^5 z^5+4 a^5 z^3+a^5 z-a^3 z^7-3 a^3 z^5-3 a^3 z^3-a^3 z+a^3 z^{-1} +2 a z^5+5 a z^3-z^3 a^{-1} +2 a z-a z^{-1} -2 z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^4 a^{10}-4 z^5 a^9+z^3 a^9-9 z^6 a^8+8 z^4 a^8-3 z^2 a^8-13 z^7 a^7+18 z^5 a^7-10 z^3 a^7+2 z a^7-12 z^8 a^6+15 z^6 a^6-2 z^4 a^6-7 z^9 a^5+21 z^5 a^5-11 z^3 a^5-2 z^{10} a^4-13 z^8 a^4+39 z^6 a^4-25 z^4 a^4+6 z^2 a^4-11 z^9 a^3+25 z^7 a^3-11 z^5 a^3+3 z^3 a^3-5 z a^3+a^3 z^{-1} -2 z^{10} a^2-4 z^8 a^2+26 z^6 a^2-27 z^4 a^2+8 z^2 a^2-a^2-4 z^9 a+11 z^7 a-6 z^5 a-2 z^3 a-z a+a z^{-1} -3 z^8+11 z^6-13 z^4+5 z^2-z^7 a^{-1} +4 z^5 a^{-1} -5 z^3 a^{-1} +2 z a^{-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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     2   -2 2                   5 1   4 0                 7 2     -5 -2               10 5       5 -4             10 8         -2 -6           11 9           2 -8         8 11             3 -10       6 10               -4 -12     3 8                 5 -14   1 6                   -5 -16   3                     3 -18 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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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