L11a380

L11a379

L11a381

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Link Presentations

[edit Notes on L11a380's Link Presentations]

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

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^3-5 u^3 v^2+3 u^3 v-u^3+2 u^2 v^4-6 u^2 v^3+7 u^2 v^2-6 u^2 v+2 u^2-u v^4+3 u v^3-5 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{11}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{17}{q^{5/2}}-\frac{18}{q^{7/2}}+\frac{15}{q^{9/2}}-\frac{12}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^7-2 z a^7+2 z^5 a^5+5 z^3 a^5+2 z a^5-z^7 a^3-3 z^5 a^3-2 z^3 a^3+a^3 z^{-1} +2 z^5 a+5 z^3 a+z a-a z^{-1} -z^3 a^{-1} -2 z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^4-a^{10} z^2+3 a^9 z^5-2 a^9 z^3+6 a^8 z^6-6 a^8 z^4+3 a^8 z^2+9 a^7 z^7-15 a^7 z^5+14 a^7 z^3-5 a^7 z+9 a^6 z^8-15 a^6 z^6+9 a^6 z^4-a^6 z^2+6 a^5 z^9-5 a^5 z^7-11 a^5 z^5+14 a^5 z^3-5 a^5 z+2 a^4 z^{10}+8 a^4 z^8-31 a^4 z^6+25 a^4 z^4-7 a^4 z^2+10 a^3 z^9-27 a^3 z^7+19 a^3 z^5-8 a^3 z^3+4 a^3 z-a^3 z^{-1} +2 a^2 z^{10}+2 a^2 z^8-21 a^2 z^6+21 a^2 z^4-6 a^2 z^2+a^2+4 a z^9-12 a z^7+z^7 a^{-1} +8 a z^5-4 z^5 a^{-1} -a z^3+5 z^3 a^{-1} +2 a z-a z^{-1} -2 z a^{-1} +3 z^8-11 z^6+12 z^4-4 z^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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     2   -2 2                   5 1   4 0                 6 2     -4 -2               9 5       4 -4             9 7         -2 -6           9 8           1 -8         7 10             3 -10       5 8               -3 -12     2 7                 5 -14   1 5                   -4 -16   2                     2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/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}^{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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