L11a373

L11a372

L11a374

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Link Presentations

[edit Notes on L11a373's Link Presentations]

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

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^4 v^3-u^4 v^2+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^3-11 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^2+v}{u^2 v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{7/2}+4 q^{5/2}-9 q^{3/2}+14 \sqrt{q}-\frac{19}{\sqrt{q}}+\frac{21}{q^{3/2}}-\frac{21}{q^{5/2}}+\frac{17}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^5 z^5-3 a^5 z^3-3 a^5 z+a^3 z^7+4 a^3 z^5+7 a^3 z^3+5 a^3 z+a^3 z^{-1} +a z^7+3 a z^5-z^5 a^{-1} +2 a z^3-2 z^3 a^{-1} -2 a z-z a^{-1} -a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^4 z^{10}-2 a^2 z^{10}-5 a^5 z^9-11 a^3 z^9-6 a z^9-5 a^6 z^8-7 a^4 z^8-11 a^2 z^8-9 z^8-3 a^7 z^7+9 a^5 z^7+21 a^3 z^7+a z^7-8 z^7 a^{-1} -a^8 z^6+12 a^6 z^6+24 a^4 z^6+28 a^2 z^6-4 z^6 a^{-2} +13 z^6+8 a^7 z^5-5 a^5 z^5-16 a^3 z^5+11 a z^5+13 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4-9 a^6 z^4-23 a^4 z^4-22 a^2 z^4+5 z^4 a^{-2} -6 z^4-6 a^7 z^3+2 a^5 z^3+9 a^3 z^3-7 a z^3-7 z^3 a^{-1} +z^3 a^{-3} -2 a^8 z^2+3 a^6 z^2+8 a^4 z^2+6 a^2 z^2-z^2 a^{-2} +2 z^2+2 a^7 z-5 a^3 z-a z+2 z a^{-1} -a^2+a^3 z^{-1} +a 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 8 3     -5 0               11 6       5 -2             11 9         -2 -4           10 10           0 -6         8 12             4 -8       5 9               -4 -10     2 8                 6 -12   1 5                   -4 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/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}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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