L11a185

L11a184

L11a186

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Link Presentations

[edit Notes on L11a185's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X14,17,15,18 X16,5,17,6 X4,15,5,16 X18,11,19,12 X22,19,7,20 X20,14,21,13 X12,22,13,21 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 6, -9, 8, -3, 5, -4, 3, -6, 7, -8, 9, -7}

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 u^2 v^3-6 u^2 v^2+4 u^2 v-u^2+2 u v^4-8 u v^3+11 u v^2-8 u v+2 u-v^4+4 v^3-6 v^2+2 v}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{18}{q^{9/2}}-\frac{16}{q^{7/2}}+\frac{11}{q^{5/2}}-\frac{7}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{4}{q^{19/2}}+\frac{7}{q^{17/2}}-\frac{12}{q^{15/2}}+\frac{16}{q^{13/2}}-\frac{18}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 \left(-z^3\right)+a^9 z^{-1} +a^7 z^5-2 a^7 z-2 a^7 z^{-1} +2 a^5 z^5+3 a^5 z^3+2 a^5 z+2 a^5 z^{-1} +a^3 z^5-2 a^3 z-a^3 z^{-1} -a z^3-a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{12}+2 z^4 a^{12}-4 z^7 a^{11}+11 z^5 a^{11}-7 z^3 a^{11}-6 z^8 a^{10}+16 z^6 a^{10}-12 z^4 a^{10}+3 z^2 a^{10}-4 z^9 a^9+2 z^7 a^9+15 z^5 a^9-18 z^3 a^9+7 z a^9-a^9 z^{-1} -z^{10} a^8-11 z^8 a^8+32 z^6 a^8-28 z^4 a^8+9 z^2 a^8-7 z^9 a^7+5 z^7 a^7+16 z^5 a^7-25 z^3 a^7+12 z a^7-2 a^7 z^{-1} -z^{10} a^6-10 z^8 a^6+22 z^6 a^6-18 z^4 a^6+7 z^2 a^6-a^6-3 z^9 a^5-6 z^7 a^5+21 z^5 a^5-22 z^3 a^5+10 z a^5-2 a^5 z^{-1} -5 z^8 a^4+4 z^6 a^4+z^4 a^4-z^2 a^4-5 z^7 a^3+8 z^5 a^3-6 z^3 a^3+4 z a^3-a^3 z^{-1} -3 z^6 a^2+5 z^4 a^2-2 z^2 a^2-z^5 a+2 z^3 a-z a }[/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 2                       1 1 0                     2   -2 -2                   5 1   4 -4                 7 3     -4 -6               9 4       5 -8             9 7         -2 -10           9 9           0 -12         7 9             2 -14       5 9               -4 -16     3 8                 5 -18   1 4                   -3 -20   3                     3 -22 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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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