L11a101

L11a100

L11a102

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Link Presentations

[edit Notes on L11a101's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X20,11,21,12 X8,19,9,20 X18,9,19,10 X10,17,11,18 X4,21,1,22 X14,6,15,5 X12,4,13,3 X22,14,5,13 X2,16,3,15
Gauss code	{1, -11, 9, -7}, {8, -1, 2, -4, 5, -6, 3, -9, 10, -8, 11, -2, 6, -5, 4, -3, 7, -10}

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-1) (v-1) \left(v^2-v+1\right) \left(2 v^2-v+2\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{7/2}+4 q^{5/2}-9 q^{3/2}+13 \sqrt{q}-\frac{18}{\sqrt{q}}+\frac{19}{q^{3/2}}-\frac{19}{q^{5/2}}+\frac{16}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^7+a z^7-a^5 z^5+4 a^3 z^5+3 a z^5-z^5 a^{-1} -3 a^5 z^3+6 a^3 z^3+2 a z^3-2 z^3 a^{-1} -2 a^5 z+3 a^3 z-z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^6-3 a^8 z^4+2 a^8 z^2+3 a^7 z^7-9 a^7 z^5+7 a^7 z^3+4 a^6 z^8-9 a^6 z^6+5 a^6 z^4-a^6 z^2+4 a^5 z^9-8 a^5 z^7+9 a^5 z^5-10 a^5 z^3+4 a^5 z+2 a^4 z^{10}+2 a^4 z^8-8 a^4 z^6+7 a^4 z^4-4 a^4 z^2+10 a^3 z^9-25 a^3 z^7+34 a^3 z^5+z^5 a^{-3} -25 a^3 z^3-z^3 a^{-3} +6 a^3 z+2 a^2 z^{10}+7 a^2 z^8-18 a^2 z^6+4 z^6 a^{-2} +14 a^2 z^4-5 z^4 a^{-2} -3 a^2 z^2+6 a z^9-6 a z^7+8 z^7 a^{-1} +a z^5-14 z^5 a^{-1} +7 z^3 a^{-1} -2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +9 z^8-16 z^6+10 z^4-2 z^2-1 }[/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                 7 3     -4 0               11 6       5 -2             10 9         -1 -4           9 9           0 -6         7 10             3 -8       4 9               -5 -10     2 7                 5 -12   1 4                   -3 -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}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\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^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/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}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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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