L10a21

L10a20

L10a22

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Link Presentations

[edit Notes on L10a21's Link Presentations]

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

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)^3 \left(v^2-v+1\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -8 q^{9/2}+12 q^{7/2}-\frac{1}{q^{7/2}}-15 q^{5/2}+\frac{4}{q^{5/2}}+16 q^{3/2}-\frac{8}{q^{3/2}}-q^{13/2}+4 q^{11/2}-16 \sqrt{q}+\frac{11}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +a z^5-4 z^5 a^{-1} +2 z^5 a^{-3} +2 a z^3-6 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +a z-3 z a^{-1} +3 z a^{-3} -z a^{-5} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 z^9 a^{-1} -2 z^9 a^{-3} -12 z^8 a^{-2} -6 z^8 a^{-4} -6 z^8-7 a z^7-12 z^7 a^{-1} -12 z^7 a^{-3} -7 z^7 a^{-5} -4 a^2 z^6+19 z^6 a^{-2} +5 z^6 a^{-4} -4 z^6 a^{-6} +6 z^6-a^3 z^5+13 a z^5+32 z^5 a^{-1} +31 z^5 a^{-3} +12 z^5 a^{-5} -z^5 a^{-7} +6 a^2 z^4-7 z^4 a^{-2} +4 z^4 a^{-4} +6 z^4 a^{-6} +z^4+a^3 z^3-8 a z^3-25 z^3 a^{-1} -23 z^3 a^{-3} -6 z^3 a^{-5} +z^3 a^{-7} -a^2 z^2-z^2 a^{-2} -3 z^2 a^{-4} -2 z^2 a^{-6} -z^2+2 a z+6 z a^{-1} +6 z a^{-3} +2 z a^{-5} +1-a z^{-1} - a^{-1} 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                     1 1 12                   3   -3 10                 5 1   4 8               7 3     -4 6             8 5       3 4           8 7         -1 2         8 8           0 0       5 10             5 -2     3 6               -3 -4   1 5                 4 -6   3                   -3 -8 1                     1

Integral Khovanov Homology (db, data source)

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