L10a109

L10a108

L10a110

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Link Presentations

[edit Notes on L10a109's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,8,17,7 X10,5,1,6 X6374 X4,9,5,10 X18,14,19,13 X20,16,11,15 X14,20,15,19 X2,11,3,12 X8,18,9,17
Gauss code	{1, -9, 4, -5, 3, -4, 2, -10, 5, -3}, {9, -1, 6, -8, 7, -2, 10, -6, 8, -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{(u-1) (v-1) \left(u^2 v^2+u^2-2 u v^2+u v-2 u+v^2+1\right)}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{11/2}-3 q^{9/2}+7 q^{7/2}-10 q^{5/2}+11 q^{3/2}-13 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{2}{q^{7/2}}+\frac{1}{q^{9/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-3} -a^3 z^3+3 z^3 a^{-3} -3 a^3 z+4 z a^{-3} -2 a^3 z^{-1} +2 a^{-3} z^{-1} -z^7 a^{-1} +2 a z^5-5 z^5 a^{-1} +8 a z^3-11 z^3 a^{-1} +12 a z-13 z a^{-1} +7 a z^{-1} -7 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} -z^2 a^{-6} +3 z^5 a^{-5} -2 z^3 a^{-5} +a^4 z^6+6 z^6 a^{-4} -4 a^4 z^4-8 z^4 a^{-4} +5 a^4 z^2+6 z^2 a^{-4} -2 a^4-2 a^{-4} +2 a^3 z^7+7 z^7 a^{-3} -6 a^3 z^5-11 z^5 a^{-3} +6 a^3 z^3+10 z^3 a^{-3} -4 a^3 z-6 z a^{-3} +2 a^3 z^{-1} +2 a^{-3} z^{-1} +2 a^2 z^8+4 z^8 a^{-2} -a^2 z^6+2 z^6 a^{-2} -11 a^2 z^4-17 z^4 a^{-2} +17 a^2 z^2+20 z^2 a^{-2} -8 a^2-8 a^{-2} +a z^9+z^9 a^{-1} +5 a z^7+10 z^7 a^{-1} -19 a z^5-27 z^5 a^{-1} +21 a z^3+27 z^3 a^{-1} -16 a z-18 z a^{-1} +7 a z^{-1} +7 a^{-1} z^{-1} +6 z^8-6 z^6-15 z^4+25 z^2-13 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                     1 -1 10                   2   2 8                 5 1   -4 6               5 2     3 4             6 5       -1 2           7 5         2 0         5 8           3 -2       4 5             -1 -4     1 5               4 -6   1 4                 -3 -8   1                   1 -10 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/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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