L10a39

L10a38

L10a40

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Link Presentations

[edit Notes on L10a39's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,10,15,9 X16,12,17,11 X10,16,11,15 X20,17,5,18 X18,7,19,8 X8,19,9,20 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 7, -8, 3, -5, 4, -2, 10, -3, 5, -4, 6, -7, 8, -6}

A Braid Representative	{{{braid_table}}}
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)^2}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -3 q^{9/2}+\frac{1}{q^{9/2}}+7 q^{7/2}-\frac{2}{q^{7/2}}-10 q^{5/2}+\frac{5}{q^{5/2}}+11 q^{3/2}-\frac{9}{q^{3/2}}+q^{11/2}-13 \sqrt{q}+\frac{10}{\sqrt{q}} }[/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{ -a z^9-z^9 a^{-1} -2 a^2 z^8-4 z^8 a^{-2} -6 z^8-2 a^3 z^7-5 a z^7-10 z^7 a^{-1} -7 z^7 a^{-3} -a^4 z^6+a^2 z^6-2 z^6 a^{-2} -6 z^6 a^{-4} +6 z^6+6 a^3 z^5+19 a z^5+27 z^5 a^{-1} +11 z^5 a^{-3} -3 z^5 a^{-5} +4 a^4 z^4+11 a^2 z^4+17 z^4 a^{-2} +8 z^4 a^{-4} -z^4 a^{-6} +15 z^4-6 a^3 z^3-21 a z^3-27 z^3 a^{-1} -10 z^3 a^{-3} +2 z^3 a^{-5} -5 a^4 z^2-17 a^2 z^2-20 z^2 a^{-2} -6 z^2 a^{-4} +z^2 a^{-6} -25 z^2+4 a^3 z+16 a z+18 z a^{-1} +6 z a^{-3} +2 a^4+8 a^2+8 a^{-2} +2 a^{-4} +13-2 a^3 z^{-1} -7 a z^{-1} -7 a^{-1} z^{-1} -2 a^{-3} 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   \ -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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