L10a158

L10a157

L10a159

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Link Presentations

[edit Notes on L10a158's Link Presentations]

Planar diagram presentation	X8192 X14,5,15,6 X20,9,13,10 X2,20,3,19 X10,4,11,3 X18,12,19,11 X16,8,17,7 X12,18,7,17 X6,13,1,14 X4,15,5,16
Gauss code	{1, -4, 5, -10, 2, -9}, {7, -1, 3, -5, 6, -8}, {9, -2, 10, -7, 8, -6, 4, -3}

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{(w-1) \left(u^2 v^2+u^2 v w-u^2 v-u^2 w+u v^2 w-u v^2+u v w^2-3 u v w+u v-u w^2+u w-v^2 w-v w^2+v w+w^2\right)}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+3 q^4-5 q^3+9 q^2-10 q+12-10 q^{-1} +9 q^{-2} -5 q^{-3} +3 q^{-4} - q^{-5} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4 \left(-z^2\right)-z^2 a^{-4} +a^2 z^4+z^4 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +a^2+ a^{-2} +2 z^4+z^2-2 z^{-2} -2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 a z^9+2 z^9 a^{-1} +4 a^2 z^8+4 z^8 a^{-2} +8 z^8+4 a^3 z^7-a z^7-z^7 a^{-1} +4 z^7 a^{-3} +3 a^4 z^6-9 a^2 z^6-9 z^6 a^{-2} +3 z^6 a^{-4} -24 z^6+a^5 z^5-8 a^3 z^5-2 a z^5-2 z^5 a^{-1} -8 z^5 a^{-3} +z^5 a^{-5} -7 a^4 z^4+9 a^2 z^4+9 z^4 a^{-2} -7 z^4 a^{-4} +32 z^4-2 a^5 z^3+3 a^3 z^3+3 a z^3+3 z^3 a^{-1} +3 z^3 a^{-3} -2 z^3 a^{-5} +3 a^4 z^2-3 a^2 z^2-3 z^2 a^{-2} +3 z^2 a^{-4} -12 z^2+2 a z+2 z a^{-1} -2 a^2-2 a^{-2} -3-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2} }[/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 11                     1 -1 9                   2   2 7                 3 1   -2 5               6 2     4 3             5 4       -1 1           7 5         2 -1         5 7           2 -3       4 5             -1 -5     2 6               4 -7   1 3                 -2 -9   2                   2 -11 1                     -1

Integral Khovanov Homology (db, data source)

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