L10a5

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Link Presentations

[edit Notes on L10a5's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X14,10,15,9 X8493 X12,5,13,6 X20,13,5,14 X16,12,17,11 X10,16,11,15 X2,18,3,17
Gauss code	{1, -10, 5, -3}, {6, -1, 2, -5, 4, -9, 8, -6, 7, -4, 9, -8, 10, -2, 3, -7}

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^4-3 v^3+3 v^2-3 v+1\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -4 q^{9/2}+\frac{1}{q^{9/2}}+8 q^{7/2}-\frac{3}{q^{7/2}}-12 q^{5/2}+\frac{6}{q^{5/2}}+14 q^{3/2}-\frac{11}{q^{3/2}}+q^{11/2}-15 \sqrt{q}+\frac{13}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -a^3 z^3+6 a z^3-6 z^3 a^{-1} +2 z^3 a^{-3} -2 a^3 z+5 a z-4 z a^{-1} +z a^{-3} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a z^9-2 z^9 a^{-1} -4 a^2 z^8-7 z^8 a^{-2} -11 z^8-3 a^3 z^7-5 a z^7-12 z^7 a^{-1} -10 z^7 a^{-3} -a^4 z^6+9 a^2 z^6+5 z^6 a^{-2} -8 z^6 a^{-4} +23 z^6+9 a^3 z^5+25 a z^5+34 z^5 a^{-1} +14 z^5 a^{-3} -4 z^5 a^{-5} +3 a^4 z^4-4 a^2 z^4+5 z^4 a^{-2} +8 z^4 a^{-4} -z^4 a^{-6} -11 z^4-9 a^3 z^3-26 a z^3-26 z^3 a^{-1} -7 z^3 a^{-3} +2 z^3 a^{-5} -2 a^4 z^2-a^2 z^2-3 z^2 a^{-2} -2 z^2 a^{-4} +4 a^3 z+10 a z+8 z a^{-1} +2 z a^{-3} +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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                     1 -1 10                   3   3 8                 5 1   -4 6               7 3     4 4             7 5       -2 2           8 7         1 0         7 9           2 -2       4 6             -2 -4     2 7               5 -6   1 4                 -3 -8   2                   2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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