L11n391

L11n390

L11n392

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Link Presentations

[edit Notes on L11n391's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X8493 X13,22,14,19 X9,20,10,21 X19,10,20,11 X21,14,22,15 X11,18,12,5 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {-8, 7, -9, 6}, {-4, -1, 2, -5, -7, 8, -10, 4, -6, 9, 11, -2, 3, 10}

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) (w-1)^2 (v w+1)}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 1-3 q^{-1} +6 q^{-2} -5 q^{-3} +7 q^{-4} -5 q^{-5} +5 q^{-6} -2 q^{-7} + q^{-9} - q^{-10} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10} z^{-2} -a^{10}+2 a^8 z^2+4 a^8 z^{-2} +6 a^8-4 a^6 z^2-5 a^6 z^{-2} -10 a^6-a^4 z^6-3 a^4 z^4+2 a^4 z^{-2} +4 a^4+a^2 z^4+2 a^2 z^2+a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^7-6 a^{11} z^5+9 a^{11} z^3-5 a^{11} z+a^{11} z^{-1} +a^{10} z^8-7 a^{10} z^6+12 a^{10} z^4-8 a^{10} z^2-a^{10} z^{-2} +3 a^{10}+a^9 z^7-12 a^9 z^5+29 a^9 z^3-20 a^9 z+5 a^9 z^{-1} +2 a^8 z^8-15 a^8 z^6+33 a^8 z^4-25 a^8 z^2-4 a^8 z^{-2} +13 a^8+a^7 z^9-2 a^7 z^7-11 a^7 z^5+35 a^7 z^3-30 a^7 z+9 a^7 z^{-1} +4 a^6 z^8-17 a^6 z^6+25 a^6 z^4-23 a^6 z^2-5 a^6 z^{-2} +16 a^6+a^5 z^9+a^5 z^7-14 a^5 z^5+19 a^5 z^3-15 a^5 z+5 a^5 z^{-1} +3 a^4 z^8-8 a^4 z^6+a^4 z^4-3 a^4 z^2-2 a^4 z^{-2} +6 a^4+3 a^3 z^7-9 a^3 z^5+4 a^3 z^3+a^2 z^6-3 a^2 z^4+3 a^2 z^2-a^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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                     2   -2 -3                   4 1   3 -5                 3 4     1 -7             1 5 2       2 -9           1 2 3         2 -11         1 6 5           0 -13       1 1 3             3 -15       1 3               -2 -17   1 1 1                 1 -19                         0 -21 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/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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