L11n412

L11n411

L11n413

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Link Presentations

[edit Notes on L11n412's Link Presentations]

Planar diagram presentation	X8192 X5,15,6,14 X10,3,11,4 X13,5,14,4 X2738 X6,9,1,10 X11,18,12,19 X17,12,18,7 X15,20,16,21 X19,22,20,13 X21,16,22,17
Gauss code	{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, -7, 8}, {-4, 2, -9, 11, -8, 7, -10, 9, -11, 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{t(1)^2 t(3)^3-2 t(1)^2 t(3)^2+t(1) t(2)^2 t(3)^2-2 t(2)^2 t(3)^2+t(1) t(3)^2+t(1)^2 t(2) t(3)^2-2 t(1) t(2) t(3)^2+t(2) t(3)^2+2 t(1)^2 t(3)-t(1) t(2)^2 t(3)+2 t(2)^2 t(3)-t(1) t(3)-t(1)^2 t(2) t(3)+2 t(1) t(2) t(3)-t(2) t(3)-t(2)^2}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-10} +2 q^{-9} -4 q^{-8} +6 q^{-7} -6 q^{-6} +8 q^{-5} -6 q^{-4} +6 q^{-3} -3 q^{-2} +2 q^{-1} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10} z^{-2} -a^{10}+3 z^2 a^8+4 a^8 z^{-2} +6 a^8-2 z^4 a^6-6 z^2 a^6-5 a^6 z^{-2} -9 a^6-z^4 a^4+2 a^4 z^{-2} +2 a^4+2 z^2 a^2+2 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^7-5 a^{11} z^5+8 a^{11} z^3-5 a^{11} z+a^{11} z^{-1} +2 a^{10} z^8-9 a^{10} z^6+12 a^{10} z^4-7 a^{10} z^2-a^{10} z^{-2} +4 a^{10}+a^9 z^9+a^9 z^7-20 a^9 z^5+35 a^9 z^3-21 a^9 z+5 a^9 z^{-1} +6 a^8 z^8-26 a^8 z^6+38 a^8 z^4-32 a^8 z^2-4 a^8 z^{-2} +17 a^8+a^7 z^9+4 a^7 z^7-28 a^7 z^5+44 a^7 z^3-33 a^7 z+9 a^7 z^{-1} +4 a^6 z^8-15 a^6 z^6+26 a^6 z^4-32 a^6 z^2-5 a^6 z^{-2} +20 a^6+4 a^5 z^7-12 a^5 z^5+18 a^5 z^3-16 a^5 z+5 a^5 z^{-1} +2 a^4 z^6-4 a^4 z^2-2 a^4 z^{-2} +6 a^4+a^3 z^5+a^3 z^3+a^3 z+3 a^2 z^2-2 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 χ -1                   2 2 -3                 3 2 -1 -5               3     3 -7             4 4     0 -9           4 2       2 -11         2 4         2 -13       4 4           0 -15     1 3             2 -17   1 3               -2 -19   1                 1 -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=-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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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