L11a418

L11a417

L11a419

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Link Presentations

[edit Notes on L11a418's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X18,10,19,9 X16,8,17,7 X22,14,11,13 X20,16,21,15 X10,18,5,17 X8,20,9,19 X14,22,15,21 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -8, 3, -7}, {11, -2, 5, -9, 6, -4, 7, -3, 8, -6, 9, -5}

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{-2 t(1) t(2)^2+2 t(1) t(3) t(2)^2-3 t(3) t(2)^2+2 t(2)^2+2 t(1) t(3)^2 t(2)-3 t(3)^2 t(2)+3 t(1) t(2)-5 t(1) t(3) t(2)+5 t(3) t(2)-2 t(2)-2 t(1) t(3)^2+2 t(3)^2+3 t(1) t(3)-2 t(3)}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^7+3 q^6-5 q^5+8 q^4-10 q^3+12 q^2-11 q+11-7 q^{-1} +5 q^{-2} -2 q^{-3} + q^{-4} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4-2 z^2 a^2+a^2 z^{-2} +z^4-2 z^2-2 z^{-2} -3+2 z^4 a^{-2} +2 z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} +z^4 a^{-4} -z^2 a^{-6} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^7 a^{-7} -4 z^5 a^{-7} +4 z^3 a^{-7} +3 z^8 a^{-6} -13 z^6 a^{-6} +16 z^4 a^{-6} -5 z^2 a^{-6} +3 z^9 a^{-5} -11 z^7 a^{-5} +11 z^5 a^{-5} -4 z^3 a^{-5} +z^{10} a^{-4} +2 z^8 a^{-4} -15 z^6 a^{-4} +a^4 z^4+14 z^4 a^{-4} -2 a^4 z^2-3 z^2 a^{-4} +a^4+5 z^9 a^{-3} -14 z^7 a^{-3} +2 a^3 z^5+10 z^5 a^{-3} -2 a^3 z^3-4 z^3 a^{-3} +z^{10} a^{-2} +2 z^8 a^{-2} +3 a^2 z^6-6 z^6 a^{-2} -3 a^2 z^4-4 z^4 a^{-2} +3 a^2 z^2+9 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -3 a^2-5 a^{-2} +2 z^9 a^{-1} +3 a z^7+z^7 a^{-1} -7 z^5 a^{-1} -5 a z^3+z^3 a^{-1} +6 a z+6 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +3 z^8-z^6-6 z^4+12 z^2+2 z^{-2} -8 }[/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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 15                       1 -1 13                     2   2 11                   3 1   -2 9                 5 2     3 7               5 3       -2 5             7 5         2 3           6 7           1 1         5 5             0 -1       3 7               4 -3     2 4                 -2 -5     3                   3 -7 1 2                     -1 -9 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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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