L11a31

L11a30

L11a32

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Link Presentations

[edit Notes on L11a31's Link Presentations]

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

A Braid Representative
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) t(2)^5-3 t(2)^5-4 t(1) t(2)^4+5 t(2)^4+5 t(1) t(2)^3-5 t(2)^3-5 t(1) t(2)^2+5 t(2)^2+5 t(1) t(2)-4 t(2)-3 t(1)+1}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{17/2}-3 q^{15/2}+7 q^{13/2}-10 q^{11/2}+13 q^{9/2}-15 q^{7/2}+14 q^{5/2}-12 q^{3/2}+8 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-7} +2 z a^{-7} -2 z^5 a^{-5} -5 z^3 a^{-5} +2 a^{-5} z^{-1} +z^7 a^{-3} +3 z^5 a^{-3} -5 z a^{-3} -4 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-6 z^3 a^{-1} +3 a z-2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^{10} a^{-2} -z^{10} a^{-4} -3 z^9 a^{-1} -7 z^9 a^{-3} -4 z^9 a^{-5} -5 z^8 a^{-2} -9 z^8 a^{-4} -6 z^8 a^{-6} -2 z^8-a z^7+11 z^7 a^{-1} +22 z^7 a^{-3} +3 z^7 a^{-5} -7 z^7 a^{-7} +30 z^6 a^{-2} +35 z^6 a^{-4} +6 z^6 a^{-6} -6 z^6 a^{-8} +7 z^6+5 a z^5-13 z^5 a^{-1} -27 z^5 a^{-3} +3 z^5 a^{-5} +9 z^5 a^{-7} -3 z^5 a^{-9} -42 z^4 a^{-2} -44 z^4 a^{-4} +2 z^4 a^{-6} +8 z^4 a^{-8} -z^4 a^{-10} -5 z^4-8 a z^3+10 z^3 a^{-1} +27 z^3 a^{-3} +2 z^3 a^{-5} -5 z^3 a^{-7} +2 z^3 a^{-9} +23 z^2 a^{-2} +27 z^2 a^{-4} -4 z^2 a^{-6} -6 z^2 a^{-8} +z^2 a^{-10} -z^2+5 a z-6 z a^{-1} -17 z a^{-3} -5 z a^{-5} +z a^{-7} -5 a^{-2} -6 a^{-4} + a^{-6} +2 a^{-8} +1-a z^{-1} + a^{-1} z^{-1} +4 a^{-3} z^{-1} +2 a^{-5} 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     2   2 14                   5 1   -4 12                 5 2     3 10               8 5       -3 8             7 5         2 6           7 8           1 4         5 7             -2 2       4 8               4 0     2 4                 -2 -2     4                   4 -4 1 2                     -1 -6 1                       1

Integral Khovanov Homology (db, data source)

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