L11a52

L11a51

L11a53

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Link Presentations

[edit Notes on L11a52's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X14,6,15,5 X10,4,11,3 X20,12,21,11 X22,14,5,13 X12,22,13,21 X16,9,17,10 X2,16,3,15 X8,17,9,18
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -11, 9, -5, 6, -8, 7, -4, 10, -9, 11, -2, 3, -6, 8, -7}

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{2 (t(1)-1) (t(2)-1) \left(t(2)^4-t(2)^3+t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{15/2}+3 q^{13/2}-6 q^{11/2}+9 q^{9/2}-11 q^{7/2}+12 q^{5/2}-12 q^{3/2}+10 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{1}{q^{7/2}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-5} -3 z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +5 z^3 a^{-3} +5 z a^{-3} +3 a^{-3} z^{-1} +z^7 a^{-1} -a z^5+4 z^5 a^{-1} -3 a z^3+3 z^3 a^{-1} -3 z a^{-1} +2 a z^{-1} -4 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{-9} +3 z^4 a^{-8} +6 z^5 a^{-7} -4 z^3 a^{-7} +z a^{-7} +9 z^6 a^{-6} -13 z^4 a^{-6} +5 z^2 a^{-6} - a^{-6} +10 z^7 a^{-5} -20 z^5 a^{-5} +9 z^3 a^{-5} -2 z a^{-5} + a^{-5} z^{-1} +8 z^8 a^{-4} -16 z^6 a^{-4} -z^4 a^{-4} +8 z^2 a^{-4} -3 a^{-4} +5 z^9 a^{-3} -9 z^7 a^{-3} -9 z^5 a^{-3} +16 z^3 a^{-3} -9 z a^{-3} +3 a^{-3} z^{-1} +2 z^{10} a^{-2} +a^2 z^8-5 a^2 z^6-18 z^6 a^{-2} +7 a^2 z^4+17 z^4 a^{-2} -2 a^2 z^2+2 z^2 a^{-2} -3 a^{-2} +3 a z^9+8 z^9 a^{-1} -17 a z^7-36 z^7 a^{-1} +31 a z^5+48 z^5 a^{-1} -18 a z^3-16 z^3 a^{-1} -a z-7 z a^{-1} +2 a z^{-1} +4 a^{-1} z^{-1} +2 z^{10}-7 z^8+2 z^6+9 z^4-3 z^2-2 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 16                       1 1 14                     2   -2 12                   4 1   3 10                 5 2     -3 8               6 4       2 6             6 5         -1 4           6 6           0 2         6 8             2 0       2 4               -2 -2     2 6                 4 -4   1 2                   -1 -6   2                     2 -8 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=-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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/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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