L11n98

L11n97

L11n99

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Link Presentations

[edit Notes on L11n98's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,21,14,20 X7,17,8,16 X9,19,10,18 X17,9,18,8 X19,11,20,10 X15,5,16,22 X21,15,22,14 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 6, -5, 7, 11, -2, -3, 9, -8, 4, -6, 5, -7, 3, -9, 8}

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(2)^7-t(1) t(2)^5+t(2)^5+2 t(1) t(2)^4-2 t(2)^4-2 t(1) t(2)^3+2 t(2)^3+t(1) t(2)^2-t(2)^2-t(1)}{\sqrt{t(1)} t(2)^{7/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{17/2}-2 q^{15/2}+4 q^{13/2}-4 q^{11/2}+4 q^{9/2}-4 q^{7/2}+2 q^{5/2}-q^{3/2}-\sqrt{q}-\frac{1}{q^{3/2}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-3} +z^5 a^{-1} -8 z^5 a^{-3} +6 z^3 a^{-1} -20 z^3 a^{-3} +3 z^3 a^{-5} +z^3 a^{-7} +10 z a^{-1} -20 z a^{-3} +7 z a^{-5} +z a^{-7} +4 a^{-1} z^{-1} -7 a^{-3} z^{-1} +3 a^{-5} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-1} -z^9 a^{-3} -z^8 a^{-2} -z^8 a^{-4} +9 z^7 a^{-1} +10 z^7 a^{-3} -z^7 a^{-7} +9 z^6 a^{-2} +9 z^6 a^{-4} -2 z^6 a^{-6} -2 z^6 a^{-8} -28 z^5 a^{-1} -33 z^5 a^{-3} -3 z^5 a^{-5} -2 z^5 a^{-9} -24 z^4 a^{-2} -26 z^4 a^{-4} +z^4 a^{-6} +2 z^4 a^{-8} -z^4 a^{-10} +37 z^3 a^{-1} +46 z^3 a^{-3} +6 z^3 a^{-5} +3 z^3 a^{-9} +23 z^2 a^{-2} +24 z^2 a^{-4} +z^2 a^{-8} +2 z^2 a^{-10} -20 z a^{-1} -29 z a^{-3} -8 z a^{-5} +z a^{-7} -7 a^{-2} -7 a^{-4} - a^{-10} +4 a^{-1} z^{-1} +7 a^{-3} z^{-1} +3 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                     1   1 14                   3 1   -2 12                 1 1     0 10               3 3       0 8           1 2 1         0 6           1 3           2 4       1 2 2             -1 2         2               2 0     1                   1 -2 1                       1 -4 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{ i=6 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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