L11n359

L11n358

L11n360

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Link Presentations

[edit Notes on L11n359's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,19,12,18 X17,9,18,8 X7,17,8,16 X15,5,16,14 X19,15,20,22 X13,20,14,21 X21,12,22,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-6, 5, -4, 3, -7, 8, -9, 7}, {10, -1, -5, 4, 11, -2, -3, 9, -8, 6}

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(3)-1) \left(t(2)^2 t(3)^3-t(2) t(3)^3-t(1) t(3)^2+t(1) t(2) t(3)^2-t(2) t(3)^2-t(2)^2 t(3)-t(1) t(2) t(3)+t(2) t(3)+t(1)-t(1) t(2)\right)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+ q^{-5} +2 q^4- q^{-4} -2 q^3+2 q^{-3} +3 q^2-2 q+2 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6-2 a^2 z^4+6 z^4+a^4 z^2-9 a^2 z^2-2 z^2 a^{-2} -z^2 a^{-4} +9 z^2+3 a^4-8 a^2- a^{-2} +6+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^3 z^9+a z^9+a^4 z^8+3 a^2 z^8+2 z^8-6 a^3 z^7-6 a z^7+z^7 a^{-1} +z^7 a^{-3} -7 a^4 z^6-22 a^2 z^6+2 z^6 a^{-2} +2 z^6 a^{-4} -15 z^6+8 a^3 z^5+5 a z^5-6 z^5 a^{-1} -2 z^5 a^{-3} +z^5 a^{-5} +16 a^4 z^4+48 a^2 z^4-7 z^4 a^{-2} -7 z^4 a^{-4} +32 z^4+2 a^3 z^3+10 a z^3+9 z^3 a^{-1} -2 z^3 a^{-3} -3 z^3 a^{-5} -16 a^4 z^2-40 a^2 z^2+5 z^2 a^{-2} +5 z^2 a^{-4} -24 z^2-7 a^3 z-10 a z-3 z a^{-1} +z a^{-3} +z a^{-5} +7 a^4+14 a^2- a^{-2} - a^{-4} +8+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - z^{-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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     1   1 7                   1 1   0 5               1 3 1     1 3             1 2 2       1 1           1 3 2         0 -1         1 2 3           2 -3       1 2 1             0 -5     1 1 2               2 -7   1 2                   1 -9                         0 -11 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{ i=3 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/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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