L11n288

L11n287

L11n289

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Link Presentations

[edit Notes on L11n288's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X11,18,12,19 X9,21,10,20 X19,5,20,10 X4,15,1,16 X17,22,18,11 X21,16,22,17
Gauss code	{1, 4, -3, -9}, {-2, -1, 5, 3, -7, 8}, {-6, 2, -4, -5, 9, 11, -10, 6, -8, 7, -11, 10}

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)^2 t(3)^4+t(1) t(2) t(3)^4+t(1) t(2)^2 t(3)^3-t(3)^3-t(1) t(2)^2 t(3)^2+t(3)^2+t(1) t(2)^2 t(3)-t(3)-t(2)+1}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-10} + q^{-9} -2 q^{-8} +3 q^{-7} -2 q^{-6} +4 q^{-5} -2 q^{-4} +3 q^{-3} - q^{-2} + q^{-1} }[/math] (db)
Signature	-6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^{10}-a^{10} z^{-2} -2 a^{10}+z^6 a^8+6 z^4 a^8+11 z^2 a^8+4 a^8 z^{-2} +9 a^8-z^8 a^6-7 z^6 a^6-17 z^4 a^6-20 z^2 a^6-5 a^6 z^{-2} -14 a^6+z^6 a^4+6 z^4 a^4+11 z^2 a^4+2 a^4 z^{-2} +7 a^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{13} z+a^{12} z^2+2 a^{11} z^3-4 a^{11} z+a^{11} z^{-1} +a^{10} z^6-2 a^{10} z^4-4 a^{10} z^2-a^{10} z^{-2} +5 a^{10}+3 a^9 z^7-16 a^9 z^5+27 a^9 z^3-21 a^9 z+5 a^9 z^{-1} +3 a^8 z^8-17 a^8 z^6+32 a^8 z^4-32 a^8 z^2-4 a^8 z^{-2} +18 a^8+a^7 z^9-2 a^7 z^7-12 a^7 z^5+34 a^7 z^3-29 a^7 z+9 a^7 z^{-1} +4 a^6 z^8-25 a^6 z^6+51 a^6 z^4-45 a^6 z^2-5 a^6 z^{-2} +21 a^6+a^5 z^9-5 a^5 z^7+4 a^5 z^5+9 a^5 z^3-13 a^5 z+5 a^5 z^{-1} +a^4 z^8-7 a^4 z^6+17 a^4 z^4-18 a^4 z^2-2 a^4 z^{-2} +9 a^4 }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ -1                   1 1 -3                     0 -5               3 1   2 -7           1 1 1     1 -9           4 2       2 -11       2 2 2         2 -13       3 2           1 -15   1 2 2             1 -17   1 2               -1 -19 1 1                 0 -21 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-7 }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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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