L11n306

L11n305

L11n307

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Link Presentations

[edit Notes on L11n306's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X15,22,16,11 X13,20,14,21 X21,14,22,15 X17,8,18,9 X7,16,8,17 X9,18,10,19 X19,10,20,5 X2536 X11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -7, 6, -8, 9}, {-11, 2, -4, 5, -3, 7, -6, 8, -9, 4, -5, 3}

A Braid Representative	{{{braid_table}}}
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{u v^2 w^4-u v^2 w^3+u v^2 w^2-u v^2 w+u v w-u v+u-v^2 w^4+v w^4-v w^3+w^3-w^2+w-1}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-10} +2 q^{-9} -3 q^{-8} +4 q^{-7} -4 q^{-6} +5 q^{-5} -3 q^{-4} +4 q^{-3} - q^{-2} + q^{-1} }[/math] (db)
Signature	-6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{10} \left(-z^2\right)-a^{10} z^{-2} -2 a^{10}+a^8 z^6+6 a^8 z^4+12 a^8 z^2+4 a^8 z^{-2} +11 a^8-a^6 z^8-7 a^6 z^6-18 a^6 z^4-24 a^6 z^2-5 a^6 z^{-2} -18 a^6+a^4 z^6+6 a^4 z^4+12 a^4 z^2+2 a^4 z^{-2} +9 a^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z a^{13}+2 z^2 a^{12}-a^{12}+z^5 a^{11}-z^3 a^{11}-2 z a^{11}+a^{11} z^{-1} +3 z^6 a^{10}-10 z^4 a^{10}+6 z^2 a^{10}-a^{10} z^{-2} +5 z^7 a^9-23 z^5 a^9+31 z^3 a^9-19 z a^9+5 a^9 z^{-1} +4 z^8 a^8-19 z^6 a^8+27 z^4 a^8-21 z^2 a^8-4 a^8 z^{-2} +13 a^8+z^9 a^7+z^7 a^7-25 z^5 a^7+50 z^3 a^7-35 z a^7+9 a^7 z^{-1} +5 z^8 a^6-29 z^6 a^6+55 z^4 a^6-46 z^2 a^6-5 a^6 z^{-2} +22 a^6+z^9 a^5-4 z^7 a^5-z^5 a^5+18 z^3 a^5-19 z a^5+5 a^5 z^{-1} +z^8 a^4-7 z^6 a^4+18 z^4 a^4-21 z^2 a^4-2 a^4 z^{-2} +11 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               4 1   3 -7             1 2     1 -9           4 2       2 -11       1 2 2         1 -13       3 3           0 -15   1 2 2             1 -17   2 3               -1 -19   1                 1 -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{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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