L11n311

L11n310

L11n312

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Link Presentations

[edit Notes on L11n311's Link Presentations]

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

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{u v^2 w^2-u v^2 w+u v w^3-2 u v w^2+u v w+u v+u w-u+v^2 w^4-v^2 w^3-v w^4-v w^3+2 v w^2-v w+w^3-w^2}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+ q^{-5} +2 q^4- q^{-4} -3 q^3+2 q^{-3} +3 q^2+ q^{-2} -3 q+3 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6-2 a^2 z^4+7 z^4+a^4 z^2-10 a^2 z^2-3 z^2 a^{-2} -z^2 a^{-4} +14 z^2+3 a^4-12 a^2-4 a^{-2} +13+2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} +4 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-5} -3 z^3 a^{-5} +z a^{-5} +a^4 z^8-7 a^4 z^6+2 z^6 a^{-4} +16 a^4 z^4-6 z^4 a^{-4} -17 a^4 z^2+2 z^2 a^{-4} -2 a^4 z^{-2} +9 a^4+a^3 z^9-6 a^3 z^7+2 z^7 a^{-3} +7 a^3 z^5-7 z^5 a^{-3} +6 a^3 z^3+6 z^3 a^{-3} -13 a^3 z-4 z a^{-3} +5 a^3 z^{-1} + a^{-3} z^{-1} +3 a^2 z^8+z^8 a^{-2} -23 a^2 z^6-4 z^6 a^{-2} +53 a^2 z^4+6 z^4 a^{-2} -49 a^2 z^2-8 z^2 a^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} +21 a^2+5 a^{-2} +a z^9-5 a z^7+3 z^7 a^{-1} -3 a z^5-18 z^5 a^{-1} +30 a z^3+33 z^3 a^{-1} -29 a z-21 z a^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} +3 z^8-22 z^6+49 z^4-42 z^2-4 z^{-2} +18 }[/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                   2 1   -1 5               1 2 1     0 3             1 3 2       0 1           1 3 2         0 -1         1 2 4           3 -3       1 3 1             1 -5     1 1 3               3 -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}^{3} }[/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}\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}^{4}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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