L11n63

L11n62

L11n64

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Link Presentations

[edit Notes on L11n63's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X20,16,21,15 X18,12,19,11 X12,20,13,19 X22,18,5,17 X16,22,17,21 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -6, -9, 3, 4, -8, 7, -5, 6, -4, 8, -7}

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)^5-t(2)^5-2 t(1) t(2)^4+4 t(2)^4+4 t(1) t(2)^3-5 t(2)^3-5 t(1) t(2)^2+4 t(2)^2+4 t(1) t(2)-2 t(2)-t(1)+1}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{11/2}-3 q^{9/2}+7 q^{7/2}-10 q^{5/2}+11 q^{3/2}-12 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{2}{q^{7/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-3} +3 z^3 a^{-3} +a^3 z+4 z a^{-3} +a^3 z^{-1} + a^{-3} z^{-1} -z^7 a^{-1} +a z^5-5 z^5 a^{-1} +2 a z^3-10 z^3 a^{-1} +a z-8 z a^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} -z^2 a^{-6} +3 z^5 a^{-5} -2 z^3 a^{-5} +6 z^6 a^{-4} -8 z^4 a^{-4} +6 z^2 a^{-4} -2 a^{-4} +7 z^7 a^{-3} +3 a^3 z^5-11 z^5 a^{-3} -8 a^3 z^3+9 z^3 a^{-3} +6 a^3 z-4 z a^{-3} -a^3 z^{-1} + a^{-3} z^{-1} +a^2 z^8+4 z^8 a^{-2} +z^6 a^{-2} -2 a^2 z^4-14 z^4 a^{-2} -a^2 z^2+16 z^2 a^{-2} +a^2-5 a^{-2} +a z^9+z^9 a^{-1} +a z^7+8 z^7 a^{-1} -2 a z^5-19 z^5 a^{-1} -4 a z^3+15 z^3 a^{-1} +2 a z-8 z a^{-1} +2 a^{-1} z^{-1} +5 z^8-5 z^6-7 z^4+8 z^2-3 }[/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 χ 12                   1 -1 10                 2   2 8               5 1   -4 6             5 2     3 4           6 5       -1 2         6 5         1 0       5 7           2 -2     3 5             -2 -4   1 5               4 -6 1 3                 -2 -8 2                   2

Integral Khovanov Homology (db, data source)

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