L10a162

L10a161

L10a163

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Link Presentations

[edit Notes on L10a162's Link Presentations]

Planar diagram presentation	X8192 X16,6,17,5 X10,20,11,19 X18,10,19,9 X20,12,13,11 X2,13,3,14 X14,3,15,4 X4758 X12,16,7,15 X6,18,1,17
Gauss code	{1, -6, 7, -8, 2, -10}, {8, -1, 4, -3, 5, -9}, {6, -7, 9, -2, 10, -4, 3, -5}

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(2)-1) (t(3) t(1)-t(1)+1) (t(1) t(3)-t(3)+1) (t(2) t(3)+1)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^7-3 q^6+6 q^5-9 q^4+12 q^3-11 q^2+12 q-8+6 q^{-1} -3 q^{-2} + q^{-3} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-4} +4 z^4 a^{-4} +5 z^2 a^{-4} + a^{-4} z^{-2} +2 a^{-4} -z^8 a^{-2} -6 z^6 a^{-2} -13 z^4 a^{-2} -12 z^2 a^{-2} -2 a^{-2} z^{-2} -5 a^{-2} +z^6+4 z^4+5 z^2+ z^{-2} +3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-8} -z^2 a^{-8} +3 z^5 a^{-7} -3 z^3 a^{-7} +5 z^6 a^{-6} -6 z^4 a^{-6} +3 z^2 a^{-6} - a^{-6} +6 z^7 a^{-5} -9 z^5 a^{-5} +6 z^3 a^{-5} +5 z^8 a^{-4} -8 z^6 a^{-4} +7 z^4 a^{-4} -3 z^2 a^{-4} - a^{-4} z^{-2} +2 a^{-4} +2 z^9 a^{-3} +3 z^7 a^{-3} -13 z^5 a^{-3} +12 z^3 a^{-3} -5 z a^{-3} +2 a^{-3} z^{-1} +9 z^8 a^{-2} +a^2 z^6-26 z^6 a^{-2} -3 a^2 z^4+29 z^4 a^{-2} +2 a^2 z^2-18 z^2 a^{-2} -2 a^{-2} z^{-2} +6 a^{-2} +2 z^9 a^{-1} +3 a z^7-9 a z^5-10 z^5 a^{-1} +6 a z^3+9 z^3 a^{-1} -5 z a^{-1} +2 a^{-1} z^{-1} +4 z^8-12 z^6+12 z^4-9 z^2- z^{-2} +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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                     1 1 13                   2   -2 11                 4 1   3 9               6 3     -3 7             6 3       3 5           6 7         1 3         6 5           1 1       3 7             4 -1     3 5               -2 -3   1 4                 3 -5   2                   -2 -7 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=3 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/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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