L10a172

L10a171

L10a173

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Link Presentations

[edit Notes on L10a172's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X20,15,17,16 X14,8,15,7 X10,12,5,11 X16,19,11,20 X8,18,9,17 X18,10,19,9 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {7, -8, 6, -3}, {9, -1, 4, -7, 8, -5}, {5, -2, 10, -4, 3, -6}

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(4)^2 t(3)^2+t(1) t(2) t(4)^2 t(3)^2-t(2) t(4)^2 t(3)^2+t(4)^2 t(3)^2-t(2) t(3)^2-t(1) t(2) t(4) t(3)^2+2 t(2) t(4) t(3)^2-t(4) t(3)^2+2 t(1) t(4)^2 t(3)-t(1) t(2) t(4)^2 t(3)-t(4)^2 t(3)-t(1) t(2) t(3)+2 t(2) t(3)-t(1) t(4) t(3)+2 t(1) t(2) t(4) t(3)-t(2) t(4) t(3)+2 t(4) t(3)-t(3)-t(1) t(4)^2-t(1)+t(1) t(2)-t(2)+2 t(1) t(4)-t(1) t(2) t(4)-t(4)+1}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 8 q^{9/2}-11 q^{7/2}+9 q^{5/2}-\frac{1}{q^{5/2}}-11 q^{3/2}+\frac{2}{q^{3/2}}-q^{15/2}+3 q^{13/2}-6 q^{11/2}+6 \sqrt{q}-\frac{6}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-5} -3 z^3 a^{-5} - a^{-5} z^{-3} -3 z a^{-5} -2 a^{-5} z^{-1} +z^7 a^{-3} +5 z^5 a^{-3} +10 z^3 a^{-3} +3 a^{-3} z^{-3} +11 z a^{-3} +7 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-8 z^3 a^{-1} +a z^{-3} -3 a^{-1} z^{-3} +3 a z-11 z a^{-1} +3 a z^{-1} -8 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{-9} +3 z^4 a^{-8} +6 z^5 a^{-7} -5 z^3 a^{-7} +2 z a^{-7} +8 z^6 a^{-6} -10 z^4 a^{-6} +2 z^2 a^{-6} +9 z^7 a^{-5} -20 z^5 a^{-5} +17 z^3 a^{-5} - a^{-5} z^{-3} -12 z a^{-5} +5 a^{-5} z^{-1} +5 z^8 a^{-4} -4 z^6 a^{-4} -17 z^4 a^{-4} +20 z^2 a^{-4} +3 a^{-4} z^{-2} -10 a^{-4} +z^9 a^{-3} +12 z^7 a^{-3} -51 z^5 a^{-3} +61 z^3 a^{-3} -3 a^{-3} z^{-3} -35 z a^{-3} +12 a^{-3} z^{-1} +7 z^8 a^{-2} -19 z^6 a^{-2} +26 z^2 a^{-2} +6 a^{-2} z^{-2} -19 a^{-2} +z^9 a^{-1} +a z^7+4 z^7 a^{-1} -5 a z^5-30 z^5 a^{-1} +10 a z^3+48 z^3 a^{-1} -a z^{-3} -3 a^{-1} z^{-3} -10 a z-31 z a^{-1} +5 a z^{-1} +12 a^{-1} z^{-1} +2 z^8-7 z^6+4 z^4+8 z^2+3 z^{-2} -10 }[/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 16                     1 1 14                   3 1 -2 12                 3     3 10               5 3     -2 8             6 3       3 6           5 7         2 4         6 4           2 2       4 9             5 0     2 2               0 -2   1 5                 4 -4   1                   -1 -6 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ r=-4 }[/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}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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