L11a537

L11a536

L11a538

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Link Presentations

[edit Notes on L11a537's Link Presentations]

Planar diagram presentation	X6172 X2536 X18,12,19,11 X10,3,11,4 X4,9,1,10 X8,18,5,17 X16,8,17,7 X14,16,9,15 X22,20,15,19 X20,13,21,14 X12,21,13,22
Gauss code	{1, -2, 4, -5}, {2, -1, 7, -6}, {5, -4, 3, -11, 10, -8}, {8, -7, 6, -3, 9, -10, 11, -9}

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)^2-2 t(2) t(4)^2 t(3)^2+t(4)^2 t(3)^2-t(1) t(3)^2-t(2) t(3)^2+2 t(1) t(4) t(3)^2-t(1) t(2) t(4) t(3)^2+3 t(2) t(4) t(3)^2-2 t(4) t(3)^2+t(3)^2+2 t(1) t(4)^2 t(3)-2 t(1) t(2) t(4)^2 t(3)+3 t(2) t(4)^2 t(3)-t(4)^2 t(3)+3 t(1) t(3)-t(1) t(2) t(3)+2 t(2) t(3)-4 t(1) t(4) t(3)+2 t(1) t(2) t(4) t(3)-4 t(2) t(4) t(3)+2 t(4) t(3)-2 t(3)-t(1) t(4)^2+t(1) t(2) t(4)^2-t(2) t(4)^2-2 t(1)+t(1) t(2)-t(2)+3 t(1) t(4)-2 t(1) t(2) t(4)+2 t(2) t(4)-t(4)+1}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{11/2}-4 q^{9/2}+9 q^{7/2}-14 q^{5/2}+17 q^{3/2}-21 \sqrt{q}+\frac{17}{\sqrt{q}}-\frac{17}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+10 a z^3-8 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-8 a^3 z+14 a z-9 z a^{-1} +2 z a^{-3} +2 a^5 z^{-1} -8 a^3 z^{-1} +11 a z^{-1} -6 a^{-1} z^{-1} + a^{-3} z^{-1} +a^5 z^{-3} -3 a^3 z^{-3} +3 a z^{-3} - a^{-1} z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} +a^5 z^7-5 a^5 z^5+4 z^5 a^{-5} +10 a^5 z^3-z^3 a^{-5} -a^5 z^{-3} -10 a^5 z+5 a^5 z^{-1} +2 a^4 z^8-6 a^4 z^6+9 z^6 a^{-4} +3 a^4 z^4-8 z^4 a^{-4} +7 a^4 z^2+3 z^2 a^{-4} +3 a^4 z^{-2} -9 a^4- a^{-4} +2 a^3 z^9+a^3 z^7+13 z^7 a^{-3} -24 a^3 z^5-19 z^5 a^{-3} +45 a^3 z^3+13 z^3 a^{-3} -3 a^3 z^{-3} -35 a^3 z-6 z a^{-3} +14 a^3 z^{-1} +2 a^{-3} z^{-1} +a^2 z^{10}+7 a^2 z^8+11 z^8 a^{-2} -23 a^2 z^6-10 z^6 a^{-2} +6 a^2 z^4-10 z^4 a^{-2} +24 a^2 z^2+14 z^2 a^{-2} +6 a^2 z^{-2} -21 a^2-6 a^{-2} +7 a z^9+5 z^9 a^{-1} -a z^7+12 z^7 a^{-1} -48 a z^5-52 z^5 a^{-1} +74 a z^3+53 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -49 a z-30 z a^{-1} +18 a z^{-1} +11 a^{-1} z^{-1} +z^{10}+16 z^8-36 z^6+2 z^4+28 z^2+3 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 12                       1 -1 10                     3   3 8                   6 1   -5 6                 8 3     5 4               10 7       -3 2             11 7         4 0           9 13           4 -2         8 8             0 -4       5 12               7 -6     2 5                 -3 -8   1 6                   5 -10   1                     -1 -12 1                       1

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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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