L11a534

L11a533

L11a535

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Link Presentations

[edit Notes on L11a534's Link Presentations]

Planar diagram presentation	X6172 X2536 X18,11,19,12 X10,3,11,4 X4,9,1,10 X16,7,17,8 X8,15,5,16 X20,14,21,13 X22,19,15,20 X12,22,13,21 X14,17,9,18
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -10, 8, -11}, {7, -6, 11, -3, 9, -8, 10, -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(2) t(4)^2 t(3)^2-t(4)^2 t(3)^2+t(1) t(3)^2-t(1) t(2) t(3)^2+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)^2-3 t(1) t(4)^2 t(3)+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(1) t(3)+2 t(1) t(2) t(3)-3 t(2) t(3)+6 t(1) t(4) t(3)-4 t(1) t(2) t(4) t(3)+6 t(2) t(4) t(3)-4 t(4) t(3)+t(3)+2 t(1) t(4)^2-t(1) t(2) t(4)^2+t(2) t(4)^2-t(4)^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)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{21}{q^{9/2}}-\frac{21}{q^{7/2}}+\frac{14}{q^{5/2}}-\frac{10}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{7}{q^{17/2}}-\frac{14}{q^{15/2}}+\frac{17}{q^{13/2}}-\frac{23}{q^{11/2}}-\sqrt{q}+\frac{4}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{11} z^{-1} +4 z a^9+4 a^9 z^{-1} +a^9 z^{-3} -6 z^3 a^7-11 z a^7-9 a^7 z^{-1} -3 a^7 z^{-3} +3 z^5 a^5+8 z^3 a^5+13 z a^5+10 a^5 z^{-1} +3 a^5 z^{-3} +z^5 a^3-2 z^3 a^3-6 z a^3-4 a^3 z^{-1} -a^3 z^{-3} -z^3 a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12} z^6-3 a^{12} z^4+3 a^{12} z^2-a^{12}+3 a^{11} z^7-8 a^{11} z^5+9 a^{11} z^3-6 a^{11} z+2 a^{11} z^{-1} +4 a^{10} z^8-4 a^{10} z^6-8 a^{10} z^4+14 a^{10} z^2-6 a^{10}+3 a^9 z^9+7 a^9 z^7-33 a^9 z^5+42 a^9 z^3-a^9 z^{-3} -30 a^9 z+11 a^9 z^{-1} +a^8 z^{10}+13 a^8 z^8-24 a^8 z^6-4 a^8 z^4+28 a^8 z^2+3 a^8 z^{-2} -18 a^8+8 a^7 z^9+7 a^7 z^7-56 a^7 z^5+74 a^7 z^3-3 a^7 z^{-3} -49 a^7 z+18 a^7 z^{-1} +a^6 z^{10}+19 a^6 z^8-37 a^6 z^6+9 a^6 z^4+24 a^6 z^2+6 a^6 z^{-2} -21 a^6+5 a^5 z^9+12 a^5 z^7-48 a^5 z^5+56 a^5 z^3-3 a^5 z^{-3} -35 a^5 z+14 a^5 z^{-1} +10 a^4 z^8-14 a^4 z^6+4 a^4 z^4+7 a^4 z^2+3 a^4 z^{-2} -9 a^4+9 a^3 z^7-16 a^3 z^5+14 a^3 z^3-a^3 z^{-3} -10 a^3 z+5 a^3 z^{-1} +4 a^2 z^6-4 a^2 z^4+a z^5-a z^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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     3   -3 -2                   7 1   6 -4                 8 4     -4 -6               13 6       7 -8             11 11         0 -10           12 10           2 -12         8 14             6 -14       6 9               -3 -16     2 9                 7 -18   1 5                   -4 -20   2                     2 -22 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/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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