L11a281

L11a280

L11a282

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Link Presentations

[edit Notes on L11a281's Link Presentations]

Planar diagram presentation	X10,1,11,2 X8,9,1,10 X12,4,13,3 X22,16,9,15 X2,17,3,18 X4,22,5,21 X14,5,15,6 X20,13,21,14 X16,12,17,11 X6,19,7,20 X18,7,19,8
Gauss code	{1, -5, 3, -6, 7, -10, 11, -2}, {2, -1, 9, -3, 8, -7, 4, -9, 5, -11, 10, -8, 6, -4}

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)^2 t(2)^5-t(1) t(2)^5+t(1)^3 t(2)^4-5 t(1)^2 t(2)^4+5 t(1) t(2)^4-t(2)^4-3 t(1)^3 t(2)^3+10 t(1)^2 t(2)^3-11 t(1) t(2)^3+3 t(2)^3+3 t(1)^3 t(2)^2-11 t(1)^2 t(2)^2+10 t(1) t(2)^2-3 t(2)^2-t(1)^3 t(2)+5 t(1)^2 t(2)-5 t(1) t(2)+t(2)-t(1)^2+t(1)}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{18}{q^{9/2}}-q^{7/2}+\frac{23}{q^{7/2}}+4 q^{5/2}-\frac{27}{q^{5/2}}-9 q^{3/2}+\frac{26}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{5}{q^{13/2}}+\frac{11}{q^{11/2}}+16 \sqrt{q}-\frac{23}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^5 z^5-a^5 z^3+a^5 z+a^5 z^{-1} +a^3 z^7+2 a^3 z^5-a^3 z^3-4 a^3 z-a^3 z^{-1} +a z^7+3 a z^5-z^5 a^{-1} +4 a z^3-2 z^3 a^{-1} +2 a z-z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -4 a^4 z^{10}-4 a^2 z^{10}-10 a^5 z^9-20 a^3 z^9-10 a z^9-10 a^6 z^8-11 a^4 z^8-12 a^2 z^8-11 z^8-5 a^7 z^7+17 a^5 z^7+43 a^3 z^7+13 a z^7-8 z^7 a^{-1} -a^8 z^6+21 a^6 z^6+40 a^4 z^6+39 a^2 z^6-4 z^6 a^{-2} +17 z^6+9 a^7 z^5-4 a^5 z^5-30 a^3 z^5-5 a z^5+11 z^5 a^{-1} -z^5 a^{-3} +a^8 z^4-10 a^6 z^4-28 a^4 z^4-33 a^2 z^4+5 z^4 a^{-2} -11 z^4-3 a^7 z^3+a^5 z^3+7 a^3 z^3-3 a z^3-5 z^3 a^{-1} +z^3 a^{-3} +a^6 z^2+5 a^4 z^2+7 a^2 z^2-z^2 a^{-2} +2 z^2-a^7 z+a^5 z+2 a^3 z+a z+z a^{-1} +a^4-a^5 z^{-1} -a^3 z^{-1} }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 10 3     -7 0               13 6       7 -2             14 11         -3 -4           13 12           1 -6         10 14             4 -8       8 13               -5 -10     4 11                 7 -12   1 7                   -6 -14   4                     4 -16 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=0 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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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