L11a369

L11a368

L11a370

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Link Presentations

[edit Notes on L11a369's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,5,17,6 X14,3,15,4 X20,8,21,7 X22,17,11,18 X18,10,19,9 X4,15,5,16 X8,20,9,19 X6,22,7,21 X2,11,3,12 X10,13,1,14
Gauss code	{1, -10, 3, -7, 2, -9, 4, -8, 6, -11}, {10, -1, 11, -3, 7, -2, 5, -6, 8, -4, 9, -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)^2 t(1)^4-t(2) t(1)^4+2 t(2)^3 t(1)^3-3 t(2)^2 t(1)^3+2 t(2) t(1)^3-t(1)^3+t(2)^4 t(1)^2-3 t(2)^3 t(1)^2+3 t(2)^2 t(1)^2-3 t(2) t(1)^2+t(1)^2-t(2)^4 t(1)+2 t(2)^3 t(1)-3 t(2)^2 t(1)+2 t(2) t(1)-t(2)^3+t(2)^2}{t(1)^2 t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{8}{q^{9/2}}-\frac{10}{q^{7/2}}-q^{5/2}+\frac{9}{q^{5/2}}+2 q^{3/2}-\frac{8}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{2}{q^{15/2}}+\frac{4}{q^{13/2}}-\frac{7}{q^{11/2}}-4 \sqrt{q}+\frac{6}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 \left(-z^3\right)-2 a^7 z+a^5 z^5+2 a^5 z^3+2 a^3 z^5+6 a^3 z^3+4 a^3 z+a^3 z^{-1} +a z^5+2 a z^3-z^3 a^{-1} -a z-a z^{-1} -2 z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^4 a^{10}+2 z^2 a^{10}-2 z^5 a^9+3 z^3 a^9-3 z^6 a^8+5 z^4 a^8-3 z^2 a^8-4 z^7 a^7+11 z^5 a^7-16 z^3 a^7+6 z a^7-3 z^8 a^6+6 z^6 a^6-5 z^4 a^6-2 z^2 a^6-2 z^9 a^5+3 z^7 a^5+z^5 a^5-7 z^3 a^5+4 z a^5-z^{10} a^4+z^8 a^4+2 z^4 a^4-4 z^9 a^3+16 z^7 a^3-27 z^5 a^3+27 z^3 a^3-9 z a^3+a^3 z^{-1} -z^{10} a^2+2 z^8 a^2+2 z^4 a^2-a^2-2 z^9 a+8 z^7 a-10 z^5 a+8 z^3 a-5 z a+a z^{-1} -2 z^8+9 z^6-11 z^4+3 z^2-z^7 a^{-1} +5 z^5 a^{-1} -7 z^3 a^{-1} +2 z a^{-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 6                       1 1 4                     1   -1 2                   3 1   2 0                 3 1     -2 -2               5 3       2 -4             5 4         -1 -6           5 4           1 -8         4 6             2 -10       3 4               -1 -12     1 4                 3 -14   1 3                   -2 -16   1                     1 -18 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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