L11a192

L11a191

L11a193

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Link Presentations

[edit Notes on L11a192's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X16,6,17,5 X22,11,7,12 X20,13,21,14 X18,15,19,16 X14,19,15,20 X12,21,13,22 X4,18,5,17 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -9, 3, -11}, {10, -1, 11, -2, 4, -8, 5, -7, 6, -3, 9, -6, 7, -5, 8, -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{3 t(2) t(1)^2-3 t(1)^2+3 t(2)^2 t(1)-7 t(2) t(1)+3 t(1)-3 t(2)^2+3 t(2)}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{7}{q^{9/2}}+\frac{7}{q^{7/2}}-\frac{7}{q^{5/2}}-q^{3/2}+\frac{5}{q^{3/2}}+\frac{1}{q^{19/2}}-\frac{2}{q^{17/2}}+\frac{3}{q^{15/2}}-\frac{5}{q^{13/2}}+\frac{6}{q^{11/2}}+2 \sqrt{q}-\frac{4}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^9+z^3 a^7+2 z^3 a^5+z a^5+2 z^3 a^3+z a^3+a^3 z^{-1} +z^3 a-z a-a z^{-1} -z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{10}+6 z^6 a^{10}-11 z^4 a^{10}+6 z^2 a^{10}-2 z^9 a^9+12 z^7 a^9-23 z^5 a^9+16 z^3 a^9-3 z a^9-z^{10} a^8+3 z^8 a^8+3 z^6 a^8-11 z^4 a^8+4 z^2 a^8-4 z^9 a^7+19 z^7 a^7-26 z^5 a^7+12 z^3 a^7-2 z a^7-z^{10} a^6+z^8 a^6+7 z^6 a^6-8 z^4 a^6+z^2 a^6-2 z^9 a^5+4 z^7 a^5+3 z^5 a^5-4 z^3 a^5+z a^5-3 z^8 a^4+7 z^6 a^4-4 z^4 a^4+3 z^2 a^4-3 z^7 a^3+3 z^5 a^3+4 z^3 a^3-4 z a^3+a^3 z^{-1} -3 z^6 a^2+2 z^4 a^2+z^2 a^2-a^2-3 z^5 a+3 z^3 a-3 z a+a z^{-1} -2 z^4+z^2-z^3 a^{-1} +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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 4                       1 1 2                     1   -1 0                   3 1   2 -2                 3 2     -1 -4               4 2       2 -6             4 4         0 -8           3 3           0 -10         3 4             1 -12       2 3               -1 -14     1 3                 2 -16   1 2                   -1 -18   1                     1 -20 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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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