L11a338

L11a337

L11a339

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Link Presentations

[edit Notes on L11a338's Link Presentations]

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

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{2 u^3 v^3-3 u^3 v^2+u^3 v-4 u^2 v^3+12 u^2 v^2-9 u^2 v+2 u^2+2 u v^3-9 u v^2+12 u v-4 u+v^2-3 v+2}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{21}{q^{9/2}}-\frac{22}{q^{7/2}}+\frac{18}{q^{5/2}}+q^{3/2}-\frac{15}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{13}{q^{13/2}}-\frac{18}{q^{11/2}}-4 \sqrt{q}+\frac{9}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^7+3 z^3 a^7+3 z a^7-z^7 a^5-4 z^5 a^5-7 z^3 a^5-4 z a^5+a^5 z^{-1} -z^7 a^3-3 z^5 a^3-3 z^3 a^3-2 z a^3-a^3 z^{-1} +z^5 a+2 z^3 a+z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-5 a^{10} z^4+2 a^{10} z^2+5 a^9 z^7-6 a^9 z^5+a^9 z^3+7 a^8 z^8-12 a^8 z^6+13 a^8 z^4-7 a^8 z^2+6 a^7 z^9-7 a^7 z^7+4 a^7 z^5+a^7 z+2 a^6 z^{10}+12 a^6 z^8-38 a^6 z^6+42 a^6 z^4-15 a^6 z^2+12 a^5 z^9-22 a^5 z^7+9 a^5 z^5+2 a^5 z^3+a^5 z^{-1} +2 a^4 z^{10}+12 a^4 z^8-39 a^4 z^6+33 a^4 z^4-9 a^4 z^2-a^4+6 a^3 z^9-6 a^3 z^7-11 a^3 z^5+11 a^3 z^3-4 a^3 z+a^3 z^{-1} +7 a^2 z^8-15 a^2 z^6+7 a^2 z^4-2 a^2 z^2+4 a z^7-9 a z^5+6 a z^3-2 a z+z^6-2 z^4+z^2 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     3   3 0                   6 1   -5 -2                 9 3     6 -4               10 7       -3 -6             12 8         4 -8           9 10           1 -10         9 12             -3 -12       5 10               5 -14     2 8                 -6 -16   1 5                   4 -18   2                     -2 -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=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-8 }[/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}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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}_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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