L11a319

L11a318

L11a320

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Link Presentations

[edit Notes on L11a319's Link Presentations]

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

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{(u-1) (v-1) \left(2 u^2 v+2 u v^2+u v+2 u+2 v\right)}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{8}{q^{9/2}}-\frac{9}{q^{7/2}}-q^{5/2}+\frac{8}{q^{5/2}}+q^{3/2}-\frac{7}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{2}{q^{15/2}}+\frac{4}{q^{13/2}}-\frac{7}{q^{11/2}}-3 \sqrt{q}+\frac{5}{\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+3 a^3 z+a z^5+3 a z^3-z^3 a^{-1} +2 a z+a z^{-1} -3 z a^{-1} - a^{-1} z^{-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^7-15 z^3 a^7+4 z a^7-3 z^8 a^6+7 z^6 a^6-8 z^4 a^6-2 z^9 a^5+5 z^7 a^5-7 z^5 a^5+3 z^3 a^5-z^{10} a^4+3 z^8 a^4-8 z^6 a^4+11 z^4 a^4-3 z^2 a^4-3 z^9 a^3+13 z^7 a^3-26 z^5 a^3+27 z^3 a^3-6 z a^3-z^{10} a^2+5 z^8 a^2-14 z^6 a^2+22 z^4 a^2-10 z^2 a^2-z^9 a+3 z^7 a-5 z^3 a+4 z a-a z^{-1} -z^8+4 z^6-3 z^4-2 z^2+1-z^7 a^{-1} +6 z^5 a^{-1} -11 z^3 a^{-1} +6 z a^{-1} - a^{-1} 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 6                       1 1 4                         0 2                   3 1   2 0                 2       -2 -2               5 3       2 -4             5 4         -1 -6           4 3           1 -8         4 5             1 -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}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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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