L10a40

L10a39

L10a41

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Link Presentations

[edit Notes on L10a40's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X18,8,19,7 X20,13,5,14 X14,19,15,20 X16,10,17,9 X10,16,11,15 X8,18,9,17 X2536 X4,11,1,12
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 6, -7, 10, -2, 4, -5, 7, -6, 8, -3, 5, -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{2 t(2)^3+4 t(1) t(2)^2-7 t(2)^2-7 t(1) t(2)+4 t(2)+2 t(1)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{5}{q^{9/2}}-q^{7/2}+\frac{6}{q^{7/2}}+2 q^{5/2}-\frac{8}{q^{5/2}}-4 q^{3/2}+\frac{9}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+6 \sqrt{q}-\frac{8}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^{-1} -3 a^5 z-a^5 z^{-1} +2 a^3 z^3-a^3 z^{-1} -z a^{-3} +3 a z^3+z^3 a^{-1} +3 a z+2 a z^{-1} -z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-3 a^7 z^3+3 a^7 z-a^7 z^{-1} +2 a^6 z^6-4 a^6 z^4+a^6+2 a^5 z^7-a^5 z^5-6 a^5 z^3+3 a^5 z-a^5 z^{-1} +2 a^4 z^8-3 a^4 z^6+3 a^4 z^4-5 a^4 z^2+2 a^4+a^3 z^9+a^3 z^7-7 a^3 z^5+z^5 a^{-3} +13 a^3 z^3-3 z^3 a^{-3} -8 a^3 z+z a^{-3} +a^3 z^{-1} +5 a^2 z^8-17 a^2 z^6+2 z^6 a^{-2} +26 a^2 z^4-5 z^4 a^{-2} -13 a^2 z^2+z^2 a^{-2} +3 a^2+a z^9+2 a z^7+3 z^7 a^{-1} -15 a z^5-9 z^5 a^{-1} +28 a z^3+9 z^3 a^{-1} -14 a z-5 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +3 z^8-10 z^6+14 z^4-7 z^2+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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                   1   -1 4                 3 1   2 2               3 1     -2 0             5 3       2 -2           5 4         -1 -4         3 4           -1 -6       3 5             2 -8     2 3               -1 -10   1 4                 3 -12   1                   -1 -14 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/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}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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