L10n74

L10n73

L10n75

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n74 at Knotilus!

Link Presentations

[edit Notes on L10n74's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X9,18,10,19 X4,17,1,18 X19,12,20,5 X11,20,12,13 X13,10,14,11
Gauss code	{1, 4, -3, -7}, {-2, -1, 5, 3, -6, 10, -9, 8}, {-10, 2, -4, -5, 7, 6, -8, 9}

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 v^3 w^2+u v^2 w^3-u v^2 w^2+v w-v-w}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-3} - q^{-4} +2 q^{-5} - q^{-6} +2 q^{-7} -2 q^{-8} +2 q^{-9} + q^{-11} }[/math] (db)
Signature	-6 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{12} z^{-2} +a^{12}-z^4 a^{10}-5 z^2 a^{10}-2 a^{10} z^{-2} -5 a^{10}+z^6 a^8+5 z^4 a^8+6 z^2 a^8+a^8 z^{-2} +2 a^8+z^6 a^6+5 z^4 a^6+6 z^2 a^6+2 a^6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^2-2 a^{14}+a^{13} z+a^{12} z^4-3 a^{12} z^2-a^{12} z^{-2} +3 a^{12}+a^{11} z^7-6 a^{11} z^5+12 a^{11} z^3-8 a^{11} z+2 a^{11} z^{-1} +a^{10} z^8-6 a^{10} z^6+13 a^{10} z^4-15 a^{10} z^2-2 a^{10} z^{-2} +9 a^{10}+2 a^9 z^7-10 a^9 z^5+14 a^9 z^3-8 a^9 z+2 a^9 z^{-1} +a^8 z^8-5 a^8 z^6+7 a^8 z^4-5 a^8 z^2-a^8 z^{-2} +3 a^8+a^7 z^7-4 a^7 z^5+2 a^7 z^3+a^7 z+a^6 z^6-5 a^6 z^4+6 a^6 z^2-2 a^6 }[/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 χ -5                 1 1 -7               1 1 0 -9             1     1 -11         1 1 1     1 -13       1 3 1       1 -15       1 1         0 -17     2 2           0 -19 1   1             2 -21 2 1               1 -23 1                 1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-7 }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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.

L10n73 L10n75