L11n411

L11n410

L11n412

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n411 at Knotilus!

Link Presentations

[edit Notes on L11n411's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,22,14,19 X7,20,8,21 X19,10,20,11 X9,16,10,17 X17,14,18,15 X15,8,16,9 X21,18,22,5 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {-5, 4, -9, 3}, {10, -1, -4, 8, -6, 5, 11, -2, -3, 7, -8, 6, -7, 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{(w-1) (w+1) \left(2 u v w^2-2 u v w+u v+w^3-2 w^2+2 w\right)}{\sqrt{u} \sqrt{v} w^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-13} +2 q^{-12} -2 q^{-11} + q^{-10} + q^{-8} +2 q^{-7} - q^{-6} +2 q^{-5} - q^{-4} + q^{-3} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{14} z^{-2} +4 a^{12} z^{-2} +5 a^{12}-z^4 a^{10}-8 z^2 a^{10}-5 a^{10} z^{-2} -12 a^{10}+z^6 a^8+5 z^4 a^8+6 z^2 a^8+2 a^8 z^{-2} +5 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^{15} z^7-5 a^{15} z^5+6 a^{15} z^3-4 a^{15} z+a^{15} z^{-1} +2 a^{14} z^8-11 a^{14} z^6+15 a^{14} z^4-8 a^{14} z^2-a^{14} z^{-2} +3 a^{14}+a^{13} z^9-3 a^{13} z^7-10 a^{13} z^5+30 a^{13} z^3-21 a^{13} z+5 a^{13} z^{-1} +4 a^{12} z^8-26 a^{12} z^6+47 a^{12} z^4-35 a^{12} z^2-4 a^{12} z^{-2} +16 a^{12}+a^{11} z^9-2 a^{11} z^7-20 a^{11} z^5+54 a^{11} z^3-39 a^{11} z+9 a^{11} z^{-1} +3 a^{10} z^8-21 a^{10} z^6+44 a^{10} z^4-43 a^{10} z^2-5 a^{10} z^{-2} +21 a^{10}+3 a^9 z^7-19 a^9 z^5+32 a^9 z^3-22 a^9 z+5 a^9 z^{-1} +a^8 z^8-5 a^8 z^6+7 a^8 z^4-10 a^8 z^2-2 a^8 z^{-2} +7 a^8+a^7 z^7-4 a^7 z^5+2 a^7 z^3+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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -5                       1 1 -7                     1 1 0 -9                 1 2     1 -11               1 1 1     1 -13             2 5 2       1 -15           2 2 3         3 -17         2 5 2           1 -19       2 2 1             1 -21     1 3 3               -1 -23   1 1                   0 -25   1                     1 -27 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=-11 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/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=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

L11n410 L11n412