L11n442

L11n441

L11n443

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n442 at Knotilus!

Link Presentations

[edit Notes on L11n442's Link Presentations]

Planar diagram presentation	X6172 X2536 X11,19,12,18 X3,11,4,10 X9,1,10,4 X7,17,8,16 X15,5,16,8 X13,20,14,21 X19,15,20,22 X21,12,22,13 X17,9,18,14
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 10, -8, 11}, {-7, 6, -11, 3, -9, 8, -10, 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 w^2 x-u v w^2+u v w x^2-4 u v w x+2 u v w-u v x^2+2 u v x-u v-u w x^2+2 u w x+u x^2-u x-v w^2 x+v w^2+2 v w x-v w+w^2 \left(-x^2\right)+2 w^2 x-w^2+2 w x^2-4 w x+w-x^2+x}{\sqrt{u} \sqrt{v} w x} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -10 q^{9/2}+10 q^{7/2}-14 q^{5/2}+10 q^{3/2}-\frac{3}{q^{3/2}}+q^{15/2}-3 q^{13/2}+6 q^{11/2}-10 \sqrt{q}+\frac{5}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ 2 z^5 a^{-3} -5 z^3 a^{-1} +7 z^3 a^{-3} -3 z^3 a^{-5} +3 a z-11 z a^{-1} +13 z a^{-3} -6 z a^{-5} +z a^{-7} +3 a z^{-1} -9 a^{-1} z^{-1} +10 a^{-3} z^{-1} -5 a^{-5} z^{-1} + a^{-7} z^{-1} +a z^{-3} -3 a^{-1} z^{-3} +3 a^{-3} z^{-3} - a^{-5} z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-8} -3 z^4 a^{-8} +3 z^2 a^{-8} - a^{-8} +3 z^7 a^{-7} -9 z^5 a^{-7} +9 z^3 a^{-7} -6 z a^{-7} +2 a^{-7} z^{-1} +3 z^8 a^{-6} -3 z^6 a^{-6} -11 z^4 a^{-6} +14 z^2 a^{-6} -6 a^{-6} +z^9 a^{-5} +10 z^7 a^{-5} -40 z^5 a^{-5} +47 z^3 a^{-5} - a^{-5} z^{-3} -30 z a^{-5} +11 a^{-5} z^{-1} +8 z^8 a^{-4} -14 z^6 a^{-4} -9 z^4 a^{-4} +28 z^2 a^{-4} +3 a^{-4} z^{-2} -18 a^{-4} +z^9 a^{-3} +14 z^7 a^{-3} -53 z^5 a^{-3} +74 z^3 a^{-3} -3 a^{-3} z^{-3} -49 z a^{-3} +18 a^{-3} z^{-1} +5 z^8 a^{-2} -7 z^6 a^{-2} -4 z^4 a^{-2} +24 z^2 a^{-2} +6 a^{-2} z^{-2} -21 a^{-2} +7 z^7 a^{-1} -22 z^5 a^{-1} +6 a z^3+42 z^3 a^{-1} -a z^{-3} -3 a^{-1} z^{-3} -10 a z-35 z a^{-1} +5 a z^{-1} +14 a^{-1} z^{-1} +3 z^6-3 z^4+7 z^2+3 z^{-2} -9 }[/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   \ -2 -1 0 1 2 3 4 5 6 7 χ 16                   1 -1 14                 2   2 12               4 1   -3 10             6 2     4 8           5 5       0 6         9 5         4 4       4 8           4 2     6 6             0 0   2 7               5 -2 1 3                 -2 -4 3                   3

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=7 }[/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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