L11n249

L11n248

L11n250

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n249 at Knotilus!

Link Presentations

[edit Notes on L11n249's Link Presentations]

Planar diagram presentation	X12,1,13,2 X9,19,10,18 X5,14,6,15 X11,6,12,7 X15,11,16,22 X7,20,8,21 X3948 X21,17,22,16 X17,4,18,5 X10,13,1,14 X19,3,20,2
Gauss code	{1, 11, -7, 9, -3, 4, -6, 7, -2, -10}, {-4, -1, 10, 3, -5, 8, -9, 2, -11, 6, -8, 5}

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^3 v^3-3 u^3 v^2+2 u^3 v-2 u^2 v^3+7 u^2 v^2-7 u^2 v+u^2+u v^3-7 u v^2+7 u v-2 u+2 v^2-3 v+1}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{3}{q^{9/2}}-4 q^{7/2}+\frac{7}{q^{7/2}}+8 q^{5/2}-\frac{12}{q^{5/2}}-12 q^{3/2}+\frac{14}{q^{3/2}}+15 \sqrt{q}-\frac{16}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z+a^5 z^{-1} -a^3 z^5-3 a^3 z^3+z^3 a^{-3} -4 a^3 z+z a^{-3} -a^3 z^{-1} +a z^7+4 a z^5-2 z^5 a^{-1} +6 a z^3-5 z^3 a^{-1} +3 a z-3 z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 a z^9-3 z^9 a^{-1} -8 a^2 z^8-6 z^8 a^{-2} -14 z^8-8 a^3 z^7-8 a z^7-4 z^7 a^{-1} -4 z^7 a^{-3} -3 a^4 z^6+13 a^2 z^6+14 z^6 a^{-2} -z^6 a^{-4} +31 z^6+13 a^3 z^5+27 a z^5+24 z^5 a^{-1} +10 z^5 a^{-3} -3 a^4 z^4-10 a^2 z^4-7 z^4 a^{-2} +2 z^4 a^{-4} -16 z^4-6 a^5 z^3-15 a^3 z^3-18 a z^3-16 z^3 a^{-1} -7 z^3 a^{-3} +a^4 z^2+3 a^2 z^2-z^2 a^{-4} +3 z^2+5 a^5 z+7 a^3 z+4 a z+3 z a^{-1} +z a^{-3} +a^4-a^5 z^{-1} -a^3 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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 3   3 6               5 1   -4 4             7 3     4 2           8 5       -3 0         8 7         1 -2       7 9           2 -4     5 7             -2 -6   2 7               5 -8 1 5                 -4 -10 3                   3

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=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/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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