L11a209

L11a208

L11a210

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a209 at Knotilus!

Link Presentations

[edit Notes on L11a209's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X22,15,7,16 X10,4,11,3 X16,6,17,5 X4,12,5,11 X12,21,13,22 X20,13,21,14 X14,19,15,20 X2,18,3,17
Gauss code	{1, -11, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -8, 9, -10, 4, -6, 11, -2, 10, -9, 8, -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{\left(2 v^2-3 v+2\right) (u v-u+1) (u v-v+1)}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{12}{q^{9/2}}-q^{7/2}+\frac{17}{q^{7/2}}+4 q^{5/2}-\frac{20}{q^{5/2}}-9 q^{3/2}+\frac{20}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{7}{q^{11/2}}+13 \sqrt{q}-\frac{19}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^7+a z^7-a^5 z^5+4 a^3 z^5+3 a z^5-z^5 a^{-1} -3 a^5 z^3+7 a^3 z^3+2 a z^3-2 z^3 a^{-1} -3 a^5 z+6 a^3 z-a z-z a^{-1} -a^5 z^{-1} +2 a^3 z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^6-3 a^8 z^4+2 a^8 z^2+3 a^7 z^7-8 a^7 z^5+5 a^7 z^3+5 a^6 z^8-13 a^6 z^6+12 a^6 z^4-7 a^6 z^2+2 a^6+5 a^5 z^9-11 a^5 z^7+12 a^5 z^5-11 a^5 z^3+4 a^5 z-a^5 z^{-1} +2 a^4 z^{10}+6 a^4 z^8-24 a^4 z^6+31 a^4 z^4-21 a^4 z^2+5 a^4+11 a^3 z^9-25 a^3 z^7+27 a^3 z^5+z^5 a^{-3} -18 a^3 z^3-z^3 a^{-3} +8 a^3 z-2 a^3 z^{-1} +2 a^2 z^{10}+10 a^2 z^8-29 a^2 z^6+4 z^6 a^{-2} +30 a^2 z^4-5 z^4 a^{-2} -14 a^2 z^2+3 a^2+6 a z^9-3 a z^7+8 z^7 a^{-1} -8 a z^5-14 z^5 a^{-1} +7 a z^3+8 z^3 a^{-1} +a z-3 z a^{-1} + a^{-1} z^{-1} +9 z^8-15 z^6+9 z^4-2 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 8 4     -4 0               11 5       6 -2             10 9         -1 -4           10 10           0 -6         7 10             3 -8       5 10               -5 -10     2 7                 5 -12   1 5                   -4 -14   2                     2 -16 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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