L11a239

L11a238

L11a240

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a239 at Knotilus!

Link Presentations

[edit Notes on L11a239's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X20,10,21,9 X22,13,7,14 X14,21,15,22 X10,16,11,15 X18,5,19,6 X16,20,17,19 X2738 X4,11,5,12 X6,17,1,18
Gauss code	{1, -9, 2, -10, 7, -11}, {9, -1, 3, -6, 10, -2, 4, -5, 6, -8, 11, -7, 8, -3, 5, -4}

A Braid Representative	{{{braid_table}}}
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{2 u^2 v^3-6 u^2 v^2+7 u^2 v-2 u^2+u v^4-7 u v^3+13 u v^2-7 u v+u-2 v^4+7 v^3-6 v^2+2 v}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{17}{q^{9/2}}+\frac{20}{q^{7/2}}+q^{5/2}-\frac{20}{q^{5/2}}-4 q^{3/2}+\frac{18}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{3}{q^{15/2}}-\frac{8}{q^{13/2}}+\frac{12}{q^{11/2}}+8 \sqrt{q}-\frac{14}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-1} -4 a^7 z-2 a^7 z^{-1} +5 a^5 z^3+6 a^5 z+2 a^5 z^{-1} -2 a^3 z^5-3 a^3 z^3-4 a^3 z-a^3 z^{-1} -a z^5+z^3 a^{-1} -a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^7-4 a^9 z^5+6 a^9 z^3-4 a^9 z+a^9 z^{-1} +3 a^8 z^8-10 a^8 z^6+10 a^8 z^4-3 a^8 z^2+4 a^7 z^9-9 a^7 z^7-2 a^7 z^5+12 a^7 z^3-7 a^7 z+2 a^7 z^{-1} +2 a^6 z^{10}+6 a^6 z^8-35 a^6 z^6+39 a^6 z^4-15 a^6 z^2+a^6+12 a^5 z^9-28 a^5 z^7+6 a^5 z^5+15 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +2 a^4 z^{10}+16 a^4 z^8-57 a^4 z^6+53 a^4 z^4-17 a^4 z^2+8 a^3 z^9-6 a^3 z^7-16 a^3 z^5+17 a^3 z^3-5 a^3 z+a^3 z^{-1} +13 a^2 z^8-24 a^2 z^6+17 a^2 z^4+z^4 a^{-2} -5 a^2 z^2+12 a z^7-16 a z^5+4 z^5 a^{-1} +6 a z^3-2 z^3 a^{-1} +a z+8 z^6-6 z^4 }[/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 1 2 3 χ 6                       1 -1 4                     3   3 2                   5 1   -4 0                 9 3     6 -2               10 6       -4 -4             10 8         2 -6           10 10           0 -8         7 10             -3 -10       5 10               5 -12     3 7                 -4 -14   1 6                   5 -16   2                     -2 -18 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/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}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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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