L11a8

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a8 at Knotilus!

Link Presentations

[edit Notes on L11a8's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X20,9,21,10 X8493 X18,22,19,21 X14,12,15,11 X12,5,13,6 X22,13,5,14 X10,19,11,20 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {8, -1, 2, -5, 4, -10, 7, -8, 9, -7, 11, -2, 3, -6, 10, -4, 6, -9}

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{(t(1)-1) (t(2)-1) \left(t(2)^4-5 t(2)^3+9 t(2)^2-5 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{7/2}+5 q^{5/2}-11 q^{3/2}+17 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{27}{q^{3/2}}-\frac{27}{q^{5/2}}+\frac{23}{q^{7/2}}-\frac{17}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^7+3 z^3 a^5+2 z a^5-a^5 z^{-1} -3 z^5 a^3-5 z^3 a^3+3 a^3 z^{-1} +z^7 a+3 z^5 a+4 z^3 a-z a-2 a z^{-1} -z^5 a^{-1} -z^3 a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^6-2 a^8 z^4+a^8 z^2+4 a^7 z^7-8 a^7 z^5+6 a^7 z^3-2 a^7 z+8 a^6 z^8-17 a^6 z^6+14 a^6 z^4-6 a^6 z^2+a^6+7 a^5 z^9-3 a^5 z^7-18 a^5 z^5+18 a^5 z^3-4 a^5 z-a^5 z^{-1} +2 a^4 z^{10}+22 a^4 z^8-59 a^4 z^6+47 a^4 z^4-16 a^4 z^2+3 a^4+15 a^3 z^9-8 a^3 z^7-35 a^3 z^5+z^5 a^{-3} +30 a^3 z^3-a^3 z-3 a^3 z^{-1} +2 a^2 z^{10}+27 a^2 z^8-61 a^2 z^6+5 z^6 a^{-2} +39 a^2 z^4-4 z^4 a^{-2} -11 a^2 z^2+3 a^2+8 a z^9+10 a z^7+11 z^7 a^{-1} -41 a z^5-15 z^5 a^{-1} +25 a z^3+7 z^3 a^{-1} +a z-2 a z^{-1} +13 z^8-15 z^6+4 z^4-2 z^2 }[/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                     4   -4 4                   7 1   6 2                 10 4     -6 0               15 7       8 -2             14 12         -2 -4           13 13           0 -6         10 14             4 -8       7 13               -6 -10     3 10                 7 -12   1 7                   -6 -14   3                     3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{13}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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