L11a2

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a2 at Knotilus!

Link Presentations

[edit Notes on L11a2's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X14,10,15,9 X8493 X10,5,11,6 X18,11,19,12 X20,14,21,13 X22,19,5,20 X12,22,13,21 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 4, -6, 7, -10, 8, -4, 11, -2, 3, -7, 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{(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^{9/2}-\frac{11}{q^{9/2}}-5 q^{7/2}+\frac{17}{q^{7/2}}+10 q^{5/2}-\frac{24}{q^{5/2}}-17 q^{3/2}+\frac{27}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{5}{q^{11/2}}+23 \sqrt{q}-\frac{27}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7-2 a^3 z^5+3 a z^5-2 z^5 a^{-1} +a^5 z^3-3 a^3 z^3+5 a z^3-3 z^3 a^{-1} +z^3 a^{-3} -a^3 z+a z+a^3 z^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} - a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^2 z^{10}-2 z^{10}-8 a^3 z^9-15 a z^9-7 z^9 a^{-1} -13 a^4 z^8-26 a^2 z^8-9 z^8 a^{-2} -22 z^8-11 a^5 z^7-8 a^3 z^7+9 a z^7+z^7 a^{-1} -5 z^7 a^{-3} -5 a^6 z^6+16 a^4 z^6+58 a^2 z^6+19 z^6 a^{-2} -z^6 a^{-4} +57 z^6-a^7 z^5+15 a^5 z^5+35 a^3 z^5+32 a z^5+23 z^5 a^{-1} +10 z^5 a^{-3} +4 a^6 z^4-5 a^4 z^4-35 a^2 z^4-12 z^4 a^{-2} +z^4 a^{-4} -39 z^4-6 a^5 z^3-21 a^3 z^3-27 a z^3-17 z^3 a^{-1} -5 z^3 a^{-3} +a^4 z^2+6 a^2 z^2+2 z^2 a^{-2} +7 z^2+a^3 z-2 z a^{-1} -z a^{-3} +1+a^3 z^{-1} +2 a z^{-1} +2 a^{-1} 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     4   4 6                   6 1   -5 4                 11 4     7 2               12 6       -6 0             15 11         4 -2           14 14           0 -4         10 13             -3 -6       7 14               7 -8     4 10                 -6 -10   1 7                   6 -12   4                     -4 -14 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{14}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/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}^{14}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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