L11a133

L11a132

L11a134

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a133 at Knotilus!

Link Presentations

[edit Notes on L11a133's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X10,5,11,6 X14,3,15,4 X18,11,19,12 X22,19,5,20 X20,14,21,13 X12,22,13,21 X2,9,3,10 X8,15,9,16
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -11, 10, -4, 6, -9, 8, -5, 11, -2, 3, -6, 7, -8, 9, -7}

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(2)^5+4 t(1) t(2)^4-6 t(2)^4-14 t(1) t(2)^3+14 t(2)^3+14 t(1) t(2)^2-14 t(2)^2-6 t(1) t(2)+4 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{25}{q^{9/2}}-\frac{22}{q^{7/2}}+\frac{15}{q^{5/2}}-\frac{10}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{5}{q^{19/2}}+\frac{10}{q^{17/2}}-\frac{16}{q^{15/2}}+\frac{22}{q^{13/2}}-\frac{25}{q^{11/2}}-\sqrt{q}+\frac{4}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^9+z a^9+a^9 z^{-1} +z^5 a^7-2 z^3 a^7-5 z a^7-3 a^7 z^{-1} +3 z^5 a^5+6 z^3 a^5+7 z a^5+4 a^5 z^{-1} +z^5 a^3-2 z^3 a^3-5 z a^3-2 a^3 z^{-1} -z^3 a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{12}+z^4 a^{12}-5 z^7 a^{11}+10 z^5 a^{11}-4 z^3 a^{11}-z a^{11}-9 z^8 a^{10}+20 z^6 a^{10}-12 z^4 a^{10}+z^2 a^{10}+a^{10}-7 z^9 a^9+4 z^7 a^9+17 z^5 a^9-14 z^3 a^9+3 z a^9-a^9 z^{-1} -2 z^{10} a^8-19 z^8 a^8+52 z^6 a^8-36 z^4 a^8+5 z^2 a^8+3 a^8-14 z^9 a^7+13 z^7 a^7+24 z^5 a^7-33 z^3 a^7+13 z a^7-3 a^7 z^{-1} -2 z^{10} a^6-21 z^8 a^6+52 z^6 a^6-38 z^4 a^6+5 z^2 a^6+3 a^6-7 z^9 a^5-5 z^7 a^5+33 z^5 a^5-36 z^3 a^5+16 z a^5-4 a^5 z^{-1} -11 z^8 a^4+17 z^6 a^4-11 z^4 a^4+z^2 a^4+2 a^4-9 z^7 a^3+15 z^5 a^3-12 z^3 a^3+7 z a^3-2 a^3 z^{-1} -4 z^6 a^2+4 z^4 a^2-z^5 a+z^3 a }[/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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     3   -3 -2                   7 1   6 -4                 9 4     -5 -6               13 6       7 -8             12 9         -3 -10           13 13           0 -12         10 13             3 -14       6 12               -6 -16     4 10                 6 -18   1 6                   -5 -20   4                     4 -22 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/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.

L11a132 L11a134