L11a181

L11a180

L11a182

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a181 at Knotilus!

Link Presentations

[edit Notes on L11a181's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X14,5,15,6 X16,8,17,7 X10,4,11,3 X22,14,7,13 X18,12,19,11 X12,18,13,17 X6,15,1,16 X4,21,5,22 X2,20,3,19
Gauss code	{1, -11, 5, -10, 3, -9}, {4, -1, 2, -5, 7, -8, 6, -3, 9, -4, 8, -7, 11, -2, 10, -6}

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{2 u^2 v^4-7 u^2 v^3+8 u^2 v^2-3 u^2 v-2 u v^4+9 u v^3-15 u v^2+9 u v-2 u-3 v^3+8 v^2-7 v+2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{11/2}+5 q^{9/2}-10 q^{7/2}+16 q^{5/2}-22 q^{3/2}+24 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{21}{q^{3/2}}-\frac{16}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7+z^7 a^{-1} -a^3 z^5+3 a z^5+2 z^5 a^{-1} -z^5 a^{-3} -2 a^3 z^3+4 a z^3-2 z^3 a^{-1} -z^3 a^{-3} -a^3 z+4 a z-6 z a^{-1} +2 z a^{-3} +2 a z^{-1} -3 a^{-1} z^{-1} + a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -4 z^{10} a^{-2} -4 z^{10}-12 a z^9-20 z^9 a^{-1} -8 z^9 a^{-3} -17 a^2 z^8-4 z^8 a^{-2} -5 z^8 a^{-4} -16 z^8-15 a^3 z^7+12 a z^7+52 z^7 a^{-1} +24 z^7 a^{-3} -z^7 a^{-5} -9 a^4 z^6+29 a^2 z^6+39 z^6 a^{-2} +15 z^6 a^{-4} +62 z^6-4 a^5 z^5+21 a^3 z^5+18 a z^5-28 z^5 a^{-1} -19 z^5 a^{-3} +2 z^5 a^{-5} -a^6 z^4+6 a^4 z^4-15 a^2 z^4-36 z^4 a^{-2} -12 z^4 a^{-4} -46 z^4+a^5 z^3-12 a^3 z^3-22 a z^3-7 z^3 a^{-1} +z^3 a^{-3} -z^3 a^{-5} -a^4 z^2+2 a^2 z^2+3 z^2 a^{-2} +6 z^2+2 a^3 z+9 a z+10 z a^{-1} +3 z a^{-3} +3 a^{-2} + a^{-4} +3-2 a z^{-1} -3 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                     4   -4 8                   6 1   5 6                 10 4     -6 4               12 6       6 2             12 10         -2 0           13 12           1 -2         9 13             4 -4       7 12               -5 -6     3 10                 7 -8   1 6                   -5 -10   3                     3 -12 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=6 }[/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.

L11a180 L11a182