L11a177

L11a176

L11a178

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a177 at Knotilus!

Link Presentations

[edit Notes on L11a177's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X16,7,17,8 X18,10,19,9 X20,11,21,12 X22,15,7,16 X12,21,13,22 X14,6,15,5 X4,14,5,13 X6,18,1,17 X2,19,3,20
Gauss code	{1, -11, 2, -9, 8, -10}, {3, -1, 4, -2, 5, -7, 9, -8, 6, -3, 10, -4, 11, -5, 7, -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{\left(v^2-v+1\right) \left(u v^2-u v+u+v-1\right) \left(u v^2-u v-v^2+v-1\right)}{u v^3} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{9}{q^{9/2}}-4 q^{7/2}+\frac{16}{q^{7/2}}+9 q^{5/2}-\frac{21}{q^{5/2}}-16 q^{3/2}+\frac{24}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{4}{q^{11/2}}+20 \sqrt{q}-\frac{25}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^7+4 a^3 z^5+5 a^3 z^3+a^3 z-a^3 z^{-1} -a z^9-6 a z^7+z^7 a^{-1} -13 a z^5+4 z^5 a^{-1} -11 a z^3+5 z^3 a^{-1} -a z+z a^{-1} +3 a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 a^2 z^{10}-3 z^{10}-8 a^3 z^9-15 a z^9-7 z^9 a^{-1} -10 a^4 z^8-11 a^2 z^8-7 z^8 a^{-2} -8 z^8-8 a^5 z^7+6 a^3 z^7+28 a z^7+10 z^7 a^{-1} -4 z^7 a^{-3} -4 a^6 z^6+13 a^4 z^6+26 a^2 z^6+14 z^6 a^{-2} -z^6 a^{-4} +24 z^6-a^7 z^5+11 a^5 z^5+a^3 z^5-20 a z^5+9 z^5 a^{-3} +5 a^6 z^4-5 a^4 z^4-14 a^2 z^4-6 z^4 a^{-2} +2 z^4 a^{-4} -12 z^4+a^7 z^3-5 a^5 z^3+2 a^3 z^3+12 a z^3-2 z^3 a^{-1} -6 z^3 a^{-3} -2 a^6 z^2-a^4 z^2-z^2 a^{-2} -z^2 a^{-4} -z^2-2 a^3 z-a z+2 z a^{-1} +z a^{-3} +a^4+3 a^2+3-a^3 z^{-1} -3 a z^{-1} -2 a^{-1} 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                     3   3 6                   6 1   -5 4                 10 3     7 2               11 7       -4 0             14 9         5 -2           11 12           1 -4         10 13             -3 -6       6 11               5 -8     3 10                 -7 -10   1 6                   5 -12   3                     -3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\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^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

L11a176 L11a178