L11a172

L11a171

L11a173

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a172 at Knotilus!

Link Presentations

[edit Notes on L11a172's Link Presentations]

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

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)^2 t(2)^6-t(1) t(2)^6-2 t(1)^2 t(2)^5+3 t(1) t(2)^5-t(2)^5+3 t(1)^2 t(2)^4-5 t(1) t(2)^4+2 t(2)^4-3 t(1)^2 t(2)^3+5 t(1) t(2)^3-3 t(2)^3+2 t(1)^2 t(2)^2-5 t(1) t(2)^2+3 t(2)^2-t(1)^2 t(2)+3 t(1) t(2)-2 t(2)-t(1)+1}{t(1) t(2)^3} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{13}{q^{9/2}}-\frac{15}{q^{7/2}}-q^{5/2}+\frac{14}{q^{5/2}}+3 q^{3/2}-\frac{13}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{10}{q^{11/2}}-6 \sqrt{q}+\frac{9}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^5 z^7-5 a^5 z^5-8 a^5 z^3-4 a^5 z+a^3 z^9+7 a^3 z^7+18 a^3 z^5+20 a^3 z^3+8 a^3 z+a^3 z^{-1} -a z^7-5 a z^5-8 a z^3-5 a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^4-a^{10} z^2+3 a^9 z^5-3 a^9 z^3+5 a^8 z^6-5 a^8 z^4+a^8 z^2+7 a^7 z^7-12 a^7 z^5+10 a^7 z^3-2 a^7 z+7 a^6 z^8-14 a^6 z^6+12 a^6 z^4-2 a^6 z^2+5 a^5 z^9-9 a^5 z^7+5 a^5 z^5-2 a^5 z^3+2 a^5 z+2 a^4 z^{10}+2 a^4 z^8-16 a^4 z^6+16 a^4 z^4-6 a^4 z^2+9 a^3 z^9-33 a^3 z^7+46 a^3 z^5-36 a^3 z^3+11 a^3 z-a^3 z^{-1} +2 a^2 z^{10}-2 a^2 z^8-9 a^2 z^6+11 a^2 z^4-5 a^2 z^2+a^2+4 a z^9-16 a z^7+z^7 a^{-1} +22 a z^5-4 z^5 a^{-1} -17 a z^3+4 z^3 a^{-1} +7 a z-a z^{-1} +3 z^8-12 z^6+13 z^4-3 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 6                       1 1 4                     2   -2 2                   4 1   3 0                 5 2     -3 -2               8 4       4 -4             7 6         -1 -6           8 7           1 -8         6 8             2 -10       4 7               -3 -12     2 6                 4 -14   1 4                   -3 -16   2                     2 -18 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

L11a171 L11a173