L10a167

L10a166

L10a168

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a167 at Knotilus!

Link Presentations

[edit Notes on L10a167's Link Presentations]

Planar diagram presentation	X6172 X2536 X18,12,19,11 X10,3,11,4 X4,9,1,10 X8,18,5,17 X16,8,17,7 X20,14,15,13 X14,16,9,15 X12,20,13,19
Gauss code	{1, -2, 4, -5}, {2, -1, 7, -6}, {5, -4, 3, -10, 8, -9}, {9, -7, 6, -3, 10, -8}

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) t(4)^2 t(3)^2+t(1) t(2) t(4)^2 t(3)^2-2 t(2) t(4)^2 t(3)^2+t(4)^2 t(3)^2+t(1) t(4) t(3)^2+t(2) t(4) t(3)^2-t(4) t(3)^2+t(1) t(4)^2 t(3)-t(1) t(2) t(4)^2 t(3)+t(2) t(4)^2 t(3)+t(1) t(3)+t(2) t(3)-2 t(1) t(4) t(3)+t(1) t(2) t(4) t(3)-2 t(2) t(4) t(3)+t(4) t(3)-t(3)-2 t(1)+t(1) t(2)-t(2)+t(1) t(4)-t(1) t(2) t(4)+t(2) t(4)+1}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 7 q^{9/2}-10 q^{7/2}+8 q^{5/2}-\frac{1}{q^{5/2}}-9 q^{3/2}+\frac{1}{q^{3/2}}-q^{15/2}+3 q^{13/2}-6 q^{11/2}+5 \sqrt{q}-\frac{5}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^7 a^{-3} -2 z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} +a z^3-9 z^3 a^{-1} +10 z^3 a^{-3} -3 z^3 a^{-5} +4 a z-14 z a^{-1} +13 z a^{-3} -3 z a^{-5} +4 a z^{-1} -11 a^{-1} z^{-1} +10 a^{-3} z^{-1} -3 a^{-5} z^{-1} +a z^{-3} -3 a^{-1} z^{-3} +3 a^{-3} z^{-3} - a^{-5} z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-1} -z^9 a^{-3} -5 z^8 a^{-2} -4 z^8 a^{-4} -z^8-a z^7-6 z^7 a^{-3} -7 z^7 a^{-5} +13 z^6 a^{-2} +4 z^6 a^{-4} -7 z^6 a^{-6} +2 z^6+6 a z^5+14 z^5 a^{-1} +27 z^5 a^{-3} +13 z^5 a^{-5} -6 z^5 a^{-7} +6 z^4 a^{-2} +11 z^4 a^{-4} +8 z^4 a^{-6} -3 z^4 a^{-8} +6 z^4-13 a z^3-30 z^3 a^{-1} -30 z^3 a^{-3} -6 z^3 a^{-5} +6 z^3 a^{-7} -z^3 a^{-9} -33 z^2 a^{-2} -16 z^2 a^{-4} -17 z^2+13 a z+28 z a^{-1} +21 z a^{-3} +3 z a^{-5} -3 z a^{-7} +24 a^{-2} +11 a^{-4} - a^{-6} +13-6 a z^{-1} -14 a^{-1} z^{-1} -12 a^{-3} z^{-1} -3 a^{-5} z^{-1} + a^{-7} z^{-1} -6 a^{-2} z^{-2} -3 a^{-4} z^{-2} -3 z^{-2} +a z^{-3} +3 a^{-1} z^{-3} +3 a^{-3} z^{-3} + a^{-5} z^{-3} }[/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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 16                     1 1 14                   3 1 -2 12                 3     3 10               4 3     -1 8             6 3       3 6           5 7         2 4         4 3           1 2       4 8             4 0     1 1               0 -2     4                 4 -4 1 1                   0 -6 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=4 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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.

L10a166 L10a168