L10a156

L10a155

L10a157

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a156 at Knotilus!

Link Presentations

[edit Notes on L10a156's Link Presentations]

Planar diagram presentation	X8192 X18,10,19,9 X6,18,1,17 X16,7,17,8 X10,4,11,3 X14,6,15,5 X4,14,5,13 X20,11,13,12 X12,15,7,16 X2,19,3,20
Gauss code	{1, -10, 5, -7, 6, -3}, {4, -1, 2, -5, 8, -9}, {7, -6, 9, -4, 3, -2, 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(3)-1) (-t(2) t(1)+t(2) t(3) t(1)+t(1)+t(2)-1) (-t(1) t(3)+t(1) t(2) t(3)-t(2) t(3)+t(3)-1)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5- q^{-5} +4 q^4+4 q^{-4} -8 q^3-8 q^{-3} +13 q^2+13 q^{-2} -15 q-15 q^{-1} +18 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^8-a^2 z^6-z^6 a^{-2} +5 z^6-3 a^2 z^4-3 z^4 a^{-2} +8 z^4-2 a^2 z^2-2 z^2 a^{-2} +3 z^2+a^2+ a^{-2} -2+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^5+z^5 a^{-5} -a^5 z^3-z^3 a^{-5} +4 a^4 z^6+4 z^6 a^{-4} -6 a^4 z^4-6 z^4 a^{-4} +2 a^4 z^2+2 z^2 a^{-4} +7 a^3 z^7+7 z^7 a^{-3} -11 a^3 z^5-11 z^5 a^{-3} +4 a^3 z^3+4 z^3 a^{-3} +7 a^2 z^8+7 z^8 a^{-2} -9 a^2 z^6-9 z^6 a^{-2} +2 a^2 z^4+2 z^4 a^{-2} +a^2 z^2+z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^2-2 a^{-2} +3 a z^9+3 z^9 a^{-1} +7 a z^7+7 z^7 a^{-1} -19 a z^5-19 z^5 a^{-1} +9 a z^3+9 z^3 a^{-1} +2 a z+2 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +14 z^8-26 z^6+16 z^4-2 z^2+2 z^{-2} -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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                     1 -1 9                   3   3 7                 5 1   -4 5               8 3     5 3             8 6       -2 1           10 7         3 -1         7 10           3 -3       6 8             -2 -5     3 8               5 -7   1 5                 -4 -9   3                   3 -11 1                     -1

Integral Khovanov Homology (db, data source)

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

L10a155 L10a157