L11a528

L11a527

L11a529

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a528 at Knotilus!

Link Presentations

[edit Notes on L11a528's Link Presentations]

Planar diagram presentation	X8192 X14,3,15,4 X22,18,13,17 X16,11,17,12 X12,15,7,16 X10,22,11,21 X20,10,21,9 X18,6,19,5 X2738 X4,13,5,14 X6,20,1,19
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 7, -6, 4, -5}, {10, -2, 5, -4, 3, -8, 11, -7, 6, -3}

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) \left(t(2) t(3)^2 t(1)^2-t(3)^2 t(1)^2+t(2) t(1)^2+t(2)^2 t(3) t(1)^2-t(2) t(3) t(1)^2+t(3) t(1)^2-t(1)^2+t(2)^2 t(1)+t(2)^2 t(3)^2 t(1)-2 t(2) t(3)^2 t(1)+t(3)^2 t(1)-2 t(2) t(1)-t(2)^2 t(3) t(1)-t(3) t(1)+t(1)-t(2)^2-t(2)^2 t(3)^2+t(2) t(3)^2+t(2)+t(2)^2 t(3)-t(2) t(3)+t(3)\right)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^6+3 q^5-6 q^4+10 q^3-13 q^2+15 q-14+14 q^{-1} -9 q^{-2} +7 q^{-3} -3 q^{-4} + q^{-5} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-2} +2 z^6-3 a^2 z^4+2 z^4 a^{-2} -z^4 a^{-4} +8 z^4+a^4 z^2-9 a^2 z^2-z^2 a^{-2} -2 z^2 a^{-4} +11 z^2+2 a^4-7 a^2-2 a^{-2} +7+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 a^2 z^{10}+2 z^{10}+3 a^3 z^9+11 a z^9+8 z^9 a^{-1} +a^4 z^8-a^2 z^8+13 z^8 a^{-2} +11 z^8-14 a^3 z^7-44 a z^7-17 z^7 a^{-1} +13 z^7 a^{-3} -5 a^4 z^6-28 a^2 z^6-36 z^6 a^{-2} +10 z^6 a^{-4} -69 z^6+20 a^3 z^5+46 a z^5-8 z^5 a^{-1} -28 z^5 a^{-3} +6 z^5 a^{-5} +10 a^4 z^4+63 a^2 z^4+28 z^4 a^{-2} -14 z^4 a^{-4} +3 z^4 a^{-6} +98 z^4-7 a^3 z^3-5 a z^3+20 z^3 a^{-1} +14 z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} -10 a^4 z^2-46 a^2 z^2-14 z^2 a^{-2} +4 z^2 a^{-4} -54 z^2-3 a^3 z-7 a z-6 z a^{-1} -2 z a^{-3} +5 a^4+13 a^2+4 a^{-2} +13+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 13                       1 -1 11                     2   2 9                   4 1   -3 7                 6 2     4 5               7 4       -3 3             8 6         2 1           8 9           1 -1         6 6             0 -3       5 10               5 -5     2 4                 -2 -7   1 5                   4 -9   2                     -2 -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=3 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/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}^{10}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

L11a527 L11a529