L11a280

L11a279

L11a281

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a280 at Knotilus!

Link Presentations

[edit Notes on L11a280's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X8,9,1,10 X22,13,9,14 X14,6,15,5 X4,22,5,21 X18,8,19,7 X20,16,21,15 X16,20,17,19 X6,18,7,17
Gauss code	{1, -2, 3, -7, 6, -11, 8, -4}, {4, -1, 2, -3, 5, -6, 9, -10, 11, -8, 10, -9, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+3 t(2) t(1)-t(1)-t(2)+2\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db)
Jones polynomial	$-q^{11/2}+3 q^{9/2}-5 q^{7/2}+8 q^{5/2}-12 q^{3/2}+13 \sqrt{q}-\frac{14}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^3 z^5-z^5 a^{-3} -3 a^3 z^3-3 z^3 a^{-3} -2 a^3 z-z a^{-3} +a z^7+z^7 a^{-1} +4 a z^5+4 z^5 a^{-1} +5 a z^3+4 z^3 a^{-1} +3 a z+a z^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$a^6 z^4-a^6 z^2+z^7 a^{-5} +3 a^5 z^5-4 z^5 a^{-5} -3 a^5 z^3+4 z^3 a^{-5} +a^5 z-z a^{-5} +3 z^8 a^{-4} +5 a^4 z^6-13 z^6 a^{-4} -5 a^4 z^4+16 z^4 a^{-4} +2 a^4 z^2-5 z^2 a^{-4} +3 z^9 a^{-3} +6 a^3 z^7-10 z^7 a^{-3} -7 a^3 z^5+7 z^5 a^{-3} +4 a^3 z^3+z^3 a^{-3} -a^3 z-z a^{-3} +z^{10} a^{-2} +5 a^2 z^8+4 z^8 a^{-2} -4 a^2 z^6-24 z^6 a^{-2} -2 a^2 z^4+28 z^4 a^{-2} +2 a^2 z^2-9 z^2 a^{-2} +3 a z^9+6 z^9 a^{-1} +a z^7-16 z^7 a^{-1} -11 a z^5+10 z^5 a^{-1} +9 a z^3-z^3 a^{-1} -4 a z-2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +z^{10}+6 z^8-20 z^6+16 z^4-5 z^2-1$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                     2   -2 8                   3 1   2 6                 5 2     -3 4               7 3       4 2             6 5         -1 0           8 7           1 -2         6 8             2 -4       4 6               -2 -6     2 6                 4 -8   1 4                   -3 -10   2                     2 -12 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

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.

L11a279 L11a281