L11a377

L11a376

L11a378

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a377 at Knotilus!

Link Presentations

[edit Notes on L11a377's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{t(2)^4 t(1)^4-t(2)^3 t(1)^4-2 t(2)^4 t(1)^3+5 t(2)^3 t(1)^3-4 t(2)^2 t(1)^3+t(2) t(1)^3+t(2)^4 t(1)^2-5 t(2)^3 t(1)^2+7 t(2)^2 t(1)^2-5 t(2) t(1)^2+t(1)^2+t(2)^3 t(1)-4 t(2)^2 t(1)+5 t(2) t(1)-2 t(1)-t(2)+1}{t(1)^2 t(2)^2}$ (db)
Jones polynomial	$q^{17/2}-3 q^{15/2}+6 q^{13/2}-10 q^{11/2}+13 q^{9/2}-15 q^{7/2}+14 q^{5/2}-13 q^{3/2}+9 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^9 a^{-3} +z^7 a^{-1} -7 z^7 a^{-3} +z^7 a^{-5} +5 z^5 a^{-1} -18 z^5 a^{-3} +5 z^5 a^{-5} +8 z^3 a^{-1} -20 z^3 a^{-3} +8 z^3 a^{-5} +5 z a^{-1} -8 z a^{-3} +4 z a^{-5} + a^{-1} z^{-1} - a^{-3} z^{-1}$ (db)
Kauffman polynomial	$z^4 a^{-10} -z^2 a^{-10} +3 z^5 a^{-9} -3 z^3 a^{-9} +z a^{-9} +5 z^6 a^{-8} -5 z^4 a^{-8} +2 z^2 a^{-8} +6 z^7 a^{-7} -6 z^5 a^{-7} +z^3 a^{-7} +z a^{-7} +6 z^8 a^{-6} -8 z^6 a^{-6} +3 z^4 a^{-6} +5 z^9 a^{-5} -10 z^7 a^{-5} +11 z^5 a^{-5} -11 z^3 a^{-5} +4 z a^{-5} +2 z^{10} a^{-4} +2 z^8 a^{-4} -16 z^6 a^{-4} +16 z^4 a^{-4} -6 z^2 a^{-4} +9 z^9 a^{-3} -32 z^7 a^{-3} +40 z^5 a^{-3} -27 z^3 a^{-3} +9 z a^{-3} - a^{-3} z^{-1} +2 z^{10} a^{-2} -z^8 a^{-2} -15 z^6 a^{-2} +20 z^4 a^{-2} -7 z^2 a^{-2} + a^{-2} +4 z^9 a^{-1} +a z^7-15 z^7 a^{-1} -4 a z^5+16 z^5 a^{-1} +4 a z^3-8 z^3 a^{-1} -a z+4 z a^{-1} - a^{-1} z^{-1} +3 z^8-12 z^6+13 z^4-4 z^2$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     2   2 14                   4 1   -3 12                 6 2     4 10               7 4       -3 8             8 6         2 6           7 8           1 4         6 7             -1 2       4 8               4 0     2 5                 -3 -2   1 4                   3 -4   2                     -2 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a376 L11a378