L11n139

L11n138

L11n140

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n139 at Knotilus!

Link Presentations

[edit Notes on L11n139's Link Presentations]

Planar diagram presentation	X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X5,13,6,12 X6718 X16,10,17,9 X20,16,21,15 X22,14,7,13 X14,22,15,21 X19,4,20,5
Gauss code	{1, 4, -3, 11, -5, -6}, {6, -1, 7, 3, -2, 5, 9, -10, 8, -7, -4, 2, -11, -8, 10, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{u^2 v^2-u^2 v+u v^2-u v+u-v+1}{u v}$ (db)
Jones polynomial	$q^{13/2}-q^{11/2}+q^{9/2}-q^{7/2}-\sqrt{q}-\frac{1}{q^{3/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^3 a^{-5} +3 z a^{-5} + a^{-5} z^{-1} -z^5 a^{-3} -5 z^3 a^{-3} -6 z a^{-3} -2 a^{-3} z^{-1} +a z+z a^{-1} +a z^{-1}$ (db)
Kauffman polynomial	$z^8 a^{-6} -7 z^6 a^{-6} +15 z^4 a^{-6} -11 z^2 a^{-6} +2 a^{-6} +z^9 a^{-5} -7 z^7 a^{-5} +15 z^5 a^{-5} -12 z^3 a^{-5} +4 z a^{-5} - a^{-5} z^{-1} +2 z^8 a^{-4} -14 z^6 a^{-4} +29 z^4 a^{-4} -21 z^2 a^{-4} +5 a^{-4} +z^9 a^{-3} -7 z^7 a^{-3} +15 z^5 a^{-3} -14 z^3 a^{-3} +8 z a^{-3} -2 a^{-3} z^{-1} +z^8 a^{-2} -7 z^6 a^{-2} +14 z^4 a^{-2} -10 z^2 a^{-2} +3 a^{-2} +a z^3-z^3 a^{-1} -3 a z+z a^{-1} +a z^{-1} -1$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 14                   1 -1 12                     0 10               1 1   0 8           1 1       0 6             1       1 4       1 2 1         0 2     1               1 0     2 1             1 -2 1                   1 -4 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L11n138 L11n140