L11n247

L11n246

L11n248

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n247 at Knotilus!

Link Presentations

[edit Notes on L11n247's Link Presentations]

Planar diagram presentation	X12,1,13,2 X3849 X14,6,15,5 X7,18,8,19 X9,21,10,20 X10,11,1,12 X6,14,7,13 X17,4,18,5 X15,11,16,22 X19,3,20,2 X21,17,22,16
Gauss code	{1, 10, -2, 8, 3, -7, -4, 2, -5, -6}, {6, -1, 7, -3, -9, 11, -8, 4, -10, 5, -11, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	0 (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+q^{5/2}-q^{3/2}-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{5/2}}}-{\frac {2}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{3}z^{3}+z^{3}a^{-3}-2a^{3}z+2za^{-3}+az^{5}-z^{5}a^{-1}+5az^{3}-5z^{3}a^{-1}+6az-6za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{4}z^{6}+z^{6}a^{-4}-4a^{4}z^{4}-4z^{4}a^{-4}+2a^{4}z^{2}+2z^{2}a^{-4}+2a^{3}z^{7}+2z^{7}a^{-3}-10a^{3}z^{5}-10z^{5}a^{-3}+11a^{3}z^{3}+11z^{3}a^{-3}-4a^{3}z-4za^{-3}+a^{2}z^{8}+z^{8}a^{-2}-5a^{2}z^{6}-5z^{6}a^{-2}+4a^{2}z^{4}+4z^{4}a^{-2}+3az^{7}+3z^{7}a^{-1}-19az^{5}-19z^{5}a^{-1}+29az^{3}+29z^{3}a^{-1}-12az-12za^{-1}+az^{-1}+a^{-1}z^{-1}+2z^{8}-12z^{6}+16z^{4}-4z^{2}-1$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ 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 χ 10                     1 -1 8                   1   1 6               1 1 1   1 4             1 2 1     0 2           2 2 1       1 0         2 4 2         0 -2       1 2 2           1 -4     1 2 1             0 -6   1 1 1               1 -8   1                   1 -10 1                     -1

Integral Khovanov Homology (db, data source)

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

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.

L11n246 L11n248