L11n319

L11n318

L11n320

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n319 at Knotilus!

Link Presentations

[edit Notes on L11n319's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,5,15,6 X11,21,12,20 X17,11,18,22 X21,17,22,16 X10,13,5,14 X8,20,9,19 X18,8,19,7 X2,9,3,10 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {3, -1, 9, -8, 10, -7}, {-4, -2, 7, -3, 11, 6, -5, -9, 8, 4, -6, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{2}w^{4}-uv^{2}w^{3}-uvw^{4}+3uvw^{3}-2uvw^{2}+2uvw-uv-uw^{3}+uw^{2}-uw+u-v^{2}w^{4}+v^{2}w^{3}-v^{2}w^{2}+v^{2}w+vw^{4}-2vw^{3}+2vw^{2}-3vw+v+w-1}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-2q^{6}+4q^{5}-7q^{4}+10q^{3}-9q^{2}+11q-7+6q^{-1}-3q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+z^{6}-13z^{4}a^{-2}+5z^{4}a^{-4}+4z^{4}-14z^{2}a^{-2}+9z^{2}a^{-4}-z^{2}a^{-6}+5z^{2}-10a^{-2}+8a^{-4}-2a^{-6}+4-5a^{-2}z^{-2}+4a^{-4}z^{-2}-a^{-6}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$2z^{9}a^{-1}+2z^{9}a^{-3}+9z^{8}a^{-2}+5z^{8}a^{-4}+4z^{8}+3az^{7}-z^{7}a^{-1}+4z^{7}a^{-5}+a^{2}z^{6}-34z^{6}a^{-2}-19z^{6}a^{-4}+z^{6}a^{-6}-13z^{6}-9az^{5}-11z^{5}a^{-1}-15z^{5}a^{-3}-13z^{5}a^{-5}-3a^{2}z^{4}+50z^{4}a^{-2}+35z^{4}a^{-4}+2z^{4}a^{-6}+14z^{4}+5az^{3}+14z^{3}a^{-1}+31z^{3}a^{-3}+25z^{3}a^{-5}+3z^{3}a^{-7}+2a^{2}z^{2}-37z^{2}a^{-2}-26z^{2}a^{-4}-4z^{2}a^{-6}-13z^{2}-11za^{-1}-24za^{-3}-18za^{-5}-5za^{-7}+15a^{-2}+12a^{-4}+3a^{-6}+7+5a^{-1}z^{-1}+9a^{-3}z^{-1}+5a^{-5}z^{-1}+a^{-7}z^{-1}-5a^{-2}z^{-2}-4a^{-4}z^{-2}-a^{-6}z^{-2}-2z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 13                   2 -2 11                 2   2 9               5 2   -3 7             5 2     3 5           5 6       1 3         6 4         2 1       3 7           4 -1     3 4             -1 -3   1 4               3 -5   2                 -2 -7 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L11n318 L11n320