L11n322

L11n321

L11n323

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n322 at Knotilus!

Link Presentations

[edit Notes on L11n322's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X9,18,10,19 X4,17,1,18 X19,13,20,22 X13,10,14,11 X21,5,22,12 X11,21,12,20
Gauss code	{1, -4, 3, -7}, {-2, -1, 5, -3, -6, 9, -11, 10}, {-9, 2, 4, -5, 7, 6, -8, 11, -10, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)^{2}(w-1)^{2}}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-2q^{3}+5q^{2}-7q+11-10q^{-1}+11q^{-2}-8q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{2}z^{6}+a^{4}z^{4}-4a^{2}z^{4}+3z^{4}+2a^{4}z^{2}-8a^{2}z^{2}-2z^{2}a^{-2}+8z^{2}+2a^{4}-7a^{2}-2a^{-2}+7+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+3a^{5}z^{7}-9a^{5}z^{5}+7a^{5}z^{3}-a^{5}z+3a^{4}z^{8}-5a^{4}z^{6}-4a^{4}z^{4}+5a^{4}z^{2}-a^{4}z^{-2}+a^{4}+a^{3}z^{9}+6a^{3}z^{7}-22a^{3}z^{5}+17a^{3}z^{3}+3z^{3}a^{-3}-6a^{3}z-2za^{-3}+2a^{3}z^{-1}+6a^{2}z^{8}-11a^{2}z^{6}+z^{6}a^{-2}+5a^{2}z^{4}+4z^{4}a^{-2}-8a^{2}z^{2}-5z^{2}a^{-2}-2a^{2}z^{-2}+7a^{2}+3a^{-2}+az^{9}+6az^{7}+3z^{7}a^{-1}-17az^{5}-4z^{5}a^{-1}+17az^{3}+10z^{3}a^{-1}-10az-7za^{-1}+2az^{-1}+3z^{8}-4z^{6}+10z^{4}-15z^{2}-z^{-2}+9$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 7                   2 -2 5                 3   3 3               4 2   -2 1             7 3     4 -1           5 6       1 -3         6 5         1 -5       4 7           3 -7     2 4             -2 -9   1 4               3 -11   2                 -2 -13 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L11n321 L11n323