L10n67

L10n66

L10n68

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n67 at Knotilus!

Link Presentations

[edit Notes on L10n67's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,16,12,17 X13,19,14,18 X17,20,18,9 X19,13,20,12 X8,16,5,15 X14,8,15,7 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 8, -7}, {10, -2, -3, 6, -4, -8, 7, 3, -5, 4, -6, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)t(3)^{3}-2t(1)t(3)^{2}+t(1)t(2)t(3)^{2}-4t(2)t(3)^{2}+t(3)^{2}+4t(1)t(3)-t(1)t(2)t(3)+2t(2)t(3)-t(3)-t(1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$2q^{2}-3q+6-6q^{-1}+6q^{-2}-5q^{-3}+5q^{-4}-2q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-3z^{2}a^{4}-3a^{4}z^{-2}-5a^{4}+2z^{4}a^{2}+6z^{2}a^{2}+4a^{2}z^{-2}+8a^{2}-4z^{2}-3z^{-2}-6+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-4a^{6}z^{4}+6a^{6}z^{2}+a^{6}z^{-2}-4a^{6}+2a^{5}z^{7}-6a^{5}z^{5}+3a^{5}z^{3}+a^{5}z-a^{5}z^{-1}+a^{4}z^{8}+3a^{4}z^{6}-21a^{4}z^{4}+25a^{4}z^{2}+3a^{4}z^{-2}-14a^{4}+6a^{3}z^{7}-18a^{3}z^{5}+11a^{3}z^{3}+a^{3}z-a^{3}z^{-1}+a^{2}z^{8}+6a^{2}z^{6}-30a^{2}z^{4}+39a^{2}z^{2}+3z^{2}a^{-2}+4a^{2}z^{-2}+a^{-2}z^{-2}-21a^{2}-4a^{-2}+4az^{7}-11az^{5}+z^{5}a^{-1}+9az^{3}+z^{3}a^{-1}+az+za^{-1}-az^{-1}-a^{-1}z^{-1}+4z^{6}-13z^{4}+23z^{2}+3z^{-2}-14$ (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 χ 5                 2 2 3               1   -1 1             5 2   3 -1           4 4     0 -3         2 3 1     0 -5       3 4         1 -7     2 2           0 -9   1 4             3 -11   1               -1 -13 1                 1

Integral Khovanov Homology (db, data source)

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

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.

L10n66 L10n68