L10n98

L10n97

L10n99

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n98 at Knotilus!

Link Presentations

[edit Notes on L10n98's Link Presentations]

Planar diagram presentation	X6172 X2536 X13,15,14,20 X10,3,11,4 X4,9,1,10 X16,7,17,8 X8,15,5,16 X11,19,12,18 X19,13,20,12 X17,9,18,14
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, -8, 9, -3, 10}, {7, -6, -10, 8, -9, 3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(4)^{2}t(3)^{2}+t(2)t(4)^{2}t(3)^{2}-t(4)^{2}t(3)^{2}-t(2)t(4)t(3)^{2}-t(1)t(4)^{2}t(3)-t(2)t(3)+t(1)t(4)t(3)+t(2)t(4)t(3)+t(1)-t(1)t(2)+t(2)-t(1)t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {4}{q^{7/2}}}-2q^{5/2}+{\frac {2}{q^{5/2}}}+2q^{3/2}-{\frac {5}{q^{3/2}}}-{\frac {1}{q^{11/2}}}-4{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-3}+a^{5}z+3a^{5}z^{-1}-3a^{3}z^{3}-3a^{3}z^{-3}-11a^{3}z-10a^{3}z^{-1}+2az^{5}+10az^{3}+3az^{-3}-2z^{3}a^{-1}-a^{-1}z^{-3}+16az+11az^{-1}-6za^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{8}-a^{2}z^{8}-a^{5}z^{7}-5a^{3}z^{7}-4az^{7}+3a^{4}z^{6}-a^{2}z^{6}-4z^{6}+6a^{5}z^{5}+25a^{3}z^{5}+17az^{5}-2z^{5}a^{-1}+4a^{4}z^{4}+19a^{2}z^{4}-z^{4}a^{-2}+14z^{4}-13a^{5}z^{3}-39a^{3}z^{3}-24az^{3}+2z^{3}a^{-1}-17a^{4}z^{2}-33a^{2}z^{2}-16z^{2}+13a^{5}z+28a^{3}z+21az+3za^{-1}-3za^{-3}+13a^{4}+24a^{2}-a^{-2}+11-6a^{5}z^{-1}-14a^{3}z^{-1}-12az^{-1}-3a^{-1}z^{-1}+a^{-3}z^{-1}-3a^{4}z^{-2}-6a^{2}z^{-2}-3z^{-2}+a^{5}z^{-3}+3a^{3}z^{-3}+3az^{-3}+a^{-1}z^{-3}$ (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 6                 2 2 4               3 3 0 2             1   1 2 0           3 4     1 -2         2 1 1     2 -4       1 4         3 -6     3 1           2 -8   1 4             3 -10                   0 -12 1                 1

Integral Khovanov Homology (db, data source)

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

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.

L10n97 L10n99