L10n17

L10n16

L10n18

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n17 at Knotilus!

Link Presentations

[edit Notes on L10n17's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X5,12,6,13 X3849 X13,17,14,16 X9,15,10,14 X15,11,16,10 X11,20,12,5 X17,2,18,3
Gauss code	{1, 10, -5, -3}, {-4, -1, 2, 5, -7, 8, -9, 4, -6, 7, -8, 6, -10, -2, 3, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-uv^{5}+2uv^{4}-3uv^{3}+uv^{2}+v^{3}-3v^{2}+2v-1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-2{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {3}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+a^{5}z^{5}+5a^{5}z^{3}+8a^{5}z+4a^{5}z^{-1}-a^{3}z^{7}-6a^{3}z^{5}-13a^{3}z^{3}-13a^{3}z-4a^{3}z^{-1}+az^{5}+4az^{3}+4az+az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{2}-a^{8}+3a^{7}z^{3}-2a^{7}z+a^{7}z^{-1}+2a^{6}z^{6}-6a^{6}z^{4}+10a^{6}z^{2}-4a^{6}+3a^{5}z^{7}-12a^{5}z^{5}+20a^{5}z^{3}-14a^{5}z+4a^{5}z^{-1}+a^{4}z^{8}+a^{4}z^{6}-13a^{4}z^{4}+18a^{4}z^{2}-7a^{4}+5a^{3}z^{7}-20a^{3}z^{5}+26a^{3}z^{3}-17a^{3}z+4a^{3}z^{-1}+a^{2}z^{8}-11a^{2}z^{4}+13a^{2}z^{2}-4a^{2}+2az^{7}-8az^{5}+9az^{3}-5az+az^{-1}+z^{6}-4z^{4}+4z^{2}-1$ (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 χ 4                 1 -1 2               1   1 0             2 1   -1 -2           3 1     2 -4         2 3       1 -6       3 2         1 -8     1 2           1 -10   2 3             -1 -12   2               2 -14 1                 -1

Integral Khovanov Homology (db, data source)

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

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.

L10n16 L10n18