L10n1

L10a174

L10n2

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n1 at Knotilus!

Link Presentations

[edit Notes on L10n1's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X11,19,12,18 X17,5,18,20 X19,13,20,12 X9,16,10,17 X13,2,14,3
Gauss code	{1, 10, -5, -3}, {-4, -1, 2, 5, -9, 4, -6, 8, -10, -2, 3, 9, -7, 6, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-uv^{5}+2uv^{4}-2uv^{3}-2v^{2}+2v-1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	${\frac {2}{q^{9/2}}}-{\frac {4}{q^{7/2}}}+{\frac {3}{q^{5/2}}}+q^{3/2}-{\frac {3}{q^{3/2}}}+{\frac {1}{q^{13/2}}}-{\frac {2}{q^{11/2}}}-2{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+z^{5}a^{5}+5z^{3}a^{5}+7za^{5}+4a^{5}z^{-1}-z^{7}a^{3}-6z^{5}a^{3}-12z^{3}a^{3}-11za^{3}-4a^{3}z^{-1}+z^{5}a+4z^{3}a+3za+az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{2}-a^{8}+2a^{7}z^{3}-3a^{7}z+a^{7}z^{-1}+a^{6}z^{6}-4a^{6}z^{4}+8a^{6}z^{2}-4a^{6}+2a^{5}z^{7}-10a^{5}z^{5}+19a^{5}z^{3}-15a^{5}z+4a^{5}z^{-1}+a^{4}z^{8}-2a^{4}z^{6}-6a^{4}z^{4}+15a^{4}z^{2}-7a^{4}+4a^{3}z^{7}-19a^{3}z^{5}+27a^{3}z^{3}-16a^{3}z+4a^{3}z^{-1}+a^{2}z^{8}-2a^{2}z^{6}-6a^{2}z^{4}+11a^{2}z^{2}-4a^{2}+2az^{7}-9az^{5}+10az^{3}-4az+az^{-1}+z^{6}-4z^{4}+3z^{2}-1$ (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             1 1   0 -2         1 3 1     1 -4         2 2       0 -6       3 2         1 -8   1 1 2           2 -10   2 2             0 -12   1               1 -14 1                 -1

Integral Khovanov Homology (db, data source)

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

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.

L10a174 L10n2