L11n79

L11n78

L11n80

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n79 at Knotilus!

Link Presentations

[edit Notes on L11n79's Link Presentations]

Planar diagram presentation	X6172 X16,9,17,10 X4,21,1,22 X11,14,12,15 X3,10,4,11 X5,13,6,12 X13,5,14,22 X15,2,16,3 X20,18,21,17 X18,8,19,7 X8,20,9,19
Gauss code	{1, 8, -5, -3}, {-6, -1, 10, -11, 2, 5, -4, 6, -7, 4, -8, -2, 9, -10, 11, -9, 3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{5}-4uv^{4}+6uv^{3}-4uv^{2}-4v^{3}+6v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {2}{q^{9/2}}}-3q^{7/2}+{\frac {4}{q^{7/2}}}+6q^{5/2}-{\frac {7}{q^{5/2}}}-8q^{3/2}+{\frac {9}{q^{3/2}}}+9{\sqrt {q}}-{\frac {11}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-4a^{3}z^{3}+z^{3}a^{-3}-5a^{3}z+2za^{-3}-2a^{3}z^{-1}+a^{-3}z^{-1}+az^{7}+5az^{5}-2z^{5}a^{-1}+9az^{3}-7z^{3}a^{-1}+7az-7za^{-1}+3az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-3a^{2}z^{8}-3z^{8}a^{-2}-6z^{8}-3a^{3}z^{7}-6az^{7}-6z^{7}a^{-1}-3z^{7}a^{-3}-a^{4}z^{6}+6a^{2}z^{6}+5z^{6}a^{-2}-z^{6}a^{-4}+13z^{6}+7a^{3}z^{5}+24az^{5}+26z^{5}a^{-1}+9z^{5}a^{-3}-2a^{4}z^{4}-10a^{2}z^{4}+6z^{4}a^{-2}+3z^{4}a^{-4}-5z^{4}-3a^{5}z^{3}-15a^{3}z^{3}-32az^{3}-27z^{3}a^{-1}-7z^{3}a^{-3}+2a^{4}z^{2}+7a^{2}z^{2}-8z^{2}a^{-2}-3z^{2}a^{-4}+4a^{5}z+10a^{3}z+16az+13za^{-1}+3za^{-3}-2a^{2}+2a^{-2}+a^{-4}-a^{5}z^{-1}-2a^{3}z^{-1}-3az^{-1}-3a^{-1}z^{-1}-a^{-3}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               4 1   -3 4             4 2     2 2           5 4       -1 0         6 4         2 -2       4 6           2 -4     3 5             -2 -6   1 4               3 -8 1 3                 -2 -10 2                   2

Integral Khovanov Homology (db, data source)

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

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.

L11n78 L11n80