L11n24

L11n23

L11n25

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n24 at Knotilus!

Link Presentations

[edit Notes on L11n24's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X9,21,10,20 X8493 X21,18,22,19 X11,14,12,15 X5,13,6,12 X13,5,14,22 X19,11,20,10 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {-8, -1, 2, -5, -4, 10, -7, 8, -9, 7, 11, -2, 3, 6, -10, 4, -6, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}+{\frac {1}{q^{9/2}}}-{\frac {3}{q^{7/2}}}-q^{5/2}+{\frac {4}{q^{5/2}}}+3q^{3/2}-{\frac {5}{q^{3/2}}}-q^{11/2}-5{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$az^{5}-a^{3}z^{3}+3az^{3}-2z^{3}a^{-1}+z^{3}a^{-3}-a^{3}z+3az-5za^{-1}+4za^{-3}-za^{-5}+2az^{-1}-4a^{-1}z^{-1}+3a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{7}a^{-5}-6z^{5}a^{-5}+9z^{3}a^{-5}-4za^{-5}+a^{-5}z^{-1}+z^{8}a^{-4}+a^{4}z^{6}-7z^{6}a^{-4}-3a^{4}z^{4}+13z^{4}a^{-4}+a^{4}z^{2}-9z^{2}a^{-4}+a^{-4}+3a^{3}z^{7}+z^{7}a^{-3}-11a^{3}z^{5}-10z^{5}a^{-3}+8a^{3}z^{3}+21z^{3}a^{-3}-3a^{3}z-14za^{-3}+3a^{-3}z^{-1}+3a^{2}z^{8}+2z^{8}a^{-2}-11a^{2}z^{6}-15z^{6}a^{-2}+7a^{2}z^{4}+31z^{4}a^{-2}-a^{2}z^{2}-20z^{2}a^{-2}+a^{2}+3a^{-2}+az^{9}+z^{9}a^{-1}-3z^{7}a^{-1}-15az^{5}-8z^{5}a^{-1}+23az^{3}+27z^{3}a^{-1}-11az-18za^{-1}+2az^{-1}+4a^{-1}z^{-1}+4z^{8}-20z^{6}+28z^{4}-13z^{2}+2$ (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 4 5 6 χ 12                       1 1 10                         0 8                 1 1 1   -1 6               2 1       1 4             2 1 1       -2 2           4 3 1         2 0         3 4             1 -2       2 3 2             1 -4     2 3                 1 -6   1 2                   -1 -8   2                     2 -10 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11n23 L11n25