L11n218

L11n217

L11n219

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n218 at Knotilus!

Link Presentations

[edit Notes on L11n218's Link Presentations]

Planar diagram presentation	X10,1,11,2 X8,9,1,10 X3,12,4,13 X22,16,9,15 X17,3,18,2 X21,4,22,5 X5,15,6,14 X13,21,14,20 X16,12,17,11 X19,7,20,6 X7,19,8,18
Gauss code	{1, 5, -3, 6, -7, 10, -11, -2}, {2, -1, 9, 3, -8, 7, 4, -9, -5, 11, -10, 8, -6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $i=2$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=6$ ${\mathbb {Z} }$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11n217 L11n219