L11a218

L11a217

L11a219

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a218 at Knotilus!

Link Presentations

[edit Notes on L11a218's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X20,10,21,9 X22,12,7,11 X10,22,11,21 X16,6,17,5 X18,16,19,15 X14,20,15,19 X2738 X4,14,5,13 X6,18,1,17
Gauss code	{1, -9, 2, -10, 6, -11}, {9, -1, 3, -5, 4, -2, 10, -8, 7, -6, 11, -7, 8, -3, 5, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{2}v^{4}-3u^{2}v^{3}+3u^{2}v^{2}-3u^{2}v+u^{2}-3uv^{4}+5uv^{3}-5uv^{2}+5uv-3u+v^{4}-3v^{3}+3v^{2}-3v+2}{uv^{2}}}$ (db)
Jones polynomial	$q^{23/2}-3q^{21/2}+6q^{19/2}-10q^{17/2}+13q^{15/2}-14q^{13/2}+14q^{11/2}-12q^{9/2}+8q^{7/2}-6q^{5/2}+2q^{3/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^{5}a^{-9}+3z^{3}a^{-9}+2za^{-9}-z^{7}a^{-7}-4z^{5}a^{-7}-5z^{3}a^{-7}-2za^{-7}+a^{-7}z^{-1}-z^{7}a^{-5}-4z^{5}a^{-5}-5z^{3}a^{-5}-4za^{-5}-3a^{-5}z^{-1}+z^{5}a^{-3}+4z^{3}a^{-3}+5za^{-3}+2a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-2z^{9}a^{-5}-6z^{9}a^{-7}-4z^{9}a^{-9}-2z^{8}a^{-4}-3z^{8}a^{-6}-8z^{8}a^{-8}-7z^{8}a^{-10}-z^{7}a^{-3}+3z^{7}a^{-5}+14z^{7}a^{-7}+3z^{7}a^{-9}-7z^{7}a^{-11}+7z^{6}a^{-4}+16z^{6}a^{-6}+29z^{6}a^{-8}+15z^{6}a^{-10}-5z^{6}a^{-12}+5z^{5}a^{-3}+9z^{5}a^{-5}-3z^{5}a^{-7}+8z^{5}a^{-9}+12z^{5}a^{-11}-3z^{5}a^{-13}-5z^{4}a^{-4}-11z^{4}a^{-6}-28z^{4}a^{-8}-16z^{4}a^{-10}+5z^{4}a^{-12}-z^{4}a^{-14}-9z^{3}a^{-3}-17z^{3}a^{-5}-5z^{3}a^{-7}-10z^{3}a^{-9}-10z^{3}a^{-11}+3z^{3}a^{-13}-3z^{2}a^{-4}-2z^{2}a^{-6}+9z^{2}a^{-8}+6z^{2}a^{-10}-z^{2}a^{-12}+z^{2}a^{-14}+7za^{-3}+10za^{-5}+3za^{-7}+2za^{-9}+2za^{-11}+3a^{-4}+3a^{-6}+a^{-8}-2a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}$ (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 8 9 χ 24                       1 -1 22                     2   2 20                   4 1   -3 18                 6 2     4 16               7 4       -3 14             7 6         1 12           7 7           0 10         5 7             -2 8       4 8               4 6     2 4                 -2 4   1 5                   4 2   1                     -1 0 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=4$ $i=6$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

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