L11n189

L11n188

L11n190

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n189 at Knotilus!

Link Presentations

[edit Notes on L11n189's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X5,16,6,17 X15,21,16,20 X21,19,22,18 X19,15,20,14 X2738 X4,11,5,12 X17,6,18,1
Gauss code	{1, -9, 2, -10, -5, 11}, {9, -1, 3, -4, 10, -2, 4, 8, -6, 5, -11, 7, -8, 6, -7, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{3}-2u^{2}v^{2}+u^{2}v+u^{2}-2uv^{3}+5uv^{2}-2uv+v^{4}+v^{3}-2v^{2}+v}{uv^{2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+4q^{5/2}-{\frac {3}{q^{5/2}}}-5q^{3/2}+{\frac {4}{q^{3/2}}}-{\frac {1}{q^{11/2}}}+6{\sqrt {q}}-{\frac {5}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$za^{5}+2a^{5}z^{-1}-z^{3}a^{3}-5za^{3}-3a^{3}z^{-1}+2z^{3}a+3za+az^{-1}-z^{5}a^{-1}-3z^{3}a^{-1}-3za^{-1}+z^{3}a^{-3}+za^{-3}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-a^{2}z^{8}-3z^{8}a^{-2}-4z^{8}-a^{5}z^{7}+4az^{7}-3z^{7}a^{-3}+6a^{2}z^{6}+11z^{6}a^{-2}-z^{6}a^{-4}+18z^{6}+7a^{5}z^{5}+3a^{3}z^{5}-4az^{5}+11z^{5}a^{-1}+11z^{5}a^{-3}+3a^{4}z^{4}-10a^{2}z^{4}-9z^{4}a^{-2}+3z^{4}a^{-4}-25z^{4}-14a^{5}z^{3}-11a^{3}z^{3}-az^{3}-12z^{3}a^{-1}-8z^{3}a^{-3}-7a^{4}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}+11z^{2}+10a^{5}z+10a^{3}z+az+2za^{-1}+za^{-3}+3a^{4}+3a^{2}+1-2a^{5}z^{-1}-3a^{3}z^{-1}-az^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     2   2 6                   2 1   -1 4                 3 2     1 2               3 2       -1 0           1 3 3         -1 -2           3 4           1 -4       1 2 2             -1 -6         3               3 -8     1 1                 0 -10 1                       1 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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.

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