L11n181

L11n180

L11n182

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n181 at Knotilus!

Link Presentations

[edit Notes on L11n181's Link Presentations]

Planar diagram presentation	X8192 X3,10,4,11 X5,14,6,15 X16,8,17,7 X22,18,7,17 X15,13,16,12 X20,10,21,9 X11,19,12,18 X13,6,14,1 X19,4,20,5 X2,21,3,22
Gauss code	{1, -11, -2, 10, -3, 9}, {4, -1, 7, 2, -8, 6, -9, 3, -6, -4, 5, 8, -10, -7, 11, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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