L11n232

L11n231

L11n233

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n232 at Knotilus!

Link Presentations

[edit Notes on L11n232's Link Presentations]

Planar diagram presentation	X10,1,11,2 X18,11,19,12 X8,9,1,10 X22,19,9,20 X20,6,21,5 X4,22,5,21 X7,15,8,14 X12,4,13,3 X13,16,14,17 X15,7,16,6 X2,18,3,17
Gauss code	{1, -11, 8, -6, 5, 10, -7, -3}, {3, -1, 2, -8, -9, 7, -10, 9, 11, -2, 4, -5, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{3}\left(-v^{2}\right)+2u^{3}v-u^{3}+u^{2}v^{2}-u^{2}v+u^{2}+uv^{3}-uv^{2}+uv-v^{3}+2v^{2}-v}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$2q^{9/2}-4q^{7/2}+4q^{5/2}-{\frac {1}{q^{5/2}}}-5q^{3/2}+{\frac {3}{q^{3/2}}}-q^{11/2}+4{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-1}+az^{3}-3z^{3}a^{-1}+2z^{3}a^{-3}+az-2za^{-1}+4za^{-3}-za^{-5}+a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{7}a^{-5}-5z^{5}a^{-5}+8z^{3}a^{-5}-5za^{-5}+a^{-5}z^{-1}+2z^{8}a^{-4}-9z^{6}a^{-4}+10z^{4}a^{-4}-2z^{2}a^{-4}-a^{-4}+z^{9}a^{-3}-z^{7}a^{-3}-11z^{5}a^{-3}+18z^{3}a^{-3}-8za^{-3}+a^{-3}z^{-1}+4z^{8}a^{-2}-17z^{6}a^{-2}+17z^{4}a^{-2}+a^{2}z^{2}-4z^{2}a^{-2}+z^{9}a^{-1}+az^{7}-z^{7}a^{-1}-4az^{5}-10z^{5}a^{-1}+5az^{3}+15z^{3}a^{-1}-2az-5za^{-1}+2z^{8}-8z^{6}+7z^{4}-z^{2}$ (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   \ -2 -1 0 1 2 3 4 5 6 χ 12                 1 1 10               1   -1 8             3 1   2 6           2 2     0 4         3 2       1 2     1 2 2         1 0     3 3           0 -2   1 2             1 -4   2               -2 -6 1                 1

Integral Khovanov Homology (db, data source)

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

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.

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