L11n231

L11n230

L11n232

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n231 at Knotilus!

Link Presentations

[edit Notes on L11n231's Link Presentations]

Planar diagram presentation	X10,1,11,2 X16,11,17,12 X5,21,6,20 X12,4,13,3 X14,8,15,7 X6,14,7,13 X17,9,18,22 X21,19,22,18 X8,9,1,10 X19,5,20,4 X2,16,3,15
Gauss code	{1, -11, 4, 10, -3, -6, 5, -9}, {9, -1, 2, -4, 6, -5, 11, -2, -7, 8, -10, 3, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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