L11n250

L11n249

L11n251

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n250 at Knotilus!

Link Presentations

[edit Notes on L11n250's Link Presentations]

Planar diagram presentation	X12,1,13,2 X9,19,10,18 X14,6,15,5 X6,12,7,11 X22,15,11,16 X20,8,21,7 X3948 X16,21,17,22 X4,18,5,17 X10,13,1,14 X19,3,20,2
Gauss code	{1, 11, -7, -9, 3, -4, 6, 7, -2, -10}, {4, -1, 10, -3, 5, -8, 9, 2, -11, -6, 8, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{3}v^{3}-3u^{3}v^{2}+2u^{3}v-u^{2}v^{3}+4u^{2}v^{2}-2u^{2}v-2uv^{2}+4uv-u+2v^{2}-3v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{13/2}-4q^{11/2}+6q^{9/2}-8q^{7/2}+8q^{5/2}-9q^{3/2}+7{\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}+az^{3}-7z^{3}a^{-1}+9z^{3}a^{-3}-4z^{3}a^{-5}+2az-5za^{-1}+8za^{-3}-4za^{-5}+za^{-7}+a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{2}a^{-8}+4z^{3}a^{-7}-za^{-7}+2z^{6}a^{-6}-z^{4}a^{-6}+z^{2}a^{-6}+5z^{7}a^{-5}-14z^{5}a^{-5}+14z^{3}a^{-5}-6za^{-5}+a^{-5}z^{-1}+5z^{8}a^{-4}-15z^{6}a^{-4}+12z^{4}a^{-4}-3z^{2}a^{-4}-a^{-4}+2z^{9}a^{-3}-17z^{5}a^{-3}+18z^{3}a^{-3}-7za^{-3}+a^{-3}z^{-1}+8z^{8}a^{-2}-30z^{6}a^{-2}+29z^{4}a^{-2}-8z^{2}a^{-2}+2z^{9}a^{-1}+az^{7}-4z^{7}a^{-1}-4az^{5}-7z^{5}a^{-1}+5az^{3}+13z^{3}a^{-1}-2az-4za^{-1}+3z^{8}-13z^{6}+16z^{4}-5z^{2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 14                   1 -1 12                 3   3 10               4 2   -2 8             4 2     2 6           4 4       0 4         5 4         1 2       3 5           2 0     2 4             -2 -2   1 3               2 -4   2                 -2 -6 1                   1

Integral Khovanov Homology (db, data source)

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

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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