L11n6

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n6 at Knotilus!

Link Presentations

[edit Notes on L11n6's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X8493 X5,11,6,10 X11,19,12,18 X13,20,14,21 X19,5,20,22 X21,12,22,13 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 6, -7, 10, -8, 4, 11, -2, 3, 7, -9, 8, -10, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(v^{2}-3v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {1}{q^{9/2}}}-3q^{7/2}+{\frac {2}{q^{7/2}}}+4q^{5/2}-{\frac {4}{q^{5/2}}}-6q^{3/2}+{\frac {6}{q^{3/2}}}+6{\sqrt {q}}-{\frac {7}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}-3za^{3}-3a^{3}z^{-1}+3z^{3}a+6za+4az^{-1}-z^{5}a^{-1}-3z^{3}a^{-1}-4za^{-1}-2a^{-1}z^{-1}+z^{3}a^{-3}+za^{-3}$ (db)
Kauffman polynomial	$a^{5}z^{3}-2a^{5}z+a^{5}z^{-1}+z^{6}a^{-4}+2a^{4}z^{4}-3z^{4}a^{-4}-3a^{4}z^{2}+z^{2}a^{-4}+a^{4}+a^{3}z^{7}+3z^{7}a^{-3}-4a^{3}z^{5}-11z^{5}a^{-3}+12a^{3}z^{3}+9z^{3}a^{-3}-12a^{3}z-3za^{-3}+3a^{3}z^{-1}+2a^{2}z^{8}+3z^{8}a^{-2}-9a^{2}z^{6}-10z^{6}a^{-2}+18a^{2}z^{4}+6z^{4}a^{-2}-12a^{2}z^{2}-z^{2}a^{-2}+3a^{2}+a^{-2}+az^{9}+z^{9}a^{-1}+2z^{7}a^{-1}-13az^{5}-20z^{5}a^{-1}+30az^{3}+28z^{3}a^{-1}-20az-13za^{-1}+4az^{-1}+2a^{-1}z^{-1}+5z^{8}-20z^{6}+25z^{4}-11z^{2}+2$ (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 10                   1 -1 8                 2   2 6               2 1   -1 4             4 2     2 2         1 3 2       0 0         5 4         1 -2       3 4           1 -4     1 3             -2 -6   1 3               2 -8   1                 -1 -10 1                   1

Integral Khovanov Homology (db, data source)

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

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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