L11n290

L11n289

L11n291

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n290 at Knotilus!

Link Presentations

[edit Notes on L11n290's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X11,18,12,19 X9,21,10,20 X19,5,20,10 X4,15,1,16 X17,22,18,11 X21,16,22,17
Gauss code	{1, -4, 3, -9}, {-2, -1, 5, -3, -7, 8}, {-6, 2, 4, -5, 9, 11, -10, 6, -8, 7, -11, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)(w-1)\left(w^{2}-w+1\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q-1+6q^{-1}-6q^{-2}+9q^{-3}-8q^{-4}+7q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{4}\right)-2a^{6}z^{2}-a^{6}z^{-2}-2a^{6}+a^{4}z^{6}+4a^{4}z^{4}+7a^{4}z^{2}+4a^{4}z^{-2}+7a^{4}-2a^{2}z^{4}-6a^{2}z^{2}-5a^{2}z^{-2}-8a^{2}+z^{2}+2z^{-2}+3$ (db)
Kauffman polynomial	$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-6z^{4}a^{8}+z^{2}a^{8}+4z^{7}a^{7}-8z^{5}a^{7}+2z^{3}a^{7}-2za^{7}+a^{7}z^{-1}+3z^{8}a^{6}-6z^{6}a^{6}+5z^{4}a^{6}-5z^{2}a^{6}-a^{6}z^{-2}+3a^{6}+z^{9}a^{5}+z^{7}a^{5}-6z^{5}a^{5}+13z^{3}a^{5}-13za^{5}+5a^{5}z^{-1}+4z^{8}a^{4}-14z^{6}a^{4}+26z^{4}a^{4}-20z^{2}a^{4}-4a^{4}z^{-2}+10a^{4}+z^{9}a^{3}-3z^{7}a^{3}+4z^{5}a^{3}+10z^{3}a^{3}-17za^{3}+9a^{3}z^{-1}+z^{8}a^{2}-5z^{6}a^{2}+16z^{4}a^{2}-18z^{2}a^{2}-5a^{2}z^{-2}+11a^{2}+z^{5}a+z^{3}a-7za+5az^{-1}+z^{4}-4z^{2}-2z^{-2}+5$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                   1 1 1                     0 -1               6 1   5 -3             4 4     0 -5           5 2       3 -7         3 4         1 -9       4 5           -1 -11     2 3             1 -13   1 4               -3 -15   2                 2 -17 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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