L11n119

L11n118

L11n120

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n119 at Knotilus!

Link Presentations

[edit Notes on L11n119's Link Presentations]

Planar diagram presentation	X6172 X3,15,4,14 X16,10,17,9 X11,21,12,20 X21,9,22,8 X7,19,8,18 X19,13,20,12 X10,16,11,15 X17,5,18,22 X2536 X13,1,14,4
Gauss code	{1, -10, -2, 11}, {10, -1, -6, 5, 3, -8, -4, 7, -11, 2, 8, -3, -9, 6, -7, 4, -5, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(2v^{2}-v+2\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$7q^{9/2}-7q^{7/2}+4q^{5/2}-3q^{3/2}+2q^{17/2}-4q^{15/2}+6q^{13/2}-7q^{11/2}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-2z^{5}a^{-5}+3z^{3}a^{-3}-8z^{3}a^{-5}+2z^{3}a^{-7}+7za^{-3}-11za^{-5}+4za^{-7}+3a^{-3}z^{-1}-5a^{-5}z^{-1}+2a^{-7}z^{-1}$ (db)
Kauffman polynomial	$3z^{4}a^{-10}-5z^{2}a^{-10}+a^{-10}+z^{7}a^{-9}+z^{5}a^{-9}-4z^{3}a^{-9}+z^{8}a^{-8}+z^{6}a^{-8}-3z^{4}a^{-8}+5z^{7}a^{-7}-11z^{5}a^{-7}+12z^{3}a^{-7}-7za^{-7}+2a^{-7}z^{-1}+z^{8}a^{-6}+4z^{6}a^{-6}-12z^{4}a^{-6}+14z^{2}a^{-6}-5a^{-6}+4z^{7}a^{-5}-12z^{5}a^{-5}+22z^{3}a^{-5}-17za^{-5}+5a^{-5}z^{-1}+3z^{6}a^{-4}-6z^{4}a^{-4}+9z^{2}a^{-4}-5a^{-4}+6z^{3}a^{-3}-10za^{-3}+3a^{-3}z^{-1}$ (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   \ 0 1 2 3 4 5 6 7 χ 18               2 -2 16             2   2 14           4 2   -2 12         3 2     1 10       4 4       0 8     3 3         0 6   1 4           3 4 2 3             -1 2 3               3

Integral Khovanov Homology (db, data source)

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

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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