L10n58

L10n57

L10n59

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n58 at Knotilus!

Link Presentations

[edit Notes on L10n58's Link Presentations]

Planar diagram presentation	X12,1,13,2 X7,17,8,16 X5,1,6,10 X3746 X9,5,10,4 X17,11,18,20 X13,19,14,18 X19,15,20,14 X2,11,3,12 X15,9,16,8
Gauss code	{1, -9, -4, 5, -3, 4, -2, 10, -5, 3}, {9, -1, -7, 8, -10, 2, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v+uv^{2}-2uv+u+v\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$8q^{9/2}-8q^{7/2}+4q^{5/2}-3q^{3/2}-q^{19/2}+3q^{17/2}-5q^{15/2}+8q^{13/2}-8q^{11/2}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-za^{-9}-a^{-9}z^{-1}+3z^{3}a^{-7}+7za^{-7}+5a^{-7}z^{-1}-2z^{5}a^{-5}-8z^{3}a^{-5}-13za^{-5}-8a^{-5}z^{-1}+3z^{3}a^{-3}+7za^{-3}+4a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-11}-2z^{3}a^{-11}+za^{-11}+3z^{6}a^{-10}-7z^{4}a^{-10}+5z^{2}a^{-10}-2a^{-10}+3z^{7}a^{-9}-3z^{5}a^{-9}-4z^{3}a^{-9}+za^{-9}+a^{-9}z^{-1}+z^{8}a^{-8}+8z^{6}a^{-8}-26z^{4}a^{-8}+24z^{2}a^{-8}-9a^{-8}+7z^{7}a^{-7}-14z^{5}a^{-7}+12z^{3}a^{-7}-9za^{-7}+5a^{-7}z^{-1}+z^{8}a^{-6}+8z^{6}a^{-6}-25z^{4}a^{-6}+31z^{2}a^{-6}-14a^{-6}+4z^{7}a^{-5}-10z^{5}a^{-5}+20z^{3}a^{-5}-19za^{-5}+8a^{-5}z^{-1}+3z^{6}a^{-4}-6z^{4}a^{-4}+12z^{2}a^{-4}-8a^{-4}+6z^{3}a^{-3}-10za^{-3}+4a^{-3}z^{-1}$ (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 8 χ 20                 1 1 18               2   -2 16             3 1   2 14           5 2     -3 12         3 3       0 10       5 5         0 8     3 3           0 6   1 5             4 4 2 3               -1 2 3                 3

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=0$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L10n57 L10n59