L10n96

L10n95

L10n97

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n96 at Knotilus!

Link Presentations

[edit Notes on L10n96's Link Presentations]

Planar diagram presentation	X6172 X2536 X13,15,14,20 X3,11,4,10 X9,1,10,4 X7,17,8,16 X15,5,16,8 X11,19,12,18 X19,13,20,12 X17,9,18,14
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -8, 9, -3, 10}, {-7, 6, -10, 8, -9, 3}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-uvw^{2}x-uvwx^{2}+2uvwx-uvw-uvx+uv+uwx^{2}-uwx+ux+vw^{2}x-vwx+vw+w^{2}x^{2}-w^{2}x-wx^{2}+2wx-w-x}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$-q^{19/2}+2q^{17/2}-5q^{15/2}+5q^{13/2}-8q^{11/2}+6q^{9/2}-7q^{7/2}+3q^{5/2}-3q^{3/2}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-2z^{5}a^{-5}+3z^{3}a^{-3}-9z^{3}a^{-5}+3z^{3}a^{-7}+8za^{-3}-16za^{-5}+9za^{-7}-za^{-9}+5a^{-3}z^{-1}-12a^{-5}z^{-1}+9a^{-7}z^{-1}-2a^{-9}z^{-1}+a^{-3}z^{-3}-3a^{-5}z^{-3}+3a^{-7}z^{-3}-a^{-9}z^{-3}$ (db)
Kauffman polynomial	$-z^{8}a^{-6}-z^{8}a^{-8}-4z^{7}a^{-5}-6z^{7}a^{-7}-2z^{7}a^{-9}-3z^{6}a^{-4}-4z^{6}a^{-6}-3z^{6}a^{-8}-2z^{6}a^{-10}+15z^{5}a^{-5}+18z^{5}a^{-7}+2z^{5}a^{-9}-z^{5}a^{-11}+9z^{4}a^{-4}+18z^{4}a^{-6}+13z^{4}a^{-8}+4z^{4}a^{-10}-6z^{3}a^{-3}-31z^{3}a^{-5}-26z^{3}a^{-7}+2z^{3}a^{-9}+3z^{3}a^{-11}-17z^{2}a^{-4}-33z^{2}a^{-6}-16z^{2}a^{-8}+13za^{-3}+28za^{-5}+21za^{-7}+3za^{-9}-3za^{-11}+13a^{-4}+24a^{-6}+11a^{-8}-a^{-10}-6a^{-3}z^{-1}-14a^{-5}z^{-1}-12a^{-7}z^{-1}-3a^{-9}z^{-1}+a^{-11}z^{-1}-3a^{-4}z^{-2}-6a^{-6}z^{-2}-3a^{-8}z^{-2}+a^{-3}z^{-3}+3a^{-5}z^{-3}+3a^{-7}z^{-3}+a^{-9}z^{-3}$ (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               1   -1 16             4 1   3 14           4 4     0 12         4 1       3 10       3 5         2 8     4 3           1 6     4             4 4 3 3               0 2 3                 3

Integral Khovanov Homology (db, data source)

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

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.

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