L10n44

L10n43

L10n45

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n44 at Knotilus!

Link Presentations

[edit Notes on L10n44's Link Presentations]

Planar diagram presentation	X8192 X9,19,10,18 X6718 X13,20,14,7 X5,13,6,12 X3,10,4,11 X15,5,16,4 X11,16,12,17 X19,14,20,15 X17,2,18,3
Gauss code	{1, 10, -6, 7, -5, -3}, {3, -1, -2, 6, -8, 5, -4, 9, -7, 8, -10, 2, -9, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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