L10n55

L10n54

L10n56

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n55 at Knotilus!

Link Presentations

[edit Notes on L10n55's Link Presentations]

Planar diagram presentation	X10,1,11,2 X11,17,12,16 X8,9,1,10 X17,9,18,20 X3,12,4,13 X7,14,8,15 X13,6,14,7 X5,18,6,19 X19,4,20,5 X15,2,16,3
Gauss code	{1, 10, -5, 9, -8, 7, -6, -3}, {3, -1, -2, 5, -7, 6, -10, 2, -4, 8, -9, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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