L10n4

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n4 at Knotilus!

Link Presentations

[edit Notes on L10n4's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X3849 X5,11,6,10 X13,5,14,20 X11,19,12,18 X19,13,20,12 X15,2,16,3
Gauss code	{1, 10, -5, -3}, {-6, -1, 2, 5, -4, 6, -8, 9, -7, 4, -10, -2, 3, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{5}-2uv^{4}+2uv^{3}-2uv^{2}-2v^{3}+2v^{2}-2v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+2q^{5/2}-3q^{3/2}+4{\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {5}{q^{3/2}}}-{\frac {4}{q^{5/2}}}+{\frac {2}{q^{7/2}}}-{\frac {2}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-a^{3}z^{5}+6az^{5}-z^{5}a^{-1}-5a^{3}z^{3}+12az^{3}-4z^{3}a^{-1}+a^{5}z-8a^{3}z+9az-4za^{-1}+2a^{5}z^{-1}-4a^{3}z^{-1}+3az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$3a^{5}z^{3}-7a^{5}z+2a^{5}z^{-1}+a^{4}z^{6}-2a^{4}z^{4}+3a^{4}z^{2}-2a^{4}+2a^{3}z^{7}-9a^{3}z^{5}+z^{5}a^{-3}+20a^{3}z^{3}-3z^{3}a^{-3}-16a^{3}z+za^{-3}+4a^{3}z^{-1}+a^{2}z^{8}-2a^{2}z^{6}+2z^{6}a^{-2}-6z^{4}a^{-2}+7a^{2}z^{2}+3z^{2}a^{-2}-3a^{2}-a^{-2}+4az^{7}+2z^{7}a^{-1}-16az^{5}-6z^{5}a^{-1}+25az^{3}+5z^{3}a^{-1}-13az-3za^{-1}+3az^{-1}+a^{-1}z^{-1}+z^{8}-z^{6}-4z^{4}+7z^{2}-3$ (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   \ -4 -3 -2 -1 0 1 2 3 4 χ 8                 1 1 6               1   -1 4             2 1   1 2           2 1     -1 0         3 2       1 -2       3 3         0 -4     1 2           -1 -6   1 3             2 -8 1 1               0 -10 2                 2

Integral Khovanov Homology (db, data source)

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

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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