L10n7

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n7 at Knotilus!

Link Presentations

[edit Notes on L10n7's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uv^{3}-3uv^{2}-3v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	${\frac {2}{q^{9/2}}}-{\frac {3}{q^{7/2}}}-q^{5/2}+{\frac {3}{q^{5/2}}}+q^{3/2}-{\frac {3}{q^{3/2}}}-{\frac {2}{q^{11/2}}}-2{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-z^{3}a^{5}-3za^{5}-a^{5}z^{-1}+z^{5}a^{3}+3z^{3}a^{3}-a^{3}z^{-1}+z^{5}a+4z^{3}a+4za+2az^{-1}-z^{3}a^{-1}-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$3a^{7}z-a^{7}z^{-1}+a^{6}z^{4}+a^{6}+2a^{5}z^{5}-4a^{5}z^{3}+3a^{5}z-a^{5}z^{-1}+2a^{4}z^{6}-4a^{4}z^{4}-a^{4}z^{2}+2a^{4}+2a^{3}z^{7}-7a^{3}z^{5}+8a^{3}z^{3}-7a^{3}z+a^{3}z^{-1}+a^{2}z^{8}-3a^{2}z^{6}+2a^{2}z^{4}-4a^{2}z^{2}+3a^{2}+3az^{7}+z^{7}a^{-1}-15az^{5}-6z^{5}a^{-1}+23az^{3}+11z^{3}a^{-1}-13az-6za^{-1}+2az^{-1}+a^{-1}z^{-1}+z^{8}-5z^{6}+7z^{4}-3z^{2}+1$ (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   \ -4 -3 -2 -1 0 1 2 3 4 χ 6                 1 1 4                   0 2             2 1   1 0           1       -1 -2         2 2       0 -4       2 2         0 -6     1 1           0 -8   1 2             1 -10 1 1               0 -12 2                 2

Integral Khovanov Homology (db, data source)

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

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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