L10n99

L10n98

L10n100

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n99 at Knotilus!

Link Presentations

[edit Notes on L10n99's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(x-1)\left(uvwx-uvw-2uvx-uwx+ux-vwx+vx+2wx+x^{2}-x\right)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$-q^{7/2}+2q^{5/2}-6q^{3/2}+7{\sqrt {q}}-{\frac {9}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {9}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {3}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-3}+2a^{5}z^{-1}+a^{3}z^{3}-3a^{3}z^{-3}-3a^{3}z-6a^{3}z^{-1}-za^{-3}-a^{-3}z^{-1}-az^{5}-az^{3}+3az^{-3}+2z^{3}a^{-1}-a^{-1}z^{-3}+2az+5az^{-1}+2za^{-1}$ (db)
Kauffman polynomial	$6a^{5}z^{3}-a^{5}z^{-3}-10a^{5}z+6a^{5}z^{-1}+3a^{4}z^{6}-6a^{4}z^{4}+15a^{4}z^{2}+3a^{4}z^{-2}-13a^{4}+4a^{3}z^{7}-9a^{3}z^{5}+z^{5}a^{-3}+19a^{3}z^{3}-3z^{3}a^{-3}-3a^{3}z^{-3}-23a^{3}z+3za^{-3}+14a^{3}z^{-1}-a^{-3}z^{-1}+a^{2}z^{8}+8a^{2}z^{6}+2z^{6}a^{-2}-24a^{2}z^{4}-3z^{4}a^{-2}+33a^{2}z^{2}+6a^{2}z^{-2}-24a^{2}+a^{-2}+7az^{7}+3z^{7}a^{-1}-15az^{5}-5z^{5}a^{-1}+19az^{3}+3z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-20az-4za^{-1}+12az^{-1}+3a^{-1}z^{-1}+z^{8}+7z^{6}-21z^{4}+18z^{2}+3z^{-2}-11$ (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               2 1 -1 4             4     4 2           3 2     -1 0         6 4       2 -2       5 7         2 -4     4 2           2 -6     5             5 -8 3 4               -1 -10 3                 3

Integral Khovanov Homology (db, data source)

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

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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