L10n87

L10n86

L10n88

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n87 at Knotilus!

Link Presentations

[edit Notes on L10n87's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,19,12,18 X17,9,18,8 X7,17,8,16 X13,15,14,20 X15,5,16,14 X19,13,20,12 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {-7, 5, -4, 3, -8, 6}, {9, -1, -5, 4, 10, -2, -3, 8, -6, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(w-1)\left(uvw^{2}+uw+u+v^{2}w^{3}+v^{2}w^{2}+vw\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{7}-q^{6}+2q^{5}-2q^{4}+2q^{3}-q^{2}+q+1+q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}-7z^{4}a^{-2}+z^{4}-14z^{2}a^{-2}+2z^{2}a^{-4}+z^{2}a^{-6}+5z^{2}-10a^{-2}+3a^{-4}+a^{-6}+6-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-8}-3z^{2}a^{-8}+a^{-8}+z^{5}a^{-7}-2z^{3}a^{-7}+z^{6}a^{-6}-3z^{4}a^{-6}+3z^{2}a^{-6}-a^{-6}+z^{5}a^{-5}-z^{3}a^{-5}+z^{4}a^{-4}-2z^{2}a^{-4}-a^{-4}z^{-2}+4a^{-4}+z^{7}a^{-3}-8z^{5}a^{-3}+19z^{3}a^{-3}-13za^{-3}+2a^{-3}z^{-1}+z^{8}a^{-2}-9z^{6}a^{-2}+26z^{4}a^{-2}-30z^{2}a^{-2}-2a^{-2}z^{-2}+14a^{-2}+z^{7}a^{-1}-8z^{5}a^{-1}+18z^{3}a^{-1}-13za^{-1}+2a^{-1}z^{-1}+z^{8}-8z^{6}+21z^{4}-22z^{2}-z^{-2}+9$ (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 5 6 χ 15                     1 1 13                   1 1 0 11                 1     1 9               1 1     0 7           1 2 1       0 5           1 2         1 3       1 2 1           0 1         2             2 -1     1                 1 -3 1                     1 -5 1                     1

Integral Khovanov Homology (db, data source)

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

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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