L10n88

L10n87

L10n89

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n88 at Knotilus!

Link Presentations

[edit Notes on L10n88's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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