L10n32

L10n31

L10n33

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n32 at Knotilus!

Link Presentations

[edit Notes on L10n32's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X7,16,8,17 X17,20,18,5 X11,19,12,18 X19,11,20,10 X9,14,10,15 X15,8,16,9 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, -3, 8, -7, 6, -5, -2, 10, 7, -8, 3, -4, 5, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	0 (db)
Jones polynomial	$-q^{7/2}+q^{5/2}-q^{3/2}-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{5/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-a^{3}z^{3}-3a^{3}z-2a^{3}z^{-1}-za^{-3}-a^{-3}z^{-1}+z^{3}a^{-1}+az^{-1}+3za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{8}-a^{2}z^{8}-a^{5}z^{7}-2a^{3}z^{7}-az^{7}+6a^{4}z^{6}+7a^{2}z^{6}-z^{6}a^{-2}+6a^{5}z^{5}+13a^{3}z^{5}+7az^{5}-z^{5}a^{-1}-z^{5}a^{-3}-10a^{4}z^{4}-14a^{2}z^{4}+4z^{4}a^{-2}-10a^{5}z^{3}-23a^{3}z^{3}-12az^{3}+5z^{3}a^{-1}+4z^{3}a^{-3}+6a^{4}z^{2}+11a^{2}z^{2}-2z^{2}a^{-2}+3z^{2}+5a^{5}z+13a^{3}z+6az-5za^{-1}-3za^{-3}-a^{4}-3a^{2}-a^{-2}-2-a^{5}z^{-1}-2a^{3}z^{-1}-az^{-1}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                       0 4                 1 1   0 2             2 1       1 0             2 1       1 -2         1 2 2         1 -4       1               -1 -6       1 1             0 -8   1 1                 0 -10                       0 -12 1                     1

Integral Khovanov Homology (db, data source)

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

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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