L10n20

L10n19

L10n21

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n20 at Knotilus!

Link Presentations

[edit Notes on L10n20's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X11,14,12,15 X8493 X5,13,6,12 X13,5,14,20 X9,16,10,17 X15,10,16,11 X2,18,3,17
Gauss code	{1, -10, 5, -3}, {-6, -1, 2, -5, -8, 9, -4, 6, -7, 4, -9, 8, 10, -2, 3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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