L10n21

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L10n22

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n21 at Knotilus!

Link Presentations

[edit Notes on L10n21's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X9,16,10,17 X17,5,18,20 X13,19,14,18 X19,15,20,14 X15,8,16,9 X2536 X4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 3, 8, -4, -2, 10, -3, -6, 7, -8, 4, -5, 6, -7, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)^{3}}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-4q^{9/2}+5q^{7/2}-6q^{5/2}+5q^{3/2}-{\frac {1}{q^{3/2}}}-q^{13/2}+3q^{11/2}-5{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-2z^{3}a^{-1}+3z^{3}a^{-3}-z^{3}a^{-5}+az-3za^{-1}+3za^{-3}-za^{-5}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-2z^{3}a^{-7}+3z^{6}a^{-6}-8z^{4}a^{-6}+3z^{2}a^{-6}+3z^{7}a^{-5}-8z^{5}a^{-5}+5z^{3}a^{-5}-2za^{-5}+z^{8}a^{-4}+z^{6}a^{-4}-7z^{4}a^{-4}+5z^{2}a^{-4}+4z^{7}a^{-3}-12z^{5}a^{-3}+15z^{3}a^{-3}-6za^{-3}+z^{8}a^{-2}-2z^{6}a^{-2}+3z^{4}a^{-2}+z^{2}a^{-2}+z^{7}a^{-1}-3z^{5}a^{-1}+az^{3}+9z^{3}a^{-1}-2az-6za^{-1}+az^{-1}+a^{-1}z^{-1}+2z^{4}-z^{2}-1$ (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   \ -2 -1 0 1 2 3 4 5 6 χ 14                 1 1 12               2   -2 10             2 1   1 8           3 2     -1 6         3 2       1 4       2 3         1 2     3 3           0 0   1 4             3 -2   1               -1 -4 1                 1

Integral Khovanov Homology (db, data source)

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

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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