L10n22

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n22 at Knotilus!

Link Presentations

[edit Notes on L10n22's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X9,16,10,17 X20,17,5,18 X18,13,19,14 X14,19,15,20 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 $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)^{3}}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	${\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {4}{q^{3/2}}}-{\frac {6}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {2}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+2a^{5}z^{3}+4a^{5}z+2a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-3a^{3}z+az^{3}-az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-4a^{8}z^{4}+5a^{8}z^{2}-2a^{8}+2a^{7}z^{7}-7a^{7}z^{5}+6a^{7}z^{3}-2a^{7}z+a^{7}z^{-1}+a^{6}z^{8}+a^{6}z^{6}-13a^{6}z^{4}+14a^{6}z^{2}-5a^{6}+5a^{5}z^{7}-16a^{5}z^{5}+14a^{5}z^{3}-7a^{5}z+2a^{5}z^{-1}+a^{4}z^{8}+2a^{4}z^{6}-12a^{4}z^{4}+11a^{4}z^{2}-3a^{4}+3a^{3}z^{7}-9a^{3}z^{5}+12a^{3}z^{3}-6a^{3}z+2a^{2}z^{6}-3a^{2}z^{4}+3a^{2}z^{2}+a^{2}+4az^{3}-az-az^{-1}+z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 χ 2                 1 -1 0               3   3 -2             3 3   0 -4           3 1     2 -6         2 3       1 -8       3 3         0 -10     1 2           1 -12   1 3             -2 -14   1               1 -16 1                 -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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