L11n384

L11n383

L11n385

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n384 at Knotilus!

Link Presentations

[edit Notes on L11n384's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X10,6,11,5 X8493 X22,18,19,17 X11,20,12,21 X19,12,20,13 X18,22,5,21 X16,10,17,9 X2,14,3,13
Gauss code	{1, -11, 5, -3}, {-8, 7, 9, -6}, {4, -1, 2, -5, 10, -4, -7, 8, 11, -2, 3, -10, 6, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)^{3}}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{6}-3q^{5}+5q^{4}-8q^{3}+11q^{2}-10q+11-7q^{-1}+6q^{-2}-2q^{-3}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}-4z^{4}a^{-2}+z^{4}a^{-4}+3z^{4}-2a^{2}z^{2}-7z^{2}a^{-2}+2z^{2}a^{-4}+7z^{2}-a^{2}-3a^{-2}+a^{-4}+3+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$2z^{9}a^{-1}+2z^{9}a^{-3}+9z^{8}a^{-2}+4z^{8}a^{-4}+5z^{8}+4az^{7}+z^{7}a^{-1}+3z^{7}a^{-5}+a^{2}z^{6}-31z^{6}a^{-2}-13z^{6}a^{-4}+z^{6}a^{-6}-16z^{6}-7az^{5}-13z^{5}a^{-1}-16z^{5}a^{-3}-10z^{5}a^{-5}+6a^{2}z^{4}+41z^{4}a^{-2}+12z^{4}a^{-4}-3z^{4}a^{-6}+32z^{4}+3a^{3}z^{3}+10az^{3}+20z^{3}a^{-1}+21z^{3}a^{-3}+8z^{3}a^{-5}-9a^{2}z^{2}-26z^{2}a^{-2}-6z^{2}a^{-4}+z^{2}a^{-6}-28z^{2}-a^{3}z-5az-9za^{-1}-7za^{-3}-2za^{-5}+2a^{2}+6a^{-2}+2a^{-4}+7-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 13                   1 1 11                 2   -2 9               3 1   2 7             5 2     -3 5           6 3       3 3         4 5         1 1       7 6           1 -1     4 8             4 -3   2 3               -1 -5   4                 4 -7 2                   -2

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-3$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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