L11n93

L11n92

L11n94

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n93 at Knotilus!

Link Presentations

[edit Notes on L11n93's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X16,8,17,7 X11,20,12,21 X17,22,18,5 X21,18,22,19 X19,10,20,11 X14,10,15,9 X8,16,9,15 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 8, 7, -4, -2, 11, -8, 9, -3, -5, 6, -7, 4, -6, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {1}{q^{9/2}}}-3q^{7/2}+{\frac {4}{q^{7/2}}}+5q^{5/2}-{\frac {7}{q^{5/2}}}-8q^{3/2}+{\frac {8}{q^{3/2}}}+9{\sqrt {q}}-{\frac {10}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{3}+z^{3}a^{-3}+za^{-3}-az^{5}-z^{5}a^{-1}-az^{3}-2z^{3}a^{-1}+az-2za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-2a^{2}z^{8}-3z^{8}a^{-2}-5z^{8}-a^{3}z^{7}-2az^{7}-4z^{7}a^{-1}-3z^{7}a^{-3}+2a^{2}z^{6}+7z^{6}a^{-2}-z^{6}a^{-4}+10z^{6}-3a^{3}z^{5}+4az^{5}+17z^{5}a^{-1}+10z^{5}a^{-3}-4a^{4}z^{4}-4a^{2}z^{4}-z^{4}a^{-2}+3z^{4}a^{-4}-4z^{4}-a^{5}z^{3}+2a^{3}z^{3}-az^{3}-13z^{3}a^{-1}-9z^{3}a^{-3}+a^{4}z^{2}+2a^{2}z^{2}-2z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}+2az+4za^{-1}+2za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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 5 χ 10                   1 -1 8                 2   2 6               3 1   -2 4             5 2     3 2           4 3       -1 0         6 5         1 -2       4 6           2 -4     3 4             -1 -6   1 4               3 -8   3                 -3 -10 1                   1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=0} $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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