L11n198

L11n197

L11n199

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n198 at Knotilus!

Link Presentations

[edit Notes on L11n198's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X5,14,6,15 X7,17,8,16 X15,21,16,20 X18,14,19,13 X21,6,22,7 X22,18,9,17 X19,5,20,4 X2,9,3,10 X8,11,1,12
Gauss code	{1, -10, 2, 9, -3, 7, -4, -11}, {10, -1, 11, -2, 6, 3, -5, 4, 8, -6, -9, 5, -7, -8}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{2}t(1)^{3}-t(2)t(1)^{3}+t(2)^{3}t(1)^{2}-2t(2)^{2}t(1)^{2}+2t(2)t(1)^{2}+2t(2)^{2}t(1)-2t(2)t(1)+t(1)-t(2)^{2}+t(2)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+3q^{5/2}-4q^{3/2}+4{\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {4}{q^{3/2}}}-{\frac {3}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {1}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{3}+z^{3}a^{-3}+3a^{3}z+2za^{-3}+2a^{3}z^{-1}-az^{5}-z^{5}a^{-1}-4az^{3}-3z^{3}a^{-1}-6az-za^{-1}-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-a^{2}z^{8}-2z^{8}a^{-2}-3z^{8}+5az^{7}+3z^{7}a^{-1}-2z^{7}a^{-3}+5a^{2}z^{6}+8z^{6}a^{-2}-z^{6}a^{-4}+14z^{6}-2a^{3}z^{5}-11az^{5}-z^{5}a^{-1}+8z^{5}a^{-3}-a^{4}z^{4}-11a^{2}z^{4}-8z^{4}a^{-2}+4z^{4}a^{-4}-22z^{4}-a^{5}z^{3}+5a^{3}z^{3}+13az^{3}-7z^{3}a^{-3}+a^{4}z^{2}+8a^{2}z^{2}+4z^{2}a^{-2}-3z^{2}a^{-4}+14z^{2}+2a^{5}z-6a^{3}z-11az-za^{-1}+2za^{-3}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums 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 \chi} (fixed $j$, alternation over $r$).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 1   1 6               2 1   -1 4             2 1     1 2           2 2       0 0         3 2         1 -2       2 3           1 -4     1 2             -1 -6     2               2 -8 1 1                 0 -10 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L11n197 L11n199