L11a196

L11a195

L11a197

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a196 at Knotilus!

Link Presentations

[edit Notes on L11a196's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X20,13,21,14 X6,12,1,11 X18,15,19,16 X4,18,5,17 X16,6,17,5 X14,19,15,20 X12,21,13,22
Gauss code	{1, -4, 2, -8, 9, -6}, {4, -1, 3, -2, 6, -11, 5, -10, 7, -9, 8, -7, 10, -5, 11, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{2}v^{2}-5u^{2}v+3u^{2}-5uv^{2}+11uv-5u+3v^{2}-5v+2}{uv}}$ (db)
Jones polynomial	$q^{11/2}-3q^{9/2}+5q^{7/2}-9q^{5/2}+11q^{3/2}-13{\sqrt {q}}+{\frac {12}{\sqrt {q}}}-{\frac {11}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+za^{-5}-2a^{3}z^{3}-2z^{3}a^{-3}-2a^{3}z-2za^{-3}-a^{-3}z^{-1}+az^{5}+z^{5}a^{-1}+az^{3}+z^{3}a^{-1}+az+2za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-6az^{9}-3z^{9}a^{-1}-3a^{4}z^{8}-4a^{2}z^{8}-4z^{8}a^{-2}-5z^{8}-a^{5}z^{7}+10a^{3}z^{7}+18az^{7}+3z^{7}a^{-1}-4z^{7}a^{-3}+13a^{4}z^{6}+26a^{2}z^{6}+3z^{6}a^{-2}-4z^{6}a^{-4}+20z^{6}+4a^{5}z^{5}-6a^{3}z^{5}-14az^{5}-z^{5}a^{-1}-3z^{5}a^{-5}-16a^{4}z^{4}-32a^{2}z^{4}+z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-19z^{4}-4a^{5}z^{3}-2a^{3}z^{3}+4az^{3}+5z^{3}a^{-1}+7z^{3}a^{-3}+4z^{3}a^{-5}+6a^{4}z^{2}+11a^{2}z^{2}+z^{2}a^{-6}+6z^{2}+a^{5}z+a^{3}z-az-4za^{-1}-5za^{-3}-2za^{-5}-a^{-2}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     2   2 8                   3 1   -2 6                 6 2     4 4               6 4       -2 2             7 5         2 0           6 7           1 -2         5 6             -1 -4       3 6               3 -6     2 5                 -3 -8   1 3                   2 -10   2                     -2 -12 1                       1

Integral Khovanov Homology (db, data source)

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