L11n192

L11n191

L11n193

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n192 at Knotilus!

Link Presentations

[edit Notes on L11n192's Link Presentations]

Planar diagram presentation	X8192 X14,9,15,10 X4758 X16,6,17,5 X18,16,19,15 X6,18,1,17 X19,7,20,22 X11,20,12,21 X21,10,22,11 X2,14,3,13 X12,4,13,3
Gauss code	{1, -10, 11, -3, 4, -6}, {3, -1, 2, 9, -8, -11, 10, -2, 5, -4, 6, -5, -7, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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