L11a209

L11a208

L11a210

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a209 at Knotilus!

Link Presentations

[edit Notes on L11a209's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X22,15,7,16 X10,4,11,3 X16,6,17,5 X4,12,5,11 X12,21,13,22 X20,13,21,14 X14,19,15,20 X2,18,3,17
Gauss code	{1, -11, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -8, 9, -10, 4, -6, 11, -2, 10, -9, 8, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {\left(2v^{2}-3v+2\right)(uv-u+1)(uv-v+1)}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {12}{q^{9/2}}}-q^{7/2}+{\frac {17}{q^{7/2}}}+4q^{5/2}-{\frac {20}{q^{5/2}}}-9q^{3/2}+{\frac {20}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {7}{q^{11/2}}}+13{\sqrt {q}}-{\frac {19}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+4a^{3}z^{5}+3az^{5}-z^{5}a^{-1}-3a^{5}z^{3}+7a^{3}z^{3}+2az^{3}-2z^{3}a^{-1}-3a^{5}z+6a^{3}z-az-za^{-1}-a^{5}z^{-1}+2a^{3}z^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+2a^{8}z^{2}+3a^{7}z^{7}-8a^{7}z^{5}+5a^{7}z^{3}+5a^{6}z^{8}-13a^{6}z^{6}+12a^{6}z^{4}-7a^{6}z^{2}+2a^{6}+5a^{5}z^{9}-11a^{5}z^{7}+12a^{5}z^{5}-11a^{5}z^{3}+4a^{5}z-a^{5}z^{-1}+2a^{4}z^{10}+6a^{4}z^{8}-24a^{4}z^{6}+31a^{4}z^{4}-21a^{4}z^{2}+5a^{4}+11a^{3}z^{9}-25a^{3}z^{7}+27a^{3}z^{5}+z^{5}a^{-3}-18a^{3}z^{3}-z^{3}a^{-3}+8a^{3}z-2a^{3}z^{-1}+2a^{2}z^{10}+10a^{2}z^{8}-29a^{2}z^{6}+4z^{6}a^{-2}+30a^{2}z^{4}-5z^{4}a^{-2}-14a^{2}z^{2}+3a^{2}+6az^{9}-3az^{7}+8z^{7}a^{-1}-8az^{5}-14z^{5}a^{-1}+7az^{3}+8z^{3}a^{-1}+az-3za^{-1}+a^{-1}z^{-1}+9z^{8}-15z^{6}+9z^{4}-2z^{2}-1$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 8 4     -4 0               11 5       6 -2             10 9         -1 -4           10 10           0 -6         7 10             3 -8       5 10               -5 -10     2 7                 5 -12   1 5                   -4 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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