L11n36

L11n35

L11n37

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n36 at Knotilus!

Link Presentations

[edit Notes on L11n36's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X11,14,12,15 X3849 X5,13,6,12 X13,5,14,22 X15,19,16,18 X9,17,10,16 X17,11,18,10 X19,2,20,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -9, 10, -4, 6, -7, 4, -8, 9, -10, 8, -11, -2, 3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{5}-4uv^{4}+5uv^{3}-3uv^{2}-3v^{3}+5v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+4q^{5/2}-7q^{3/2}+8{\sqrt {q}}-{\frac {9}{\sqrt {q}}}+{\frac {8}{q^{3/2}}}-{\frac {7}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {2}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-a^{3}z^{5}+5az^{5}-2z^{5}a^{-1}-4a^{3}z^{3}+9az^{3}-8z^{3}a^{-1}+z^{3}a^{-3}+a^{5}z-5a^{3}z+8az-9za^{-1}+3za^{-3}+a^{5}z^{-1}-2a^{3}z^{-1}+3az^{-1}-3a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$3a^{5}z^{3}-4a^{5}z+a^{5}z^{-1}+a^{4}z^{6}+z^{6}a^{-4}+2a^{4}z^{4}-4z^{4}a^{-4}-2a^{4}z^{2}+4z^{2}a^{-4}-a^{-4}+3a^{3}z^{7}+2z^{7}a^{-3}-7a^{3}z^{5}-7z^{5}a^{-3}+13a^{3}z^{3}+7z^{3}a^{-3}-9a^{3}z-4za^{-3}+2a^{3}z^{-1}+a^{-3}z^{-1}+3a^{2}z^{8}+2z^{8}a^{-2}-8a^{2}z^{6}-4z^{6}a^{-2}+14a^{2}z^{4}-3z^{4}a^{-2}-9a^{2}z^{2}+6z^{2}a^{-2}+2a^{2}-2a^{-2}+az^{9}+z^{9}a^{-1}+3az^{7}+2z^{7}a^{-1}-15az^{5}-15z^{5}a^{-1}+24az^{3}+21z^{3}a^{-1}-15az-14za^{-1}+3az^{-1}+3a^{-1}z^{-1}+5z^{8}-14z^{6}+13z^{4}-5z^{2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 1   1 6               3 1   -2 4             4 1     3 2           4 3       -1 0         5 4         1 -2       4 5           1 -4     3 4             -1 -6   1 4               3 -8 1 3                 -2 -10 2                   2

Integral Khovanov Homology (db, data source)

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

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.

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