L11n156

L11n155

L11n157

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n156 at Knotilus!

Link Presentations

[edit Notes on L11n156's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X11,21,12,20 X5,12,6,13 X19,4,20,5 X14,18,15,17 X16,8,17,7 X22,16,7,15 X18,14,19,13 X6,21,1,22
Gauss code	{1, -2, 3, 6, -5, -11}, {8, -1, 2, -3, -4, 5, 10, -7, 9, -8, 7, -10, -6, 4, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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