L11n100

L11n99

L11n101

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n100 at Knotilus!

Link Presentations

[edit Notes on L11n100's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X20,13,21,14 X7,17,8,16 X19,9,20,8 X9,19,10,18 X17,11,18,10 X22,15,5,16 X14,21,15,22 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, -6, 7, 11, -2, 3, -9, 8, 4, -7, 6, -5, -3, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uv^{3}-3uv^{2}+uv-u-v^{3}+v^{2}-3v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	${\frac {3}{q^{9/2}}}-{\frac {4}{q^{7/2}}}-q^{5/2}+{\frac {4}{q^{5/2}}}+2q^{3/2}-{\frac {5}{q^{3/2}}}-{\frac {2}{q^{11/2}}}-3{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-a^{5}z^{3}-3a^{5}z-2a^{5}z^{-1}+a^{3}z^{5}+3a^{3}z^{3}+2a^{3}z+2a^{3}z^{-1}+az^{5}+3az^{3}-z^{3}a^{-1}+az-az^{-1}-2za^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-a^{4}z^{8}-3a^{2}z^{8}-2z^{8}+4a^{3}z^{7}+3az^{7}-z^{7}a^{-1}+2a^{4}z^{6}+12a^{2}z^{6}+10z^{6}-4a^{5}z^{5}-9a^{3}z^{5}+5z^{5}a^{-1}-a^{6}z^{4}+a^{4}z^{4}-12a^{2}z^{4}-14z^{4}+11a^{5}z^{3}+17a^{3}z^{3}-az^{3}-7z^{3}a^{-1}-a^{6}z^{2}-2a^{4}z^{2}+4a^{2}z^{2}+5z^{2}-3a^{7}z-9a^{5}z-10a^{3}z-2az+2za^{-1}+a^{4}+a^{7}z^{-1}+2a^{5}z^{-1}+2a^{3}z^{-1}+az^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 χ 6                 1 1 4               1   -1 2             2 1   1 0           2 1     -1 -2         3 2       1 -4       2 3         1 -6     2 2           0 -8   1 2             1 -10 1 2               -1 -12 2                 2

Integral Khovanov Homology (db, data source)

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

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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