L11n87

L11n86

L11n88

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n87 at Knotilus!

Link Presentations

[edit Notes on L11n87's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X14,8,15,7 X15,20,16,21 X9,19,10,18 X19,9,20,8 X17,22,18,5 X21,16,22,17 X10,14,11,13 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 6, -5, -9, 11, -2, 9, -3, -4, 8, -7, 5, -6, 4, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{3}-uv^{2}-u-v^{3}-v+1}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{7/2}}}-q^{5/2}+{\frac {1}{q^{5/2}}}+2q^{3/2}-{\frac {2}{q^{3/2}}}-2{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}-a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}+az^{5}+4az^{3}-z^{3}a^{-1}+3az-2za^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-a^{4}z^{8}-3a^{2}z^{8}-2z^{8}+5a^{3}z^{7}+4az^{7}-z^{7}a^{-1}+6a^{4}z^{6}+17a^{2}z^{6}+11z^{6}-5a^{3}z^{5}+5z^{5}a^{-1}-9a^{4}z^{4}-25a^{2}z^{4}-16z^{4}+a^{5}z^{3}+a^{3}z^{3}-6az^{3}-6z^{3}a^{-1}+a^{6}z^{2}+4a^{4}z^{2}+10a^{2}z^{2}+7z^{2}+a^{5}z+az+2za^{-1}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 χ 6                 1 1 4               1   -1 2             1 1   0 0         1 2 1     0 -2       1 2 1       0 -4       1 2         1 -6     2 1           1 -8 1   1             2 -10 1 1               0

Integral Khovanov Homology (db, data source)

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

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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