L11n321

L11n320

L11n322

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n321 at Knotilus!

Link Presentations

[edit Notes on L11n321's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X9,18,10,19 X17,1,18,4 X19,13,20,22 X13,10,14,11 X21,5,22,12 X11,21,12,20
Gauss code	{1, -4, -3, 7}, {-2, -1, 5, 3, -6, 9, -11, 10}, {-9, 2, 4, -5, -7, 6, -8, 11, -10, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-uv^{2}w^{3}+2uv^{2}w^{2}-uv^{2}w+uvw^{3}-4uvw^{2}+2uvw+uw^{2}-uw+v^{3}w^{2}-v^{3}w-2v^{2}w^{2}+4v^{2}w-v^{2}+vw^{2}-2vw+v}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-2q^{3}+5q^{2}-6q+9-8q^{-1}+9q^{-2}-6q^{-3}+4q^{-4}-2q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{2}z^{6}+a^{4}z^{4}-5a^{2}z^{4}+3z^{4}+3a^{4}z^{2}-11a^{2}z^{2}-2z^{2}a^{-2}+9z^{2}+3a^{4}-9a^{2}-2a^{-2}+8+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-4a^{6}z^{4}+4a^{6}z^{2}-a^{6}+2a^{5}z^{7}-7a^{5}z^{5}+6a^{5}z^{3}-a^{5}z+2a^{4}z^{8}-5a^{4}z^{6}+2a^{4}z^{4}-2a^{4}z^{2}-a^{4}z^{-2}+3a^{4}+a^{3}z^{9}+a^{3}z^{7}-10a^{3}z^{5}+13a^{3}z^{3}+3z^{3}a^{-3}-8a^{3}z-2za^{-3}+2a^{3}z^{-1}+5a^{2}z^{8}-16a^{2}z^{6}+z^{6}a^{-2}+24a^{2}z^{4}+4z^{4}a^{-2}-23a^{2}z^{2}-7z^{2}a^{-2}-2a^{2}z^{-2}+11a^{2}+3a^{-2}+az^{9}+2az^{7}+3z^{7}a^{-1}-9az^{5}-6z^{5}a^{-1}+15az^{3}+11z^{3}a^{-1}-12az-7za^{-1}+2az^{-1}+3z^{8}-9z^{6}+22z^{4}-24z^{2}-z^{-2}+11$ (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 7                   2 -2 5                 3   3 3               3 2   -1 1             6 3     3 -1           5 6       1 -3         4 3         1 -5       2 5           3 -7     2 4             -2 -9   1 3               2 -11   1                 -1 -13 1                   1

Integral Khovanov Homology (db, data source)

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

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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