L11n327

L11n326

L11n328

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n327 at Knotilus!

Link Presentations

[edit Notes on L11n327's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{2}w^{3}-2uv^{2}w^{2}-uvw^{3}+3uvw^{2}-2uvw-uw^{2}+uw+v^{3}\left(-w^{2}\right)+v^{3}w+2v^{2}w^{2}-3v^{2}w+v^{2}+2vw-v}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$2q^{5}-4q^{4}+6q^{3}-6q^{2}+8q-6+6q^{-1}-3q^{-2}+2q^{-3}-q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$2z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}-a^{2}z^{4}-3z^{4}a^{-2}-3a^{2}z^{2}-10z^{2}a^{-2}-2a^{-2}z^{-2}-2a^{2}-9a^{-2}+z^{6}+5z^{4}+10z^{2}+z^{-2}+8$ (db)
Kauffman polynomial	$3z^{2}a^{-6}-a^{-6}+z^{5}a^{-5}+3z^{3}a^{-5}-za^{-5}+3z^{6}a^{-4}-3z^{4}a^{-4}-a^{-4}z^{-2}+3a^{-4}+a^{3}z^{7}+4z^{7}a^{-3}-5a^{3}z^{5}-10z^{5}a^{-3}+7a^{3}z^{3}+12z^{3}a^{-3}-2a^{3}z-8za^{-3}+2a^{-3}z^{-1}+2a^{2}z^{8}+3z^{8}a^{-2}-10a^{2}z^{6}-8z^{6}a^{-2}+16a^{2}z^{4}+12z^{4}a^{-2}-10a^{2}z^{2}-19z^{2}a^{-2}-2a^{-2}z^{-2}+3a^{2}+11a^{-2}+az^{9}+z^{9}a^{-1}-az^{7}+2z^{7}a^{-1}-9az^{5}-15z^{5}a^{-1}+16az^{3}+18z^{3}a^{-1}-7az-12za^{-1}+2a^{-1}z^{-1}+5z^{8}-21z^{6}+31z^{4}-26z^{2}-z^{-2}+11$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 11                   2 2 9                 3 1 -2 7               3 1   2 5             3 3     0 3           5 3       2 1         4 6         2 -1       2 2           0 -3     1 4             3 -5   1 2               -1 -7   1                 1 -9 1                   -1

Integral Khovanov Homology (db, data source)

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

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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