L11n328

L11n327

L11n329

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n328 at Knotilus!

Link Presentations

[edit Notes on L11n328's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X8493 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 X4,17,1,18
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 $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)(w-1)(vw+1)}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{5}-2q^{4}-q^{-4}+4q^{3}+2q^{-3}-4q^{2}-3q^{-2}+6q+5q^{-1}-4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-4}+a^{-4}z^{-2}+2a^{-4}-a^{2}z^{4}-2z^{4}a^{-2}-3a^{2}z^{2}-7z^{2}a^{-2}-2a^{-2}z^{-2}-2a^{2}-7a^{-2}+z^{6}+5z^{4}+9z^{2}+z^{-2}+7$ (db)
Kauffman polynomial	$az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+2z^{8}a^{-2}+4z^{8}+a^{3}z^{7}-2az^{7}-z^{7}a^{-1}+2z^{7}a^{-3}-10a^{2}z^{6}-7z^{6}a^{-2}+z^{6}a^{-4}-18z^{6}-5a^{3}z^{5}-6az^{5}-8z^{5}a^{-1}-7z^{5}a^{-3}+15a^{2}z^{4}+9z^{4}a^{-2}-2z^{4}a^{-4}+26z^{4}+7a^{3}z^{3}+13az^{3}+14z^{3}a^{-1}+10z^{3}a^{-3}+2z^{3}a^{-5}-9a^{2}z^{2}-10z^{2}a^{-2}+3z^{2}a^{-4}+z^{2}a^{-6}-21z^{2}-2a^{3}z-7az-10za^{-1}-6za^{-3}-za^{-5}+3a^{2}+7a^{-2}+a^{-4}-a^{-6}+9+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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                   1 1 9                 1   -1 7               3 1   2 5             3 3     0 3           3 2 1     2 1         3 5         2 -1       2 2 1         1 -3     1 3             2 -5   1 2               -1 -7   1                 1 -9 1                   -1

Integral Khovanov Homology (db, data source)

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

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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