L11n396

L11n395

L11n397

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n396 at Knotilus!

Link Presentations

[edit Notes on L11n396's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X9,22,10,19 X8493 X21,17,22,16 X11,5,12,18 X5,21,6,20 X17,11,18,10 X19,12,20,13 X2,14,3,13
Gauss code	{1, -11, 5, -3}, {-10, 8, -6, 4}, {-8, -1, 2, -5, -4, 9, -7, 10, 11, -2, 3, 6, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$0$ (db)
Jones polynomial	$q^{6}-2q^{5}+2q^{4}-2q^{3}+q^{2}+q+1+3q^{-1}-2q^{-2}+2q^{-3}-q^{-4}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}+z^{6}-a^{2}z^{4}-6z^{4}a^{-2}+z^{4}a^{-4}+6z^{4}-3a^{2}z^{2}-11z^{2}a^{-2}+3z^{2}a^{-4}+11z^{2}-2a^{2}-6a^{-2}+2a^{-4}+6+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$az^{9}+2z^{9}a^{-1}+z^{9}a^{-3}+2a^{2}z^{8}+5z^{8}a^{-2}+2z^{8}a^{-4}+5z^{8}+a^{3}z^{7}-3az^{7}-9z^{7}a^{-1}-3z^{7}a^{-3}+2z^{7}a^{-5}-11a^{2}z^{6}-33z^{6}a^{-2}-10z^{6}a^{-4}+z^{6}a^{-6}-33z^{6}-5a^{3}z^{5}-8az^{5}-4z^{5}a^{-1}-10z^{5}a^{-3}-9z^{5}a^{-5}+17a^{2}z^{4}+59z^{4}a^{-2}+11z^{4}a^{-4}-4z^{4}a^{-6}+61z^{4}+6a^{3}z^{3}+20az^{3}+32z^{3}a^{-1}+26z^{3}a^{-3}+8z^{3}a^{-5}-14a^{2}z^{2}-40z^{2}a^{-2}-8z^{2}a^{-4}+2z^{2}a^{-6}-44z^{2}-2a^{3}z-10az-18za^{-1}-14za^{-3}-4za^{-5}+4a^{2}+12a^{-2}+4a^{-4}+13-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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 5 6 χ 13                       1 1 11                     1   -1 9                 1 2 1   0 7               1 2 1     0 5             2 3 2       -1 3           3 3 2         2 1         2 6 2           2 -1       2 3 5             4 -3     1 2                 1 -5   1 2 1                 0 -7   1                     1 -9 1                       -1

Integral Khovanov Homology (db, data source)

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

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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