L11n406

L11n405

L11n407

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n406 at Knotilus!

Link Presentations

[edit Notes on L11n406's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X11,22,12,19 X10,4,11,3 X5,21,6,20 X21,5,22,18 X19,12,20,13 X14,9,15,10 X2,14,3,13 X8,15,9,16
Gauss code	{1, -10, 5, -3}, {-8, 6, -7, 4}, {-6, -1, 2, -11, 9, -5, -4, 8, 10, -9, 11, -2, 3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$0$ (db)
Jones polynomial	$q^{3}-q^{2}+2q+q^{-1}+2q^{-4}-q^{-5}+q^{-6}-q^{-7}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-2a^{6}+z^{4}a^{4}+4z^{2}a^{4}+4a^{4}+a^{2}z^{-2}-z^{4}-4z^{2}-2z^{-2}-4+z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$a^{7}z^{7}-6a^{7}z^{5}+10a^{7}z^{3}-4a^{7}z+a^{6}z^{8}-6a^{6}z^{6}+11a^{6}z^{4}-10a^{6}z^{2}+4a^{6}+2a^{5}z^{7}-12a^{5}z^{5}+18a^{5}z^{3}-8a^{5}z+2a^{4}z^{8}-14a^{4}z^{6}+30a^{4}z^{4}-28a^{4}z^{2}+8a^{4}+a^{3}z^{9}-6a^{3}z^{7}+8a^{3}z^{5}-2a^{3}z^{3}+2a^{2}z^{8}-13a^{2}z^{6}+z^{6}a^{-2}+22a^{2}z^{4}-5z^{4}a^{-2}-12a^{2}z^{2}+6z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-4a^{-2}+az^{9}-6az^{7}+z^{7}a^{-1}+10az^{5}-4z^{5}a^{-1}-10az^{3}+8az+4za^{-1}-2az^{-1}-2a^{-1}z^{-1}+z^{8}-4z^{6}-2z^{4}+12z^{2}+2z^{-2}-7$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                         0 3                   2 1   1 1               3 1       2 -1             2 5 2       1 -3           1 2 2         1 -5         1 2 1           0 -7       2 2 2             2 -9       1                 1 -11   1 2 1                 0 -13                         0 -15 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $i=1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{5}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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