L11n334

L11n333

L11n335

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n334 at Knotilus!

Link Presentations

[edit Notes on L11n334's Link Presentations]

Planar diagram presentation	X6172 X11,16,12,17 X8493 X2,18,3,17 X5,14,6,15 X18,7,19,8 X15,12,16,5 X13,20,14,21 X9,13,10,22 X21,11,22,10 X4,19,1,20
Gauss code	{1, -4, 3, -11}, {-5, -1, 6, -3, -9, 10, -2, 7}, {-8, 5, -7, 2, 4, -6, 11, 8, -10, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$0$ (db)
Jones polynomial	$q^{3}-2q^{2}+2q-1+2q^{-1}+q^{-2}+2q^{-4}-2q^{-5}+2q^{-6}-q^{-7}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-a^{6}+z^{4}a^{4}+3z^{2}a^{4}+a^{4}z^{-2}+2a^{4}-2a^{2}z^{-2}-a^{2}-z^{4}-3z^{2}+z^{-2}-1+z^{2}a^{-2}+a^{-2}$ (db)
Kauffman polynomial	$a^{7}z^{7}-5a^{7}z^{5}+6a^{7}z^{3}-2a^{7}z+2a^{6}z^{8}-11a^{6}z^{6}+16a^{6}z^{4}-9a^{6}z^{2}+3a^{6}+a^{5}z^{9}-4a^{5}z^{7}-2a^{5}z^{5}+12a^{5}z^{3}-7a^{5}z+3a^{4}z^{8}-20a^{4}z^{6}+37a^{4}z^{4}-27a^{4}z^{2}-a^{4}z^{-2}+9a^{4}+a^{3}z^{9}-5a^{3}z^{7}+13a^{3}z^{3}-10a^{3}z+2a^{3}z^{-1}+2a^{2}z^{8}-14a^{2}z^{6}+z^{6}a^{-2}+26a^{2}z^{4}-4z^{4}a^{-2}-20a^{2}z^{2}+3z^{2}a^{-2}-2a^{2}z^{-2}+7a^{2}-a^{-2}+2az^{7}+2z^{7}a^{-1}-12az^{5}-9z^{5}a^{-1}+15az^{3}+8z^{3}a^{-1}-6az-za^{-1}+2az^{-1}+z^{8}-4z^{6}+z^{4}+z^{2}-z^{-2}+1$ (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                     1   -1 3                   1 1   0 1               2 2 1     1 -1             1 3 1       1 -3           2 2 3         3 -5         2 4 1           1 -7       1 1 2             2 -9     1 2 1               0 -11   1 1                   0 -13   1                     1 -15 1                       -1

Integral Khovanov Homology (db, data source)

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

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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