L11n333

L11n332

L11n334

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n333 at Knotilus!

Link Presentations

[edit Notes on L11n333's Link Presentations]

Planar diagram presentation	X6172 X11,16,12,17 X3849 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 X19,1,20,4
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 $u$, $v$, $w$, ...)	$-{\frac {(v+w-1)(vw-v-w)(v-uw)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-7}+2q^{-6}-3q^{-5}+6q^{-4}-5q^{-3}+q^{2}+6q^{-2}-2q-5q^{-1}+5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-a^{6}+z^{4}a^{4}+2z^{2}a^{4}+a^{4}z^{-2}+3a^{4}+z^{4}a^{2}-2a^{2}z^{-2}-3a^{2}-2z^{2}+z^{-2}+a^{-2}$ (db)
Kauffman polynomial	$a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+5a^{4}z^{8}+3a^{2}z^{8}+a^{7}z^{7}-a^{5}z^{7}+a^{3}z^{7}+3az^{7}-10a^{6}z^{6}-23a^{4}z^{6}-12a^{2}z^{6}+z^{6}-5a^{7}z^{5}-10a^{5}z^{5}-16a^{3}z^{5}-11az^{5}+16a^{6}z^{4}+36a^{4}z^{4}+19a^{2}z^{4}-z^{4}+7a^{7}z^{3}+17a^{5}z^{3}+23a^{3}z^{3}+15az^{3}+2z^{3}a^{-1}-11a^{6}z^{2}-31a^{4}z^{2}-20a^{2}z^{2}+z^{2}a^{-2}+z^{2}-2a^{7}z-7a^{5}z-12a^{3}z-8az-za^{-1}+3a^{6}+11a^{4}+11a^{2}-a^{-2}+3+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 5                   1 1 3                 1   -1 1               4 1   3 -1             4 4     0 -3           2 2 1     1 -5         3 4         1 -7       3 2           1 -9     1 4             3 -11   1 2               -1 -13   1                 1 -15 1                   -1

Integral Khovanov Homology (db, data source)

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

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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