L11n336

L11n335

L11n337

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n336 at Knotilus!

Link Presentations

[edit Notes on L11n336's Link Presentations]

Planar diagram presentation	X6172 X16,12,17,11 X8493 X2,18,3,17 X14,6,15,5 X18,7,19,8 X12,16,5,15 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 $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)(w-1)\left(v^{2}+vw+w^{2}\right)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{7}+3q^{6}-4q^{5}+8q^{4}-7q^{3}+8q^{2}+q^{-2}-7q-3q^{-1}+6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$2z^{4}a^{-2}+z^{4}a^{-4}+4z^{2}a^{-2}-z^{2}a^{-6}-3z^{2}+a^{2}+3a^{-2}-2a^{-4}-2+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}$ (db)
Kauffman polynomial	$2z^{9}a^{-3}+2z^{9}a^{-5}+5z^{8}a^{-2}+8z^{8}a^{-4}+3z^{8}a^{-6}+5z^{7}a^{-1}-z^{7}a^{-3}-5z^{7}a^{-5}+z^{7}a^{-7}-17z^{6}a^{-2}-33z^{6}a^{-4}-14z^{6}a^{-6}+2z^{6}-17z^{5}a^{-1}-16z^{5}a^{-3}-3z^{5}a^{-5}-4z^{5}a^{-7}+15z^{4}a^{-2}+39z^{4}a^{-4}+20z^{4}a^{-6}-4z^{4}+3az^{3}+19z^{3}a^{-1}+18z^{3}a^{-3}+6z^{3}a^{-5}+4z^{3}a^{-7}+a^{2}z^{2}-3z^{2}a^{-2}-17z^{2}a^{-4}-9z^{2}a^{-6}+6z^{2}-2az-6za^{-1}-2za^{-3}+2za^{-5}-a^{2}-a^{-2}-2a^{-4}-2a^{-6}-1-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 15                   1 -1 13                 2   2 11               2 1   -1 9             6 2     4 7           3 4       1 5         5 4         1 3     1 3 3           1 1     4 5             -1 -1   1 4               3 -3   2                 -2 -5 1                   1

Integral Khovanov Homology (db, data source)

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

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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