L11n335

L11n334

L11n336

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n335 at Knotilus!

Link Presentations

[edit Notes on L11n335'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 X20,14,21,13 X22,9,13,10 X10,21,11,22 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 $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)(w-1)(vw-2v-2w+1)}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{2}+5q-9+13q^{-1}-16q^{-2}+17q^{-3}-14q^{-4}+12q^{-5}-6q^{-6}+3q^{-7}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}+2z^{2}a^{6}-2a^{6}z^{-2}-a^{6}-3z^{4}a^{4}-4z^{2}a^{4}+a^{4}z^{-2}+z^{6}a^{2}+2z^{4}a^{2}+2z^{2}a^{2}-z^{4}+1$ (db)
Kauffman polynomial	$2a^{5}z^{9}+2a^{3}z^{9}+5a^{6}z^{8}+12a^{4}z^{8}+7a^{2}z^{8}+3a^{7}z^{7}+9a^{5}z^{7}+15a^{3}z^{7}+9az^{7}-9a^{6}z^{6}-18a^{4}z^{6}-4a^{2}z^{6}+5z^{6}-3a^{7}z^{5}-23a^{5}z^{5}-36a^{3}z^{5}-15az^{5}+z^{5}a^{-1}+6a^{8}z^{4}+16a^{6}z^{4}+7a^{4}z^{4}-9a^{2}z^{4}-6z^{4}+7a^{7}z^{3}+21a^{5}z^{3}+19a^{3}z^{3}+5az^{3}-11a^{8}z^{2}-16a^{6}z^{2}-3a^{4}z^{2}+2a^{2}z^{2}-9a^{7}z-9a^{5}z+6a^{8}+9a^{6}+3a^{4}+1+2a^{7}z^{-1}+2a^{5}z^{-1}-a^{8}z^{-2}-2a^{6}z^{-2}-a^{4}z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                   1 -1 3                 4   4 1               5 1   -4 -1             8 4     4 -3           10 7       -3 -5         7 6         1 -7       7 10           3 -9     5 7             -2 -11   1 7               6 -13 2 5                 -3 -15 3                   3

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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