L11a329

L11a328

L11a330

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a329 at Knotilus!

Link Presentations

[edit Notes on L11a329's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X20,5,21,6 X16,8,17,7 X18,14,19,13 X22,16,9,15 X8,18,1,17 X14,22,15,21 X6,9,7,10 X4,19,5,20
Gauss code	{1, -2, 3, -11, 4, -10, 5, -8}, {10, -1, 2, -3, 6, -9, 7, -5, 8, -6, 11, -4, 9, -7}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {\left(v^{2}-v+1\right)(uv+1)(uv-u+1)(uv-v+1)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+7q^{5/2}-11q^{3/2}+14{\sqrt {q}}-{\frac {18}{\sqrt {q}}}+{\frac {17}{q^{3/2}}}-{\frac {15}{q^{5/2}}}+{\frac {11}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+5a^{3}z^{5}+9a^{3}z^{3}+7a^{3}z+2a^{3}z^{-1}-az^{9}-7az^{7}+z^{7}a^{-1}-19az^{5}+5z^{5}a^{-1}-25az^{3}+9z^{3}a^{-1}-16az+7za^{-1}-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-5a^{3}z^{9}-10az^{9}-5z^{9}a^{-1}-6a^{4}z^{8}-4a^{2}z^{8}-5z^{8}a^{-2}-3z^{8}-5a^{5}z^{7}+9a^{3}z^{7}+31az^{7}+14z^{7}a^{-1}-3z^{7}a^{-3}-3a^{6}z^{6}+11a^{4}z^{6}+19a^{2}z^{6}+14z^{6}a^{-2}-z^{6}a^{-4}+20z^{6}-a^{7}z^{5}+8a^{5}z^{5}-10a^{3}z^{5}-45az^{5}-18z^{5}a^{-1}+8z^{5}a^{-3}+5a^{6}z^{4}-9a^{4}z^{4}-28a^{2}z^{4}-12z^{4}a^{-2}+3z^{4}a^{-4}-29z^{4}+2a^{7}z^{3}-3a^{5}z^{3}+7a^{3}z^{3}+34az^{3}+18z^{3}a^{-1}-4z^{3}a^{-3}-a^{6}z^{2}+4a^{4}z^{2}+14a^{2}z^{2}+7z^{2}a^{-2}-2z^{2}a^{-4}+18z^{2}-a^{7}z+2a^{5}z-6a^{3}z-17az-8za^{-1}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     2   2 6                   5 1   -4 4                 6 2     4 2               8 5       -3 0             10 6         4 -2           8 9           1 -4         7 9             -2 -6       4 8               4 -8     3 7                 -4 -10   1 5                   4 -12   2                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a328 L11a330