L11a287

L11a286

L11a288

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a287 at Knotilus!

Link Presentations

[edit Notes on L11a287's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(uv+1)\left(uv^{2}-uv+u+v-1\right)\left(uv^{2}-uv-v^{2}+v-1\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$14q^{9/2}-16q^{7/2}+15q^{5/2}-{\frac {1}{q^{5/2}}}-14q^{3/2}+{\frac {3}{q^{3/2}}}+q^{17/2}-3q^{15/2}+6q^{13/2}-11q^{11/2}+10{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-5}+5z^{5}a^{-5}+8z^{3}a^{-5}+3za^{-5}-a^{-5}z^{-1}-z^{9}a^{-3}-7z^{7}a^{-3}-18z^{5}a^{-3}-19z^{3}a^{-3}-5za^{-3}+a^{-3}z^{-1}+z^{7}a^{-1}+5z^{5}a^{-1}+8z^{3}a^{-1}+4za^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}a^{-4}-4z^{9}a^{-1}-9z^{9}a^{-3}-5z^{9}a^{-5}+z^{8}a^{-2}-3z^{8}a^{-4}-7z^{8}a^{-6}-3z^{8}-az^{7}+15z^{7}a^{-1}+29z^{7}a^{-3}+6z^{7}a^{-5}-7z^{7}a^{-7}+11z^{6}a^{-2}+14z^{6}a^{-4}+10z^{6}a^{-6}-5z^{6}a^{-8}+12z^{6}+4az^{5}-19z^{5}a^{-1}-35z^{5}a^{-3}+9z^{5}a^{-7}-3z^{5}a^{-9}-12z^{4}a^{-2}-7z^{4}a^{-4}-4z^{4}a^{-6}+4z^{4}a^{-8}-z^{4}a^{-10}-14z^{4}-4az^{3}+13z^{3}a^{-1}+26z^{3}a^{-3}+z^{3}a^{-5}-5z^{3}a^{-7}+3z^{3}a^{-9}+4z^{2}a^{-2}-2z^{2}a^{-6}-z^{2}a^{-8}+z^{2}a^{-10}+4z^{2}-5za^{-1}-5za^{-3}-za^{-7}-za^{-9}+a^{-4}-a^{-3}z^{-1}-a^{-5}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     2   2 14                   4 1   -3 12                 7 2     5 10               8 5       -3 8             8 6         2 6           7 8           1 4         7 8             -1 2       4 8               4 0     2 6                 -4 -2   1 4                   3 -4   2                     -2 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a286 L11a288