L11a200

L11a199

L11a201

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a200 at Knotilus!

Link Presentations

[edit Notes on L11a200's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X16,5,17,6 X22,18,7,17 X18,12,19,11 X20,14,21,13 X12,20,13,19 X14,22,15,21 X6718 X4,15,5,16
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 6, -8, 7, -9, 11, -4, 5, -6, 8, -7, 9, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{6}-2u^{2}v^{5}+2u^{2}v^{4}-2u^{2}v^{3}+2u^{2}v^{2}+2uv^{5}-3uv^{4}+3uv^{3}-3uv^{2}+2uv+2v^{4}-2v^{3}+2v^{2}-2v+1}{uv^{3}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {4}{q^{9/2}}}-2q^{7/2}+{\frac {6}{q^{7/2}}}+4q^{5/2}-{\frac {9}{q^{5/2}}}-6q^{3/2}+{\frac {9}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+8{\sqrt {q}}-{\frac {10}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{9}+a^{3}z^{7}-8az^{7}+z^{7}a^{-1}+6a^{3}z^{5}-24az^{5}+6z^{5}a^{-1}+12a^{3}z^{3}-33az^{3}+12z^{3}a^{-1}+9a^{3}z-21az+9za^{-1}+3a^{3}z^{-1}-5az^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+a^{7}z+2a^{6}z^{6}-5a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{7}-9a^{5}z^{5}+8a^{5}z^{3}-3a^{5}z+3a^{4}z^{8}-9a^{4}z^{6}+z^{6}a^{-4}+9a^{4}z^{4}-4z^{4}a^{-4}-a^{4}z^{2}+4z^{2}a^{-4}-a^{-4}+3a^{3}z^{9}-13a^{3}z^{7}+2z^{7}a^{-3}+26a^{3}z^{5}-7z^{5}a^{-3}-22a^{3}z^{3}+5z^{3}a^{-3}+12a^{3}z-3a^{3}z^{-1}+a^{2}z^{10}+2z^{8}a^{-2}-11a^{2}z^{6}-5z^{6}a^{-2}+29a^{2}z^{4}-20a^{2}z^{2}+2z^{2}a^{-2}+5a^{2}+5az^{9}+2z^{9}a^{-1}-25az^{7}-7z^{7}a^{-1}+55az^{5}+12z^{5}a^{-1}-54az^{3}-16z^{3}a^{-1}+25az+9za^{-1}-5az^{-1}-2a^{-1}z^{-1}+z^{10}-z^{8}-6z^{6}+19z^{4}-20z^{2}+5$ (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                     1   1 6                   3 1   -2 4                 3 1     2 2               5 3       -2 0             5 3         2 -2           5 6           1 -4         4 4             0 -6       2 5               3 -8     2 4                 -2 -10     2                   2 -12 1 2                     -1 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a199 L11a201