L11n22

L11n21

L11n23

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n22 at Knotilus!

Link Presentations

[edit Notes on L11n22's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X9,21,10,20 X3849 X21,18,22,19 X11,14,12,15 X5,13,6,12 X13,5,14,22 X19,11,20,10 X15,2,16,3
Gauss code	{1, 11, -5, -3}, {-8, -1, 2, 5, -4, 10, -7, 8, -9, 7, -11, -2, 3, 6, -10, 4, -6, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{5}-4t(1)t(2)^{4}+4t(1)t(2)^{3}+4t(2)^{2}-4t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {3}{q^{9/2}}}-{\frac {6}{q^{7/2}}}-q^{5/2}+{\frac {6}{q^{5/2}}}+2q^{3/2}-{\frac {5}{q^{3/2}}}+{\frac {1}{q^{13/2}}}-{\frac {3}{q^{11/2}}}-4{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-za^{7}+z^{5}a^{5}+4z^{3}a^{5}+3za^{5}+2a^{5}z^{-1}-z^{7}a^{3}-5z^{5}a^{3}-8z^{3}a^{3}-8za^{3}-4a^{3}z^{-1}+2z^{5}a+8z^{3}a+7za+3az^{-1}-z^{3}a^{-1}-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{2}+3a^{7}z^{3}-3a^{7}z+a^{6}z^{6}-a^{6}z^{4}+a^{6}z^{2}+a^{6}+3a^{5}z^{7}-12a^{5}z^{5}+20a^{5}z^{3}-15a^{5}z+2a^{5}z^{-1}+3a^{4}z^{8}-11a^{4}z^{6}+11a^{4}z^{4}-4a^{4}z^{2}+2a^{4}+a^{3}z^{9}+2a^{3}z^{7}-23a^{3}z^{5}+37a^{3}z^{3}-23a^{3}z+4a^{3}z^{-1}+5a^{2}z^{8}-21a^{2}z^{6}+23a^{2}z^{4}-9a^{2}z^{2}+3a^{2}+az^{9}+z^{7}a^{-1}-16az^{5}-5z^{5}a^{-1}+28az^{3}+8z^{3}a^{-1}-16az-5za^{-1}+3az^{-1}+a^{-1}z^{-1}+2z^{8}-9z^{6}+11z^{4}-5z^{2}+1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                   1 1 4                 1   -1 2               3 1   2 0             2 1     -1 -2         1 4 3       0 -4         4 3         -1 -6       3 3           0 -8   1 2 4             3 -10   2 2               0 -12   2                 2 -14 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

L11n21 L11n23