L11n14

L11n13

L11n15

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n14 at Knotilus!

Link Presentations

[edit Notes on L11n14's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X17,1,18,4 X5,10,6,11 X3849 X11,22,12,5 X13,20,14,21 X19,14,20,15 X21,12,22,13 X9,18,10,19 X2,16,3,15
Gauss code	{1, -11, -5, 3}, {-4, -1, 2, 5, -10, 4, -6, 9, -7, 8, 11, -2, -3, 10, -8, 7, -9, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {3(t(1)-1)(t(2)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {3}{q^{5/2}}}+{\frac {2}{q^{3/2}}}+{\frac {1}{q^{19/2}}}-{\frac {1}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {3}{q^{11/2}}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}(-z)-a^{9}z^{-1}+a^{7}z^{3}+2a^{7}z+2a^{7}z^{-1}+a^{5}z^{3}-a^{5}z^{-1}+a^{3}z^{3}+a^{3}z+a^{3}z^{-1}-2az-az^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{10}+7z^{6}a^{10}-16z^{4}a^{10}+13z^{2}a^{10}-3a^{10}-z^{9}a^{9}+6z^{7}a^{9}-11z^{5}a^{9}+7z^{3}a^{9}-3za^{9}+a^{9}z^{-1}-3z^{8}a^{8}+18z^{6}a^{8}-35z^{4}a^{8}+26z^{2}a^{8}-7a^{8}-z^{9}a^{7}+4z^{7}a^{7}-4z^{5}a^{7}+4z^{3}a^{7}-6za^{7}+2a^{7}z^{-1}-2z^{8}a^{6}+9z^{6}a^{6}-14z^{4}a^{6}+13z^{2}a^{6}-4a^{6}-2z^{7}a^{5}+5z^{5}a^{5}+z^{3}a^{5}-3za^{5}+a^{5}z^{-1}-2z^{6}a^{4}+4z^{4}a^{4}-2z^{5}a^{3}+4z^{3}a^{3}-3za^{3}+a^{3}z^{-1}-z^{4}a^{2}-a^{2}-3za+az^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ 0                   2 2 -2                 2 2 0 -4               1     1 -6             2 2     0 -8           2 1       1 -10         1 2         1 -12       2 2           0 -14       1             1 -16   1 2               -1 -18                     0 -20 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

L11n13 L11n15