L11n226

L11n225

L11n227

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n226 at Knotilus!

Link Presentations

[edit Notes on L11n226's Link Presentations]

Planar diagram presentation	X10,1,11,2 X18,11,19,12 X6,9,7,10 X7,21,8,20 X19,1,20,8 X22,15,9,16 X3,12,4,13 X16,6,17,5 X13,4,14,5 X14,21,15,22 X2,18,3,17
Gauss code	{1, -11, -7, 9, 8, -3, -4, 5}, {3, -1, 2, 7, -9, -10, 6, -8, 11, -2, -5, 4, 10, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

L11n225 L11n227