L11n331

L11n330

L11n332

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n331 at Knotilus!

Link Presentations

[edit Notes on L11n331's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X13,18,14,19 X9,13,10,22 X11,21,12,20 X19,5,20,12 X21,11,22,10 X4,17,1,18
Gauss code	{1, -4, 3, -11}, {-2, -1, 5, -3, -7, 10, -8, 9}, {-6, 2, 4, -5, 11, 6, -9, 8, -10, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)(t(2)+t(3))}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{-5}+q^{4}+2q^{-4}-q^{3}-3q^{-3}+4q^{2}+5q^{-2}-4q-5q^{-1}+6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}\left(-z^{2}\right)+a^{-4}z^{-2}-a^{4}+a^{-4}+a^{2}z^{4}+2a^{2}z^{2}-2z^{2}a^{-2}-2a^{-2}z^{-2}+2a^{2}-3a^{-2}+z^{4}+z^{2}+z^{-2}+1$ (db)
Kauffman polynomial	$az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+z^{8}a^{-2}+3z^{8}+2a^{3}z^{7}-3az^{7}-5z^{7}a^{-1}+2a^{4}z^{6}-7a^{2}z^{6}-6z^{6}a^{-2}-15z^{6}+a^{5}z^{5}-5a^{3}z^{5}+2az^{5}+9z^{5}a^{-1}+z^{5}a^{-3}-6a^{4}z^{4}+8a^{2}z^{4}+17z^{4}a^{-2}+z^{4}a^{-4}+30z^{4}-3a^{5}z^{3}+a^{3}z^{3}+5az^{3}-z^{3}a^{-3}+3a^{4}z^{2}-2a^{2}z^{2}-20z^{2}a^{-2}-4z^{2}a^{-4}-21z^{2}+a^{5}z+a^{3}z-3az-6za^{-1}-3za^{-3}-a^{4}+9a^{-2}+4a^{-4}+7+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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 9                   1 1 7                     0 5               4 1   3 3             2 2     0 1           4 2       2 -1         3 4         1 -3       2 2           0 -5     1 3             2 -7   1 2               -1 -9   1                 1 -11 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

L11n330 L11n332