L11n340

L11n339

L11n341

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n340 at Knotilus!

Link Presentations

[edit Notes on L11n340's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,12,19,11 X7,14,8,15 X13,8,14,9 X22,20,13,19 X20,16,21,15 X16,22,17,21 X12,18,5,17 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, 3, -9}, {-5, 4, 7, -8, 9, -3, 6, -7, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(v-1)(w-1)\left(-uvw^{2}+2uvw-uv-uw+v^{2}w+vw^{2}-2vw+v\right)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}-7q^{3}+11q^{2}-13q+14-12q^{-1}+11q^{-2}-5q^{-3}+3q^{-4}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{-4}+2a^{4}z^{-2}+2a^{4}-a^{-4}+a^{2}z^{4}+2z^{4}a^{-2}-2a^{2}z^{2}+3z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-7a^{2}+a^{-2}-z^{6}-2z^{4}+4z^{-2}+5$ (db)
Kauffman polynomial	$z^{5}a^{-5}-2z^{3}a^{-5}+za^{-5}+3z^{6}a^{-4}+6a^{4}z^{4}-5z^{4}a^{-4}-13a^{4}z^{2}+3z^{2}a^{-4}-2a^{4}z^{-2}+9a^{4}-a^{-4}+3a^{3}z^{7}+5z^{7}a^{-3}-6a^{3}z^{5}-8z^{5}a^{-3}+14a^{3}z^{3}+5z^{3}a^{-3}-16a^{3}z-3za^{-3}+5a^{3}z^{-1}+a^{-3}z^{-1}+4a^{2}z^{8}+4z^{8}a^{-2}-8a^{2}z^{6}+18a^{2}z^{4}-10z^{4}a^{-2}-26a^{2}z^{2}+6z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+18a^{2}+az^{9}+z^{9}a^{-1}+11az^{7}+13z^{7}a^{-1}-32az^{5}-35z^{5}a^{-1}+43az^{3}+36z^{3}a^{-1}-32az-20za^{-1}+9az^{-1}+5a^{-1}z^{-1}+8z^{8}-11z^{6}+7z^{4}-10z^{2}-4z^{-2}+11$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 11                   1 -1 9                 2   2 7               5 1   -4 5             6 2     4 3           7 5       -2 1         7 6         1 -1       7 9           2 -3     4 5             -1 -5   1 7               6 -7 2 4                 -2 -9 3                   3

Integral Khovanov Homology (db, data source)

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

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.

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