L11n344

L11n343

L11n345

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n344 at Knotilus!

Link Presentations

[edit Notes on L11n344's Link Presentations]

Planar diagram presentation	X6172 X2,16,3,15 X3,10,4,11 X5,14,6,15 X11,22,12,13 X13,12,14,5 X21,1,22,4 X20,17,21,18 X16,7,17,8 X8,20,9,19 X18,10,19,9
Gauss code	{1, -2, -3, 7}, {-4, -1, 9, -10, 11, 3, -5, 6}, {-6, 4, 2, -9, 8, -11, 10, -8, -7, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{2}w^{3}-3uv^{2}w^{2}+2uv^{2}w-uvw^{3}+3uvw^{2}-3uvw-uw^{2}+uw+v^{3}\left(-w^{2}\right)+v^{3}w+3v^{2}w^{2}-3v^{2}w+v^{2}-2vw^{2}+3vw-v}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$2q-3+7q^{-1}-9q^{-2}+11q^{-3}-9q^{-4}+9q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{4}\right)-2a^{6}z^{2}-2a^{6}+a^{4}z^{6}+4a^{4}z^{4}+8a^{4}z^{2}+a^{4}z^{-2}+7a^{4}-3a^{2}z^{4}-9a^{2}z^{2}-2a^{2}z^{-2}-9a^{2}+2z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-6z^{4}a^{8}+3z^{2}a^{8}-a^{8}+4z^{7}a^{7}-6z^{5}a^{7}-z^{3}a^{7}+za^{7}+3z^{8}a^{6}-2z^{6}a^{6}-4z^{4}a^{6}+2z^{2}a^{6}+z^{9}a^{5}+4z^{7}a^{5}-10z^{5}a^{5}+9z^{3}a^{5}-3za^{5}+5z^{8}a^{4}-12z^{6}a^{4}+21z^{4}a^{4}-21z^{2}a^{4}-a^{4}z^{-2}+9a^{4}+z^{9}a^{3}+z^{7}a^{3}-4z^{5}a^{3}+11z^{3}a^{3}-8za^{3}+2a^{3}z^{-1}+2z^{8}a^{2}-7z^{6}a^{2}+22z^{4}a^{2}-28z^{2}a^{2}-2a^{2}z^{-2}+13a^{2}+z^{7}a-z^{5}a+3z^{3}a-5za+2az^{-1}+3z^{4}-8z^{2}-z^{-2}+6$ (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 3                   2 2 1                 1   -1 -1               6 2   4 -3             6 4     -2 -5           5 3       2 -7         4 6         2 -9       5 5           0 -11     2 5             3 -13   1 4               -3 -15   2                 2 -17 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

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