L11a409

L11a408

L11a410

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a409 at Knotilus!

Link Presentations

[edit Notes on L11a409's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X8,18,9,17 X14,8,15,7 X18,10,19,9 X10,12,5,11 X22,19,11,20 X20,15,21,16 X16,21,17,22 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -3, 5, -6}, {6, -2, 11, -4, 8, -9, 3, -5, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{2}w^{4}-2uv^{2}w^{3}+2uv^{2}w^{2}-uvw^{4}+4uvw^{3}-5uvw^{2}+2uvw-2uw^{3}+4uw^{2}-3uw+u-v^{2}w^{4}+3v^{2}w^{3}-4v^{2}w^{2}+2v^{2}w-2vw^{3}+5vw^{2}-4vw+v-2w^{2}+2w-1}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+7q^{5}-11q^{4}-q^{-4}+16q^{3}+3q^{-3}-16q^{2}-7q^{-2}+18q+11q^{-1}-14$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-15z^{4}a^{-2}+4z^{4}a^{-4}+9z^{4}-3a^{2}z^{2}-19z^{2}a^{-2}+6z^{2}a^{-4}+15z^{2}-3a^{2}-13a^{-2}+4a^{-4}+12-a^{2}z^{-2}-5a^{-2}z^{-2}+2a^{-4}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-8}-z^{2}a^{-8}+3z^{5}a^{-7}-2z^{3}a^{-7}+6z^{6}a^{-6}-7z^{4}a^{-6}+5z^{2}a^{-6}-a^{-6}+8z^{7}a^{-5}-10z^{5}a^{-5}+5z^{3}a^{-5}+9z^{8}a^{-4}-18z^{6}a^{-4}+21z^{4}a^{-4}-17z^{2}a^{-4}-2a^{-4}z^{-2}+8a^{-4}+5z^{9}a^{-3}+a^{3}z^{7}+z^{7}a^{-3}-4a^{3}z^{5}-26z^{5}a^{-3}+6a^{3}z^{3}+33z^{3}a^{-3}-4a^{3}z-18za^{-3}+a^{3}z^{-1}+5a^{-3}z^{-1}+z^{10}a^{-2}+3a^{2}z^{8}+16z^{8}a^{-2}-11a^{2}z^{6}-58z^{6}a^{-2}+13a^{2}z^{4}+72z^{4}a^{-2}-7a^{2}z^{2}-50z^{2}a^{-2}-a^{2}z^{-2}-5a^{-2}z^{-2}+3a^{2}+20a^{-2}+3az^{9}+8z^{9}a^{-1}-4az^{7}-12z^{7}a^{-1}-15az^{5}-24z^{5}a^{-1}+30az^{3}+50z^{3}a^{-1}-19az-33za^{-1}+5az^{-1}+9a^{-1}z^{-1}+z^{10}+10z^{8}-45z^{6}+56z^{4}-34z^{2}-4z^{-2}+15$ (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 5 6 χ 15                       1 1 13                     3 1 -2 11                   4     4 9                 7 3     -4 7               9 4       5 5             8 8         0 3           10 8           2 1         6 10             4 -1       5 8               -3 -3     2 6                 4 -5   1 5                   -4 -7   2                     2 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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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