L11a510

L11a509

L11a511

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a510 at Knotilus!

Link Presentations

[edit Notes on L11a510's Link Presentations]

Planar diagram presentation	X8192 X14,3,15,4 X22,16,13,15 X10,20,11,19 X18,10,19,9 X20,12,21,11 X12,22,7,21 X16,6,17,5 X2738 X4,13,5,14 X6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 5, -4, 6, -7}, {10, -2, 3, -8, 11, -5, 4, -6, 7, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{2}w^{3}-u^{2}v^{2}w^{2}-u^{2}vw^{3}+2u^{2}vw^{2}-u^{2}vw-u^{2}w^{2}+2u^{2}w-u^{2}-2uv^{2}w^{3}+2uv^{2}w^{2}-uv^{2}w+uvw^{3}-2uvw^{2}+2uvw-uv+uw^{2}-2uw+2u+v^{2}w^{3}-2v^{2}w^{2}+v^{2}w+vw^{2}-2vw+v+w-1}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{9}+3q^{8}-5q^{7}+8q^{6}-10q^{5}+11q^{4}-10q^{3}+10q^{2}+q^{-2}-6q-2q^{-1}+5$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{8}a^{-4}-2z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-10z^{4}a^{-2}+13z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}-15z^{2}a^{-2}+13z^{2}a^{-4}-4z^{2}a^{-6}+4z^{2}-8a^{-2}+5a^{-4}-a^{-6}+4-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{3}a^{-11}+3z^{4}a^{-10}-z^{2}a^{-10}+5z^{5}a^{-9}-3z^{3}a^{-9}+7z^{6}a^{-8}-9z^{4}a^{-8}+2z^{2}a^{-8}+8z^{7}a^{-7}-16z^{5}a^{-7}+7z^{3}a^{-7}-za^{-7}+7z^{8}a^{-6}-17z^{6}a^{-6}+9z^{4}a^{-6}-4z^{2}a^{-6}+a^{-6}+4z^{9}a^{-5}-6z^{7}a^{-5}-13z^{5}a^{-5}+17z^{3}a^{-5}-3za^{-5}+z^{10}a^{-4}+7z^{8}a^{-4}-42z^{6}a^{-4}+59z^{4}a^{-4}-32z^{2}a^{-4}-a^{-4}z^{-2}+8a^{-4}+6z^{9}a^{-3}-24z^{7}a^{-3}+22z^{5}a^{-3}+4z^{3}a^{-3}-8za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+z^{8}a^{-2}-24z^{6}a^{-2}+51z^{4}a^{-2}-38z^{2}a^{-2}-2a^{-2}z^{-2}+12a^{-2}+2z^{9}a^{-1}-10z^{7}a^{-1}+14z^{5}a^{-1}-2z^{3}a^{-1}-6za^{-1}+2a^{-1}z^{-1}+z^{8}-6z^{6}+13z^{4}-13z^{2}-z^{-2}+6$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 19                       1 -1 17                     2   2 15                   3 1   -2 13                 5 2     3 11               6 4       -2 9             5 4         1 7           6 7           1 5         4 4             0 3       3 7               4 1     2 3                 -1 -1   1 4                   3 -3   1                     -1 -5 1                       1

Integral Khovanov Homology (db, data source)

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

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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