L11a509

L11a508

L11a510

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a509 at Knotilus!

Link Presentations

[edit Notes on L11a509's Link Presentations]

Planar diagram presentation	X8192 X20,10,21,9 X14,5,15,6 X12,14,7,13 X16,8,17,7 X22,18,13,17 X10,4,11,3 X18,11,19,12 X6,15,1,16 X4,20,5,19 X2,21,3,22
Gauss code	{1, -11, 7, -10, 3, -9}, {5, -1, 2, -7, 8, -4}, {4, -3, 9, -5, 6, -8, 10, -2, 11, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(3)-1)(-t(2)t(1)+t(2)t(3)t(1)-t(3)t(1)+t(1)+t(2)+t(3)-1)(-t(1)t(2)+t(1)t(3)t(2)-t(3)t(2)+t(2)-t(1)t(3)+t(3)-1)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{6}-5q^{5}+12q^{4}-20q^{3}+28q^{2}-31q+33-27q^{-1}+21q^{-2}-12q^{-3}+5q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+4z^{6}-2a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}+6z^{4}-a^{2}z^{2}-3z^{2}a^{-2}+z^{2}a^{-4}+2z^{2}+a^{2}+a^{-2}-2+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-6}-z^{4}a^{-6}+5z^{7}a^{-5}+a^{5}z^{5}-8z^{5}a^{-5}+3z^{3}a^{-5}+11z^{8}a^{-4}+5a^{4}z^{6}-22z^{6}a^{-4}-3a^{4}z^{4}+14z^{4}a^{-4}-4z^{2}a^{-4}+12z^{9}a^{-3}+12a^{3}z^{7}-18z^{7}a^{-3}-13a^{3}z^{5}+z^{5}a^{-3}+3a^{3}z^{3}+4z^{3}a^{-3}+5z^{10}a^{-2}+18a^{2}z^{8}+18z^{8}a^{-2}-28a^{2}z^{6}-62z^{6}a^{-2}+16a^{2}z^{4}+49z^{4}a^{-2}-3a^{2}z^{2}-11z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{2}-2a^{-2}+15az^{9}+27z^{9}a^{-1}-15az^{7}-50z^{7}a^{-1}-4az^{5}+19z^{5}a^{-1}+4az^{3}+2z^{3}a^{-1}+2az+2za^{-1}-2az^{-1}-2a^{-1}z^{-1}+5z^{10}+25z^{8}-72z^{6}+53z^{4}-10z^{2}+2z^{-2}-3$ (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 13                       1 1 11                     4   -4 9                   8 1   7 7                 12 4     -8 5               16 8       8 3             16 13         -3 1           17 15           2 -1         12 18             6 -3       9 15               -6 -5     4 13                 9 -7   1 8                   -7 -9   4                     4 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=-1$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{18}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{17}$ $r=1$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{16}$ ${\mathbb {Z} }^{16}$ $r=2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{16}$ $r=3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a508 L11a510