L11a516

L11a515

L11a517

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a516 at Knotilus!

Link Presentations

[edit Notes on L11a516's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(3)-1)\left(t(3)^{2}t(2)^{2}+t(1)t(3)t(2)^{2}-2t(3)t(2)^{2}+t(2)^{2}+t(1)t(3)^{2}t(2)-t(3)^{2}t(2)+t(1)t(2)-2t(1)t(3)t(2)+2t(3)t(2)-t(2)-t(1)t(3)^{2}-t(1)+2t(1)t(3)-t(3)\right)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{4}-4q^{3}+9q^{2}-15q+21-22q^{-1}+24q^{-2}-19q^{-3}+15q^{-4}-9q^{-5}+4q^{-6}-q^{-7}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-a^{6}+3z^{4}a^{4}+6z^{2}a^{4}+a^{4}z^{-2}+3a^{4}-2z^{6}a^{2}-6z^{4}a^{2}-7z^{2}a^{2}-2a^{2}z^{-2}-4a^{2}-z^{6}-z^{4}+z^{2}+z^{-2}+2+z^{4}a^{-2}+z^{2}a^{-2}$ (db)
Kauffman polynomial	$a^{7}z^{7}-3a^{7}z^{5}+3a^{7}z^{3}-a^{7}z+4a^{6}z^{8}-13a^{6}z^{6}+14a^{6}z^{4}-7a^{6}z^{2}+2a^{6}+6a^{5}z^{9}-16a^{5}z^{7}+7a^{5}z^{5}+6a^{5}z^{3}-3a^{5}z+3a^{4}z^{10}+7a^{4}z^{8}-48a^{4}z^{6}+60a^{4}z^{4}+z^{4}a^{-4}-29a^{4}z^{2}-a^{4}z^{-2}+7a^{4}+17a^{3}z^{9}-44a^{3}z^{7}+24a^{3}z^{5}+4z^{5}a^{-3}+3a^{3}z^{3}-z^{3}a^{-3}-4a^{3}z+2a^{3}z^{-1}+3a^{2}z^{10}+19a^{2}z^{8}-73a^{2}z^{6}+9z^{6}a^{-2}+74a^{2}z^{4}-7z^{4}a^{-2}-33a^{2}z^{2}+2z^{2}a^{-2}-2a^{2}z^{-2}+7a^{2}+11az^{9}-13az^{7}+14z^{7}a^{-1}-8az^{5}-18z^{5}a^{-1}+8az^{3}+7z^{3}a^{-1}-2az+2az^{-1}+16z^{8}-29z^{6}+20z^{4}-9z^{2}-z^{-2}+3$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                       1 1 7                     3   -3 5                   6 1   5 3                 9 3     -6 1               12 6       6 -1             12 11         -1 -3           12 10           2 -5         9 14             5 -7       6 10               -4 -9     3 9                 6 -11   1 6                   -5 -13   3                     3 -15 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a515 L11a517