L11a512

L11a511

L11a513

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a512 at Knotilus!

Link Presentations

[edit Notes on L11a512's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{2}t(3)^{3}-t(2)^{2}t(3)^{3}+t(1)^{2}t(2)t(3)^{3}-2t(1)t(2)t(3)^{3}+t(2)t(3)^{3}+t(1)^{2}t(3)^{2}+t(1)^{2}t(2)^{2}t(3)^{2}-4t(1)t(2)^{2}t(3)^{2}+2t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}-3t(1)^{2}t(2)t(3)^{2}+7t(1)t(2)t(3)^{2}-3t(2)t(3)^{2}+t(3)^{2}-2t(1)^{2}t(3)-t(1)^{2}t(2)^{2}t(3)+2t(1)t(2)^{2}t(3)-t(2)^{2}t(3)+4t(1)t(3)+3t(1)^{2}t(2)t(3)-7t(1)t(2)t(3)+3t(2)t(3)-t(3)+t(1)^{2}-t(1)-t(1)^{2}t(2)+2t(1)t(2)-t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-11}+4q^{-10}-8q^{-9}+13q^{-8}-17q^{-7}+20q^{-6}-18q^{-5}+17q^{-4}-11q^{-3}+7q^{-2}-3q^{-1}+1$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{10}\left(-z^{2}\right)-a^{10}+3a^{8}z^{4}+7a^{8}z^{2}+a^{8}z^{-2}+4a^{8}-2a^{6}z^{6}-7a^{6}z^{4}-10a^{6}z^{2}-2a^{6}z^{-2}-8a^{6}-a^{4}z^{6}-a^{4}z^{4}+4a^{4}z^{2}+a^{4}z^{-2}+5a^{4}+a^{2}z^{4}+2a^{2}z^{2}$ (db)
Kauffman polynomial	$a^{13}z^{5}-a^{13}z^{3}+4a^{12}z^{6}-6a^{12}z^{4}+2a^{12}z^{2}+7a^{11}z^{7}-11a^{11}z^{5}+5a^{11}z^{3}-a^{11}z+7a^{10}z^{8}-7a^{10}z^{6}-a^{10}z^{4}+a^{10}+4a^{9}z^{9}+6a^{9}z^{7}-23a^{9}z^{5}+18a^{9}z^{3}-3a^{9}z+a^{8}z^{10}+14a^{8}z^{8}-36a^{8}z^{6}+39a^{8}z^{4}-24a^{8}z^{2}-a^{8}z^{-2}+8a^{8}+8a^{7}z^{9}-9a^{7}z^{7}-6a^{7}z^{5}+14a^{7}z^{3}-8a^{7}z+2a^{7}z^{-1}+a^{6}z^{10}+12a^{6}z^{8}-40a^{6}z^{6}+53a^{6}z^{4}-37a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+4a^{5}z^{9}-5a^{5}z^{7}-3a^{5}z^{5}+7a^{5}z^{3}-6a^{5}z+2a^{5}z^{-1}+5a^{4}z^{8}-14a^{4}z^{6}+16a^{4}z^{4}-13a^{4}z^{2}-a^{4}z^{-2}+6a^{4}+3a^{3}z^{7}-8a^{3}z^{5}+5a^{3}z^{3}+a^{2}z^{6}-3a^{2}z^{4}+2a^{2}z^{2}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                       1 1 -1                     2   -2 -3                   5 1   4 -5                 7 3     -4 -7               10 4       6 -9             9 8         -1 -11           11 9           2 -13         7 10             3 -15       6 10               -4 -17     3 8                 5 -19   1 5                   -4 -21   3                     3 -23 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-5$ $i=-3$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-6$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=-5$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a511 L11a513