L11a283

L11a282

L11a284

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a283 at Knotilus!

Link Presentations

[edit Notes on L11a283's Link Presentations]

Planar diagram presentation	X10,1,11,2 X20,11,21,12 X8,9,1,10 X22,17,9,18 X12,4,13,3 X18,8,19,7 X6,14,7,13 X14,6,15,5 X4,16,5,15 X16,21,17,22 X2,20,3,19
Gauss code	{1, -11, 5, -9, 8, -7, 6, -3}, {3, -1, 2, -5, 7, -8, 9, -10, 4, -6, 11, -2, 10, -4}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{5}-t(1)t(2)^{5}+t(1)^{3}t(2)^{4}-4t(1)^{2}t(2)^{4}+5t(1)t(2)^{4}-t(2)^{4}-2t(1)^{3}t(2)^{3}+7t(1)^{2}t(2)^{3}-7t(1)t(2)^{3}+2t(2)^{3}+2t(1)^{3}t(2)^{2}-7t(1)^{2}t(2)^{2}+7t(1)t(2)^{2}-2t(2)^{2}-t(1)^{3}t(2)+5t(1)^{2}t(2)-4t(1)t(2)+t(2)-t(1)^{2}+t(1)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$3q^{9/2}-{\frac {4}{q^{9/2}}}-7q^{7/2}+{\frac {9}{q^{7/2}}}+11q^{5/2}-{\frac {14}{q^{5/2}}}-17q^{3/2}+{\frac {18}{q^{3/2}}}-q^{11/2}+{\frac {1}{q^{11/2}}}+19{\sqrt {q}}-{\frac {20}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}+z^{7}a^{-1}-a^{3}z^{5}+3az^{5}+4z^{5}a^{-1}-z^{5}a^{-3}-2a^{3}z^{3}+2az^{3}+7z^{3}a^{-1}-3z^{3}a^{-3}-a^{3}z-az+7za^{-1}-3za^{-3}-az^{-1}+3a^{-1}z^{-1}-2a^{-3}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}+z^{7}a^{-5}+4a^{5}z^{5}-4z^{5}a^{-5}-a^{5}z^{3}+5z^{3}a^{-5}-2za^{-5}+3z^{8}a^{-4}+9a^{4}z^{6}-11z^{6}a^{-4}-8a^{4}z^{4}+12z^{4}a^{-4}+3a^{4}z^{2}-3z^{2}a^{-4}+4z^{9}a^{-3}+13a^{3}z^{7}-12z^{7}a^{-3}-18a^{3}z^{5}+10z^{5}a^{-3}+9a^{3}z^{3}-4z^{3}a^{-3}-a^{3}z+5za^{-3}-2a^{-3}z^{-1}+2z^{10}a^{-2}+12a^{2}z^{8}+3z^{8}a^{-2}-16a^{2}z^{6}-23z^{6}a^{-2}+4a^{2}z^{4}+26z^{4}a^{-2}-a^{2}z^{2}-11z^{2}a^{-2}+a^{2}+3a^{-2}+7az^{9}+11z^{9}a^{-1}-2az^{7}-28z^{7}a^{-1}-14az^{5}+22z^{5}a^{-1}+4az^{3}-15z^{3}a^{-1}+2az+10za^{-1}-az^{-1}-3a^{-1}z^{-1}+2z^{10}+12z^{8}-37z^{6}+27z^{4}-12z^{2}+3$ (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 12                       1 1 10                     2   -2 8                   5 1   4 6                 7 3     -4 4               10 4       6 2             9 7         -2 0           11 10           1 -2         8 10             2 -4       6 10               -4 -6     3 8                 5 -8   1 6                   -5 -10   3                     3 -12 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a282 L11a284