L11a382

L11a381

L11a383

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a382 at Knotilus!

Link Presentations

[edit Notes on L11a382's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {\left(t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+2t(1)t(2)-t(2)-t(1)+1\right)\left(t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(1)+2t(2)t(1)-t(1)+1\right)}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+6q^{5/2}-10q^{3/2}+13{\sqrt {q}}-{\frac {16}{\sqrt {q}}}+{\frac {15}{q^{3/2}}}-{\frac {14}{q^{5/2}}}+{\frac {10}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{9}+a^{3}z^{7}-7az^{7}+z^{7}a^{-1}+5a^{3}z^{5}-18az^{5}+5z^{5}a^{-1}+8a^{3}z^{3}-20az^{3}+8z^{3}a^{-1}+4a^{3}z-9az+4za^{-1}+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-2a^{7}z^{3}+a^{7}z+3a^{6}z^{6}-6a^{6}z^{4}+3a^{6}z^{2}+4a^{5}z^{7}-5a^{5}z^{5}-a^{5}z^{3}+a^{5}z+4a^{4}z^{8}-3a^{4}z^{6}+z^{6}a^{-4}-3a^{4}z^{4}-3z^{4}a^{-4}+a^{4}z^{2}+z^{2}a^{-4}+4a^{3}z^{9}-8a^{3}z^{7}+3z^{7}a^{-3}+14a^{3}z^{5}-9z^{5}a^{-3}-14a^{3}z^{3}+5z^{3}a^{-3}+5a^{3}z-za^{-3}-a^{3}z^{-1}+2a^{2}z^{10}+5z^{8}a^{-2}-6a^{2}z^{6}-16z^{6}a^{-2}+12a^{2}z^{4}+14z^{4}a^{-2}-6a^{2}z^{2}-5z^{2}a^{-2}+a^{2}+9az^{9}+5z^{9}a^{-1}-31az^{7}-16z^{7}a^{-1}+47az^{5}+18z^{5}a^{-1}-30az^{3}-10z^{3}a^{-1}+9az+3za^{-1}-az^{-1}+2z^{10}+z^{8}-17z^{6}+26z^{4}-10z^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     2   2 6                   4 1   -3 4                 6 2     4 2               7 4       -3 0             9 6         3 -2           7 8           1 -4         7 8             -1 -6       4 8               4 -8     2 6                 -4 -10   1 4                   3 -12   2                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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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