L11a282

L11a281

L11a283

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a282 at Knotilus!

Link Presentations

[edit Notes on L11a282'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 X14,6,15,5 X4,14,5,13 X6,16,7,15 X16,21,17,22 X2,20,3,19
Gauss code	{1, -11, 5, -8, 7, -9, 6, -3}, {3, -1, 2, -5, 8, -7, 9, -10, 4, -6, 11, -2, 10, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{3}t(2)^{5}-t(1)^{2}t(2)^{5}-2t(1)^{3}t(2)^{4}+4t(1)^{2}t(2)^{4}-t(1)t(2)^{4}+2t(1)^{3}t(2)^{3}-6t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-t(2)^{3}-t(1)^{3}t(2)^{2}+5t(1)^{2}t(2)^{2}-6t(1)t(2)^{2}+2t(2)^{2}-t(1)^{2}t(2)+4t(1)t(2)-2t(2)-t(1)+1}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+3q^{11/2}-6q^{9/2}+9q^{7/2}-13q^{5/2}+14q^{3/2}-15{\sqrt {q}}+{\frac {12}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{9}a^{-1}-az^{7}+7z^{7}a^{-1}-z^{7}a^{-3}-5az^{5}+18z^{5}a^{-1}-5z^{5}a^{-3}-8az^{3}+21z^{3}a^{-1}-8z^{3}a^{-3}-5az+12za^{-1}-5za^{-3}-az^{-1}+3a^{-1}z^{-1}-2a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-5az^{9}-9z^{9}a^{-1}-4z^{9}a^{-3}-5a^{2}z^{8}+z^{8}a^{-2}-4z^{8}a^{-4}-3a^{3}z^{7}+18az^{7}+35z^{7}a^{-1}+10z^{7}a^{-3}-4z^{7}a^{-5}-a^{4}z^{6}+17a^{2}z^{6}+7z^{6}a^{-2}+5z^{6}a^{-4}-3z^{6}a^{-6}+17z^{6}+9a^{3}z^{5}-24az^{5}-57z^{5}a^{-1}-17z^{5}a^{-3}+6z^{5}a^{-5}-z^{5}a^{-7}+3a^{4}z^{4}-16a^{2}z^{4}-18z^{4}a^{-2}+6z^{4}a^{-6}-31z^{4}-4a^{3}z^{3}+17az^{3}+39z^{3}a^{-1}+16z^{3}a^{-3}+2z^{3}a^{-7}-a^{4}z^{2}+7a^{2}z^{2}+11z^{2}a^{-2}-2z^{2}a^{-6}+17z^{2}-6az-15za^{-1}-8za^{-3}-za^{-7}-a^{2}-3a^{-2}-3+az^{-1}+3a^{-1}z^{-1}+2a^{-3}z^{-1}$ (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 14                       1 1 12                     2   -2 10                   4 1   3 8                 6 3     -3 6               7 3       4 4             7 6         -1 2           8 7           1 0         5 8             3 -2       4 7               -3 -4     2 5                 3 -6   1 4                   -3 -8   2                     2 -10 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

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