L11a284

L11a283

L11a285

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a284 at Knotilus!

Link Presentations

[edit Notes on L11a284's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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