L11a255

L11a254

L11a256

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a255 at Knotilus!

Link Presentations

[edit Notes on L11a255's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)(t(1)t(2)+1)^{2}\left(t(2)^{2}-t(2)+1\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+3q^{11/2}-6q^{9/2}+10q^{7/2}-13q^{5/2}+15q^{3/2}-16{\sqrt {q}}+{\frac {12}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-3}-5z^{5}a^{-3}-8z^{3}a^{-3}-4za^{-3}+z^{9}a^{-1}-az^{7}+7z^{7}a^{-1}-5az^{5}+18z^{5}a^{-1}-8az^{3}+20z^{3}a^{-1}-4az+8za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-2z^{3}a^{-7}+3z^{6}a^{-6}-6z^{4}a^{-6}+z^{2}a^{-6}+5z^{7}a^{-5}-11z^{5}a^{-5}+6z^{3}a^{-5}-za^{-5}+6z^{8}a^{-4}+a^{4}z^{6}-16z^{6}a^{-4}-3a^{4}z^{4}+18z^{4}a^{-4}+2a^{4}z^{2}-7z^{2}a^{-4}+5z^{9}a^{-3}+3a^{3}z^{7}-14z^{7}a^{-3}-9a^{3}z^{5}+20z^{5}a^{-3}+6a^{3}z^{3}-11z^{3}a^{-3}-a^{3}z+4za^{-3}+2z^{10}a^{-2}+4a^{2}z^{8}+z^{8}a^{-2}-10a^{2}z^{6}-15z^{6}a^{-2}+4a^{2}z^{4}+29z^{4}a^{-2}-13z^{2}a^{-2}+4az^{9}+9z^{9}a^{-1}-11az^{7}-33z^{7}a^{-1}+13az^{5}+54z^{5}a^{-1}-13az^{3}-38z^{3}a^{-1}+4az+10za^{-1}+az^{-1}+a^{-1}z^{-1}+2z^{10}-z^{8}-7z^{6}+12z^{4}-7z^{2}-1$ (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 2     -4 6               7 4       3 4             8 6         -2 2           8 7           1 0         6 10             4 -2       4 6               -2 -4     2 6                 4 -6   1 4                   -3 -8   2                     2 -10 1                       -1

Integral Khovanov Homology (db, data source)

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

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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