L11a250

L11a249

L11a251

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a250 at Knotilus!

Link Presentations

[edit Notes on L11a250's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{3}v^{5}-3u^{3}v^{4}+3u^{3}v^{3}-u^{3}v^{2}-u^{2}v^{5}+4u^{2}v^{4}-8u^{2}v^{3}+5u^{2}v^{2}-u^{2}v-uv^{4}+5uv^{3}-8uv^{2}+4uv-u-v^{3}+3v^{2}-3v+1}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-q^{5/2}+4q^{3/2}-7{\sqrt {q}}+{\frac {10}{\sqrt {q}}}-{\frac {15}{q^{3/2}}}+{\frac {16}{q^{5/2}}}-{\frac {17}{q^{7/2}}}+{\frac {15}{q^{9/2}}}-{\frac {11}{q^{11/2}}}+{\frac {7}{q^{13/2}}}-{\frac {4}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{3}z^{9}-a^{5}z^{7}+6a^{3}z^{7}-az^{7}-4a^{5}z^{5}+11a^{3}z^{5}-4az^{5}-3a^{5}z^{3}+4a^{3}z^{3}-3az^{3}+2a^{5}z-5a^{3}z+az+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}+4a^{9}z^{5}-3a^{9}z^{3}+7a^{8}z^{6}-7a^{8}z^{4}+a^{8}z^{2}+8a^{7}z^{7}-7a^{7}z^{5}-3a^{7}z^{3}+2a^{7}z+8a^{6}z^{8}-10a^{6}z^{6}-a^{6}z^{4}+3a^{6}z^{2}+7a^{5}z^{9}-15a^{5}z^{7}+13a^{5}z^{5}-7a^{5}z^{3}-a^{5}z+a^{5}z^{-1}+3a^{4}z^{10}+a^{4}z^{8}-21a^{4}z^{6}+21a^{4}z^{4}-2a^{4}z^{2}-a^{4}+13a^{3}z^{9}-48a^{3}z^{7}+54a^{3}z^{5}-17a^{3}z^{3}-4a^{3}z+a^{3}z^{-1}+3a^{2}z^{10}-3a^{2}z^{8}-19a^{2}z^{6}+27a^{2}z^{4}-6a^{2}z^{2}+6az^{9}-24az^{7}+z^{7}a^{-1}+27az^{5}-3z^{5}a^{-1}-9az^{3}+z^{3}a^{-1}-az+4z^{8}-15z^{6}+13z^{4}-2z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     3   -3 2                   4 1   3 0                 6 3     -3 -2               9 4       5 -4             8 7         -1 -6           9 8           1 -8         6 8             2 -10       5 9               -4 -12     3 7                 4 -14   1 4                   -3 -16   3                     3 -18 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a249 L11a251