L11a231

L11a230

L11a232

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a231 at Knotilus!

Link Presentations

[edit Notes on L11a231's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X22,13,7,14 X14,9,15,10 X10,21,11,22 X18,5,19,6 X20,16,21,15 X16,20,17,19 X2738 X4,11,5,12 X6,17,1,18
Gauss code	{1, -9, 2, -10, 6, -11}, {9, -1, 4, -5, 10, -2, 3, -4, 7, -8, 11, -6, 8, -7, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2u^{2}v^{3}-7u^{2}v^{2}+6u^{2}v-2u^{2}+uv^{4}-8uv^{3}+13uv^{2}-8uv+u-2v^{4}+6v^{3}-7v^{2}+2v}{uv^{2}}}$ (db)
Jones polynomial	${\frac {21}{q^{9/2}}}-{\frac {19}{q^{7/2}}}+{\frac {14}{q^{5/2}}}-{\frac {9}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {7}{q^{17/2}}}-{\frac {13}{q^{15/2}}}+{\frac {17}{q^{13/2}}}-{\frac {21}{q^{11/2}}}-{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4a^{9}z+3a^{9}z^{-1}-5a^{7}z^{3}-6a^{7}z-2a^{7}z^{-1}+2a^{5}z^{5}+2a^{5}z^{3}+a^{5}z+a^{3}z^{5}-a^{3}z^{3}-2a^{3}z-az^{3}$ (db)
Kauffman polynomial	$a^{12}z^{6}-3a^{12}z^{4}+3a^{12}z^{2}-a^{12}+3a^{11}z^{7}-8a^{11}z^{5}+7a^{11}z^{3}-3a^{11}z+a^{11}z^{-1}+4a^{10}z^{8}-5a^{10}z^{6}-6a^{10}z^{4}+9a^{10}z^{2}-3a^{10}+4a^{9}z^{9}-2a^{9}z^{7}-11a^{9}z^{5}+15a^{9}z^{3}-12a^{9}z+3a^{9}z^{-1}+2a^{8}z^{10}+6a^{8}z^{8}-17a^{8}z^{6}+8a^{8}z^{4}+3a^{8}z^{2}-3a^{8}+11a^{7}z^{9}-20a^{7}z^{7}+9a^{7}z^{5}+6a^{7}z^{3}-7a^{7}z+2a^{7}z^{-1}+2a^{6}z^{10}+12a^{6}z^{8}-34a^{6}z^{6}+31a^{6}z^{4}-8a^{6}z^{2}+7a^{5}z^{9}-7a^{5}z^{7}-2a^{5}z^{5}+5a^{5}z^{3}+10a^{4}z^{8}-19a^{4}z^{6}+15a^{4}z^{4}-5a^{4}z^{2}+8a^{3}z^{7}-13a^{3}z^{5}+6a^{3}z^{3}-2a^{3}z+4a^{2}z^{6}-5a^{2}z^{4}+az^{5}-az^{3}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     3   -3 -2                   6 1   5 -4                 9 4     -5 -6               10 5       5 -8             11 9         -2 -10           10 10           0 -12         7 11             4 -14       6 10               -4 -16     2 8                 6 -18   1 5                   -4 -20   2                     2 -22 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=-5$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a230 L11a232