L11a240

L11a239

L11a241

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a240 at Knotilus!

Link Presentations

[edit Notes on L11a240's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{4}-3u^{2}v^{3}+4u^{2}v^{2}-4u^{2}v+3u^{2}-3uv^{4}+7uv^{3}-7uv^{2}+7uv-3u+3v^{4}-4v^{3}+4v^{2}-3v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+8q^{7/2}-13q^{5/2}+17q^{3/2}-18{\sqrt {q}}+{\frac {18}{\sqrt {q}}}-{\frac {15}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+9az^{3}-7z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-7a^{3}z+7az-5za^{-1}+za^{-3}+2a^{5}z^{-1}-3a^{3}z^{-1}+az^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-8az^{9}-5z^{9}a^{-1}-2a^{4}z^{8}-8a^{2}z^{8}-10z^{8}a^{-2}-16z^{8}-a^{5}z^{7}+8a^{3}z^{7}+14az^{7}-6z^{7}a^{-1}-11z^{7}a^{-3}+6a^{4}z^{6}+34a^{2}z^{6}+12z^{6}a^{-2}-8z^{6}a^{-4}+48z^{6}+5a^{5}z^{5}-4a^{3}z^{5}+5az^{5}+32z^{5}a^{-1}+14z^{5}a^{-3}-4z^{5}a^{-5}-3a^{4}z^{4}-35a^{2}z^{4}-z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}-41z^{4}-9a^{5}z^{3}-3a^{3}z^{3}-9az^{3}-23z^{3}a^{-1}-6z^{3}a^{-3}+2z^{3}a^{-5}-4a^{4}z^{2}+8a^{2}z^{2}-z^{2}a^{-2}-2z^{2}a^{-4}+13z^{2}+7a^{5}z+5a^{3}z+3za^{-1}+za^{-3}+3a^{4}+3a^{2}+1-2a^{5}z^{-1}-3a^{3}z^{-1}-az^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   5 1   -4 6                 8 3     5 4               9 5       -4 2             9 8         1 0           10 10           0 -2         5 8             -3 -4       5 10               5 -6     2 5                 -3 -8     5                   5 -10 1 2                     -1 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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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