L11a268

L11a267

L11a269

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a268 at Knotilus!

Link Presentations

[edit Notes on L11a268's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{3}v^{3}-3u^{3}v^{2}+3u^{3}v-u^{3}-3u^{2}v^{3}+9u^{2}v^{2}-10u^{2}v+3u^{2}+3uv^{3}-10uv^{2}+9uv-3u-v^{3}+3v^{2}-3v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {14}{q^{9/2}}}-q^{7/2}+{\frac {18}{q^{7/2}}}+4q^{5/2}-{\frac {21}{q^{5/2}}}-8q^{3/2}+{\frac {21}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {8}{q^{11/2}}}+13{\sqrt {q}}-{\frac {19}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+3z^{3}a^{5}+4za^{5}+a^{5}z^{-1}-3z^{5}a^{3}-8z^{3}a^{3}-7za^{3}-a^{3}z^{-1}+z^{7}a+4z^{5}a+7z^{3}a+3za-z^{5}a^{-1}-2z^{3}a^{-1}-za^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-4a^{5}z^{9}-8a^{3}z^{9}-4az^{9}-6a^{6}z^{8}-15a^{4}z^{8}-16a^{2}z^{8}-7z^{8}-4a^{7}z^{7}-4a^{5}z^{7}-3a^{3}z^{7}-10az^{7}-7z^{7}a^{-1}-a^{8}z^{6}+12a^{6}z^{6}+36a^{4}z^{6}+32a^{2}z^{6}-4z^{6}a^{-2}+5z^{6}+10a^{7}z^{5}+29a^{5}z^{5}+40a^{3}z^{5}+33az^{5}+11z^{5}a^{-1}-z^{5}a^{-3}+2a^{8}z^{4}-5a^{6}z^{4}-22a^{4}z^{4}-17a^{2}z^{4}+6z^{4}a^{-2}+4z^{4}-8a^{7}z^{3}-29a^{5}z^{3}-42a^{3}z^{3}-29az^{3}-7z^{3}a^{-1}+z^{3}a^{-3}-a^{8}z^{2}+3a^{4}z^{2}+a^{2}z^{2}-2z^{2}a^{-2}-3z^{2}+2a^{7}z+11a^{5}z+14a^{3}z+7az+2za^{-1}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   5 1   4 2                 8 3     -5 0               11 5       6 -2             11 9         -2 -4           10 10           0 -6         8 11             3 -8       6 10               -4 -10     3 9                 6 -12   1 5                   -4 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

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