L11a367

L11a366

L11a368

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a367 at Knotilus!

Link Presentations

[edit Notes on L11a367's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{4}v^{4}-u^{4}v^{3}-u^{3}v^{4}+3u^{3}v^{3}-u^{3}v^{2}-u^{2}v^{3}+3u^{2}v^{2}-u^{2}v-uv^{2}+3uv-u-v+1}{u^{2}v^{2}}}$ (db)
Jones polynomial	${\sqrt {q}}-{\frac {2}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {4}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {5}{q^{13/2}}}+{\frac {4}{q^{15/2}}}-{\frac {3}{q^{17/2}}}+{\frac {2}{q^{19/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{7}z^{7}+6a^{7}z^{5}+11a^{7}z^{3}+7a^{7}z+a^{7}z^{-1}-a^{5}z^{9}-8a^{5}z^{7}-23a^{5}z^{5}-29a^{5}z^{3}-14a^{5}z-a^{5}z^{-1}+a^{3}z^{7}+6a^{3}z^{5}+10a^{3}z^{3}+4a^{3}z$ (db)
Kauffman polynomial	$a^{13}z^{3}-a^{13}z+2a^{12}z^{4}-2a^{12}z^{2}+2a^{11}z^{5}-a^{11}z^{3}+2a^{10}z^{6}-2a^{10}z^{4}+a^{10}z^{2}+2a^{9}z^{7}-4a^{9}z^{5}+3a^{9}z^{3}-a^{9}z+2a^{8}z^{8}-6a^{8}z^{6}+5a^{8}z^{4}-2a^{8}z^{2}+2a^{7}z^{9}-9a^{7}z^{7}+15a^{7}z^{5}-16a^{7}z^{3}+8a^{7}z-a^{7}z^{-1}+a^{6}z^{10}-3a^{6}z^{8}-2a^{6}z^{6}+9a^{6}z^{4}-7a^{6}z^{2}+a^{6}+4a^{5}z^{9}-24a^{5}z^{7}+48a^{5}z^{5}-42a^{5}z^{3}+15a^{5}z-a^{5}z^{-1}+a^{4}z^{10}-4a^{4}z^{8}+10a^{4}z^{4}-6a^{4}z^{2}+2a^{3}z^{9}-13a^{3}z^{7}+27a^{3}z^{5}-21a^{3}z^{3}+5a^{3}z+a^{2}z^{8}-6a^{2}z^{6}+10a^{2}z^{4}-4a^{2}z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 2                       1 -1 0                     1   1 -2                   1 1   0 -4                 3 1     2 -6               2 2       0 -8             3 2         1 -10           2 2           0 -12         3 3             0 -14       1 2               1 -16     2 3                 -1 -18   1 2                   1 -20   1                     -1 -22 1                       1

Integral Khovanov Homology (db, data source)

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

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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