L11a368

L11a367

L11a369

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a368 at Knotilus!

Link Presentations

[edit Notes on L11a368's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{4}v^{3}-u^{4}v^{2}+u^{3}v^{4}-3u^{3}v^{3}+4u^{3}v^{2}-u^{3}v-u^{2}v^{4}+4u^{2}v^{3}-5u^{2}v^{2}+4u^{2}v-u^{2}-uv^{3}+4uv^{2}-3uv+u-v^{2}+v}{u^{2}v^{2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {5}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {12}{q^{7/2}}}+{\frac {11}{q^{9/2}}}-{\frac {10}{q^{11/2}}}+{\frac {7}{q^{13/2}}}-{\frac {4}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+4z^{3}a^{7}+4za^{7}+a^{7}z^{-1}-z^{7}a^{5}-5z^{5}a^{5}-9z^{3}a^{5}-7za^{5}-a^{5}z^{-1}-z^{7}a^{3}-4z^{5}a^{3}-4z^{3}a^{3}-za^{3}+z^{5}a+3z^{3}a+za$ (db)
Kauffman polynomial	$a^{11}z^{5}-3a^{11}z^{3}+2a^{11}z+2a^{10}z^{6}-5a^{10}z^{4}+3a^{10}z^{2}+2a^{9}z^{7}-2a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+2a^{8}z^{8}-2a^{8}z^{6}+a^{8}z^{4}-a^{8}z^{2}+2a^{7}z^{9}-5a^{7}z^{7}+13a^{7}z^{5}-15a^{7}z^{3}+7a^{7}z-a^{7}z^{-1}+a^{6}z^{10}-3a^{6}z^{6}+10a^{6}z^{4}-7a^{6}z^{2}+a^{6}+5a^{5}z^{9}-18a^{5}z^{7}+31a^{5}z^{5}-24a^{5}z^{3}+9a^{5}z-a^{5}z^{-1}+a^{4}z^{10}+2a^{4}z^{8}-13a^{4}z^{6}+17a^{4}z^{4}-6a^{4}z^{2}+3a^{3}z^{9}-8a^{3}z^{7}+5a^{3}z^{5}-a^{3}z^{3}+4a^{2}z^{8}-13a^{2}z^{6}+10a^{2}z^{4}-2a^{2}z^{2}+3az^{7}-10az^{5}+7az^{3}-az+z^{6}-3z^{4}+z^{2}$ (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 4                       1 -1 2                     2   2 0                   3 1   -2 -2                 5 2     3 -4               6 4       -2 -6             6 4         2 -8           5 6           1 -10         5 6             -1 -12       2 5               3 -14     2 5                 -3 -16   1 3                   2 -18   1                     -1 -20 1                       1

Integral Khovanov Homology (db, data source)

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

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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