L11a366

L11a365

L11a367

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a366 at Knotilus!

Link Presentations

[edit Notes on L11a366's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{4}v^{3}-u^{4}v^{2}+u^{3}v^{4}-3u^{3}v^{3}+4u^{3}v^{2}-2u^{3}v-u^{2}v^{4}+4u^{2}v^{3}-5u^{2}v^{2}+4u^{2}v-u^{2}-2uv^{3}+4uv^{2}-3uv+u-v^{2}+v}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {2}{q^{9/2}}}+6q^{7/2}-{\frac {4}{q^{7/2}}}-9q^{5/2}+{\frac {7}{q^{5/2}}}+11q^{3/2}-{\frac {10}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-13{\sqrt {q}}+{\frac {11}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{3}z^{5}+z^{5}a^{-3}+4a^{3}z^{3}+3z^{3}a^{-3}+4a^{3}z+2za^{-3}+a^{3}z^{-1}-az^{7}-z^{7}a^{-1}-5az^{5}-4z^{5}a^{-1}-9az^{3}-4z^{3}a^{-1}-7az-az^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-z^{2}a^{-6}+a^{5}z^{7}-5a^{5}z^{5}+3z^{5}a^{-5}+7a^{5}z^{3}-3z^{3}a^{-5}-2a^{5}z+2a^{4}z^{8}-9a^{4}z^{6}+5z^{6}a^{-4}+12a^{4}z^{4}-6z^{4}a^{-4}-5a^{4}z^{2}+2z^{2}a^{-4}+2a^{3}z^{9}-7a^{3}z^{7}+6z^{7}a^{-3}+7a^{3}z^{5}-10z^{5}a^{-3}-5a^{3}z^{3}+9z^{3}a^{-3}+4a^{3}z-3za^{-3}-a^{3}z^{-1}+a^{2}z^{10}+5z^{8}a^{-2}-8a^{2}z^{6}-8z^{6}a^{-2}+10a^{2}z^{4}+6z^{4}a^{-2}-6a^{2}z^{2}-z^{2}a^{-2}+a^{2}+5az^{9}+3z^{9}a^{-1}-17az^{7}-3z^{7}a^{-1}+24az^{5}-z^{5}a^{-1}-23az^{3}+z^{3}a^{-1}+10az+za^{-1}-az^{-1}+z^{10}+3z^{8}-12z^{6}+11z^{4}-5z^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     2   2 8                   4 1   -3 6                 5 2     3 4               6 4       -2 2             7 5         2 0           5 7           2 -2         5 6             -1 -4       3 6               3 -6     1 4                 -3 -8   1 3                   2 -10   1                     -1 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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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