L11a365

L11a364

L11a366

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a365 at Knotilus!

Link Presentations

[edit Notes on L11a365's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 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 X4,15,5,16
Gauss code	{1, -2, 3, -11, 4, -5, 6, -9, 8, -10}, {5, -1, 2, -3, 11, -4, 7, -6, 9, -8, 10, -7}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {\left(t(2)^{2}t(1)^{2}-t(2)t(1)^{2}+t(2)t(1)-t(1)+1\right)\left(t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+t(1)t(2)-t(2)+1\right)}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+3q^{5/2}-5q^{3/2}+6{\sqrt {q}}-{\frac {8}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {7}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+6a^{3}z^{5}+11a^{3}z^{3}+6a^{3}z+a^{3}z^{-1}-az^{9}-8az^{7}+z^{7}a^{-1}-23az^{5}+6z^{5}a^{-1}-28az^{3}+11z^{3}a^{-1}-13az+6za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+a^{7}z+2a^{6}z^{6}-6a^{6}z^{4}+3a^{6}z^{2}+2a^{5}z^{7}-5a^{5}z^{5}+2a^{5}z^{3}+2a^{4}z^{8}-6a^{4}z^{6}+z^{6}a^{-4}+6a^{4}z^{4}-4z^{4}a^{-4}-a^{4}z^{2}+3z^{2}a^{-4}+2a^{3}z^{9}-9a^{3}z^{7}+2z^{7}a^{-3}+18a^{3}z^{5}-8z^{5}a^{-3}-14a^{3}z^{3}+7z^{3}a^{-3}+6a^{3}z-za^{-3}-a^{3}z^{-1}+a^{2}z^{10}-3a^{2}z^{8}+2z^{8}a^{-2}+a^{2}z^{6}-7z^{6}a^{-2}+9a^{2}z^{4}+5z^{4}a^{-2}-6a^{2}z^{2}-z^{2}a^{-2}+a^{2}+4az^{9}+2z^{9}a^{-1}-22az^{7}-9z^{7}a^{-1}+48az^{5}+16z^{5}a^{-1}-42az^{3}-16z^{3}a^{-1}+14az+6za^{-1}-az^{-1}+z^{10}-3z^{8}+z^{6}+6z^{4}-6z^{2}$ (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 10                       1 -1 8                     1   1 6                   2 1   -1 4                 3 1     2 2               3 2       -1 0             5 3         2 -2           3 4           1 -4         4 4             0 -6       2 4               2 -8     1 3                 -2 -10   1 2                   1 -12   1                     -1 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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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