L11a369

L11a368

L11a370

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a369 at Knotilus!

Link Presentations

[edit Notes on L11a369's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{2}t(1)^{4}-t(2)t(1)^{4}+2t(2)^{3}t(1)^{3}-3t(2)^{2}t(1)^{3}+2t(2)t(1)^{3}-t(1)^{3}+t(2)^{4}t(1)^{2}-3t(2)^{3}t(1)^{2}+3t(2)^{2}t(1)^{2}-3t(2)t(1)^{2}+t(1)^{2}-t(2)^{4}t(1)+2t(2)^{3}t(1)-3t(2)^{2}t(1)+2t(2)t(1)-t(2)^{3}+t(2)^{2}}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	${\frac {8}{q^{9/2}}}-{\frac {10}{q^{7/2}}}-q^{5/2}+{\frac {9}{q^{5/2}}}+2q^{3/2}-{\frac {8}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {2}{q^{15/2}}}+{\frac {4}{q^{13/2}}}-{\frac {7}{q^{11/2}}}-4{\sqrt {q}}+{\frac {6}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-2a^{7}z+a^{5}z^{5}+2a^{5}z^{3}+2a^{3}z^{5}+6a^{3}z^{3}+4a^{3}z+a^{3}z^{-1}+az^{5}+2az^{3}-z^{3}a^{-1}-az-az^{-1}-2za^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{10}+2z^{2}a^{10}-2z^{5}a^{9}+3z^{3}a^{9}-3z^{6}a^{8}+5z^{4}a^{8}-3z^{2}a^{8}-4z^{7}a^{7}+11z^{5}a^{7}-16z^{3}a^{7}+6za^{7}-3z^{8}a^{6}+6z^{6}a^{6}-5z^{4}a^{6}-2z^{2}a^{6}-2z^{9}a^{5}+3z^{7}a^{5}+z^{5}a^{5}-7z^{3}a^{5}+4za^{5}-z^{10}a^{4}+z^{8}a^{4}+2z^{4}a^{4}-4z^{9}a^{3}+16z^{7}a^{3}-27z^{5}a^{3}+27z^{3}a^{3}-9za^{3}+a^{3}z^{-1}-z^{10}a^{2}+2z^{8}a^{2}+2z^{4}a^{2}-a^{2}-2z^{9}a+8z^{7}a-10z^{5}a+8z^{3}a-5za+az^{-1}-2z^{8}+9z^{6}-11z^{4}+3z^{2}-z^{7}a^{-1}+5z^{5}a^{-1}-7z^{3}a^{-1}+2za^{-1}$ (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 6                       1 1 4                     1   -1 2                   3 1   2 0                 3 1     -2 -2               5 3       2 -4             5 4         -1 -6           5 4           1 -8         4 6             2 -10       3 4               -1 -12     1 4                 3 -14   1 3                   -2 -16   1                     1 -18 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a368 L11a370