L11n365

L11n364

L11n366

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n365 at Knotilus!

Link Presentations

[edit Notes on L11n365's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X22,15,17,16 X9,18,10,19 X17,8,18,9 X20,13,21,14 X14,21,15,22 X16,19,5,20 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {-6, 5, 9, -7, 8, -4}, {10, -1, 3, 6, -5, -2, 11, -3, 7, -8, 4, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(w-1)\left(-v^{2}w+vw^{3}-vw^{2}+vw-v+w^{2}\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-1+4q^{-1}-5q^{-2}+8q^{-3}-8q^{-4}+8q^{-5}-6q^{-6}+5q^{-7}-2q^{-8}+q^{-9}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+a^{8}z^{-2}+2a^{8}-2a^{6}z^{4}-6a^{6}z^{2}-2a^{6}z^{-2}-6a^{6}+a^{4}z^{6}+4a^{4}z^{4}+6a^{4}z^{2}+a^{4}z^{-2}+3a^{4}-a^{2}z^{4}-a^{2}z^{2}+a^{2}$ (db)
Kauffman polynomial	$z^{6}a^{10}-4z^{4}a^{10}+5z^{2}a^{10}-2a^{10}+2z^{7}a^{9}-6z^{5}a^{9}+3z^{3}a^{9}+za^{9}+2z^{8}a^{8}-4z^{6}a^{8}-z^{4}a^{8}-z^{2}a^{8}-a^{8}z^{-2}+3a^{8}+z^{9}a^{7}+z^{7}a^{7}-9z^{5}a^{7}+9z^{3}a^{7}-7za^{7}+2a^{7}z^{-1}+5z^{8}a^{6}-17z^{6}a^{6}+25z^{4}a^{6}-23z^{2}a^{6}-2a^{6}z^{-2}+11a^{6}+z^{9}a^{5}+z^{7}a^{5}-8z^{5}a^{5}+14z^{3}a^{5}-9za^{5}+2a^{5}z^{-1}+3z^{8}a^{4}-12z^{6}a^{4}+26z^{4}a^{4}-21z^{2}a^{4}-a^{4}z^{-2}+7a^{4}+2z^{7}a^{3}-5z^{5}a^{3}+9z^{3}a^{3}-2za^{3}+4z^{4}a^{2}-4z^{2}a^{2}+z^{3}a-za$ (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 χ 1                   1 -1 -1                 3   3 -3               4 3   -1 -5             4 1     3 -7           4 4       0 -9         4 4         0 -11       2 4           2 -13     3 4             -1 -15   1 4               3 -17   1                 -1 -19 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L11n364 L11n366