L11n408

L11n407

L11n409

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n408 at Knotilus!

Link Presentations

[edit Notes on L11n408's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X7,16,8,17 X17,19,18,22 X11,20,12,21 X19,10,20,11 X21,5,22,18 X9,14,10,15 X15,8,16,9 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {-6, 5, -7, 4}, {10, -1, -3, 9, -8, 6, -5, -2, 11, 8, -9, 3, -4, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

L11n407 L11n409