L11n422

L11n421

L11n423

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n422 at Knotilus!

Link Presentations

[edit Notes on L11n422's Link Presentations]

Planar diagram presentation	X8192 X16,8,17,7 X3,10,4,11 X2,18,3,17 X9,19,10,18 X20,12,21,11 X14,6,15,5 X22,15,13,16 X6,14,1,13 X19,5,20,4 X12,22,7,21
Gauss code	{1, -4, -3, 10, 7, -9}, {2, -1, -5, 3, 6, -11}, {9, -7, 8, -2, 4, 5, -10, -6, 11, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{2}w^{2}-2u^{2}v^{2}w-u^{2}vw^{2}+2u^{2}vw-u^{2}v-uv^{2}w^{2}+uv^{2}w+3uvw^{2}-3uvw-uw^{2}+uw+vw^{3}-2vw^{2}+vw+2w^{2}-w}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{9}-2q^{8}+5q^{7}-6q^{6}+8q^{5}-8q^{4}+8q^{3}-5q^{2}+4q-1$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}-z^{4}a^{-2}+4z^{4}a^{-4}-2z^{4}a^{-6}-z^{2}a^{-2}+6z^{2}a^{-4}-6z^{2}a^{-6}+z^{2}a^{-8}+a^{-2}+3a^{-4}-6a^{-6}+2a^{-8}+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{9}a^{-5}+z^{9}a^{-7}+2z^{8}a^{-4}+5z^{8}a^{-6}+3z^{8}a^{-8}+z^{7}a^{-3}+z^{7}a^{-5}+2z^{7}a^{-7}+2z^{7}a^{-9}-5z^{6}a^{-4}-17z^{6}a^{-6}-11z^{6}a^{-8}+z^{6}a^{-10}+z^{5}a^{-3}-8z^{5}a^{-5}-15z^{5}a^{-7}-6z^{5}a^{-9}+4z^{4}a^{-2}+11z^{4}a^{-4}+25z^{4}a^{-6}+14z^{4}a^{-8}-4z^{4}a^{-10}+z^{3}a^{-1}+14z^{3}a^{-5}+18z^{3}a^{-7}+3z^{3}a^{-9}-3z^{2}a^{-2}-10z^{2}a^{-4}-23z^{2}a^{-6}-12z^{2}a^{-8}+4z^{2}a^{-10}-9za^{-5}-9za^{-7}-a^{-2}+5a^{-4}+11a^{-6}+5a^{-8}-a^{-10}+2a^{-5}z^{-1}+2a^{-7}z^{-1}-a^{-4}z^{-2}-2a^{-6}z^{-2}-a^{-8}z^{-2}$ (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   \ -1 0 1 2 3 4 5 6 7 8 χ 19                   1 1 17                 2 1 -1 15               3     3 13             3 2     -1 11           5 3       2 9         3 3         0 7       5 5           0 5     2 5             3 3   2 3               -1 1   3                 3 -1 1                   -1

Integral Khovanov Homology (db, data source)

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

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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