L11n423

L11n422

L11n424

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n423 at Knotilus!

Link Presentations

[edit Notes on L11n423's Link Presentations]

Planar diagram presentation	X8192 X16,8,17,7 X10,4,11,3 X2,18,3,17 X18,9,19,10 X20,12,21,11 X5,14,6,15 X15,13,16,22 X13,6,14,1 X4,19,5,20 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 $u$, $v$, $w$, ...)	${\frac {(w-1)\left(u^{2}v-u^{2}+uv^{2}w^{2}-uvw^{2}-uvw-uv+u-v^{2}w^{2}+vw^{2}\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{4}-q^{-4}+3q^{3}+3q^{-3}-3q^{2}-3q^{-2}+5q+5q^{-1}-4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}-a^{2}z^{4}-z^{4}a^{-2}+4z^{4}-2a^{2}z^{2}-2z^{2}a^{-2}+3z^{2}+a^{2}+a^{-2}-2+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{7}+z^{7}a^{-3}-4a^{3}z^{5}-4z^{5}a^{-3}+3a^{3}z^{3}+3z^{3}a^{-3}+3a^{2}z^{8}+3z^{8}a^{-2}-15a^{2}z^{6}-15z^{6}a^{-2}+20a^{2}z^{4}+20z^{4}a^{-2}-7a^{2}z^{2}-7z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{2}-2a^{-2}+2az^{9}+2z^{9}a^{-1}-8az^{7}-8z^{7}a^{-1}+5az^{5}+5z^{5}a^{-1}+az^{3}+z^{3}a^{-1}+2az+2za^{-1}-2az^{-1}-2a^{-1}z^{-1}+6z^{8}-30z^{6}+40z^{4}-14z^{2}+2z^{-2}-3$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 χ 9                 1 -1 7               2   2 5             2 2   0 3           3 2 1   2 1         3 4       1 -1       2 2 1       1 -3     2 4           2 -5   1 1             0 -7   2               2 -9 1                 -1

Integral Khovanov Homology (db, data source)

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

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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