L11n424

L11n423

L11n425

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n424 at Knotilus!

Link Presentations

[edit Notes on L11n424's Link Presentations]

Planar diagram presentation	X8192 X7,16,8,17 X5,14,6,15 X3,10,4,11 X13,4,14,5 X2,18,3,17 X18,9,19,10 X21,7,22,12 X11,13,12,22 X15,20,16,21 X19,1,20,6
Gauss code	{1, -6, -4, 5, -3, 11}, {-2, -1, 7, 4, -9, 8}, {-5, 3, -10, 2, 6, -7, -11, 10, -8, 9}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(w-1)(uw-v-w+1)(uvw-uv-uw+v)}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{-8}+3q^{-7}-6q^{-6}+9q^{-5}-10q^{-4}+12q^{-3}-9q^{-2}+2q+8q^{-1}-4$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{6}-2z^{2}a^{6}-2a^{6}+z^{6}a^{4}+4z^{4}a^{4}+8z^{2}a^{4}+a^{4}z^{-2}+6a^{4}-3z^{4}a^{2}-8z^{2}a^{2}-2a^{2}z^{-2}-7a^{2}+2z^{2}+z^{-2}+3$ (db)
Kauffman polynomial	$z^{5}a^{9}-2z^{3}a^{9}+3z^{6}a^{8}-6z^{4}a^{8}+z^{2}a^{8}+5z^{7}a^{7}-12z^{5}a^{7}+8z^{3}a^{7}-2za^{7}+5z^{8}a^{6}-14z^{6}a^{6}+18z^{4}a^{6}-12z^{2}a^{6}+4a^{6}+2z^{9}a^{5}-8z^{5}a^{5}+14z^{3}a^{5}-6za^{5}+8z^{8}a^{4}-28z^{6}a^{4}+49z^{4}a^{4}-38z^{2}a^{4}-a^{4}z^{-2}+13a^{4}+2z^{9}a^{3}-4z^{7}a^{3}+6z^{5}a^{3}+2z^{3}a^{3}-7za^{3}+2a^{3}z^{-1}+3z^{8}a^{2}-11z^{6}a^{2}+28z^{4}a^{2}-32z^{2}a^{2}-2a^{2}z^{-2}+13a^{2}+z^{7}a+z^{5}a-2z^{3}a-3za+2az^{-1}+3z^{4}-7z^{2}-z^{-2}+5$ (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 χ 3                   2 2 1                 2   -2 -1               6 2   4 -3             5 4     -1 -5           7 4       3 -7         5 7         2 -9       4 5           -1 -11     2 5             3 -13   1 4               -3 -15   2                 2 -17 1                   -1

Integral Khovanov Homology (db, data source)

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

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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