L11n357

L11n356

L11n358

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n357 at Knotilus!

Link Presentations

[edit Notes on L11n357's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,19,12,18 X7,16,8,17 X15,8,16,9 X17,15,18,22 X13,21,14,20 X19,13,20,12 X21,5,22,14 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, -6, 3, -8, 7, -9, 6}, {10, -1, -4, 5, 11, -2, -3, 8, -7, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(w-1)\left(-uv^{2}w-uv+u+v^{2}w^{3}-vw^{3}-w^{2}\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{-5}-q^{4}-q^{-4}+2q^{3}+3q^{-3}-2q^{2}-3q^{-2}+4q+4q^{-1}-3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$a^{4}z^{2}+a^{4}z^{-2}+3a^{4}-2a^{2}z^{4}-z^{4}a^{-2}-8a^{2}z^{2}-3z^{2}a^{-2}-2a^{2}z^{-2}-8a^{2}-a^{-2}+z^{6}+5z^{4}+8z^{2}+z^{-2}+6$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+a^{4}z^{8}+4a^{2}z^{8}+3z^{8}-5a^{3}z^{7}-2az^{7}+3z^{7}a^{-1}-7a^{4}z^{6}-24a^{2}z^{6}+2z^{6}a^{-2}-15z^{6}+5a^{3}z^{5}-8az^{5}-12z^{5}a^{-1}+z^{5}a^{-3}+17a^{4}z^{4}+47a^{2}z^{4}-6z^{4}a^{-2}+24z^{4}+5a^{3}z^{3}+18az^{3}+11z^{3}a^{-1}-2z^{3}a^{-3}-17a^{4}z^{2}-38a^{2}z^{2}+3z^{2}a^{-2}+2z^{2}a^{-4}-20z^{2}-7a^{3}z-10az-3za^{-1}+za^{-3}+za^{-5}+7a^{4}+14a^{2}-a^{-2}-a^{-4}+8+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 9                   1 -1 7                 1   1 5               2 2   0 3             2 1 1   2 1           2 3       1 -1         2 2 1       1 -3       2 3           1 -5     1 1             0 -7   1 3               2 -9                     0 -11 1                   1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $i=3$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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