L11n280

L11n279

L11n281

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n280 at Knotilus!

Link Presentations

[edit Notes on L11n280's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X9,18,10,19 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 X19,10,20,5
Gauss code	{1, -4, 3, -10}, {-2, -1, 5, -3, -6, 11}, {-8, 2, 4, -5, 10, 9, -7, 6, -11, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)(w-1)\left(w^{2}-w+1\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{3}+2q^{2}-5q+8-7q^{-1}+9q^{-2}-6q^{-3}+6q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{2}z^{6}+a^{4}z^{4}-4a^{2}z^{4}+2z^{4}+2a^{4}z^{2}-7a^{2}z^{2}-z^{2}a^{-2}+6z^{2}+2a^{4}-8a^{2}-2a^{-2}+8+2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+3a^{5}z^{7}-9a^{5}z^{5}+5a^{5}z^{3}+3a^{4}z^{8}-7a^{4}z^{6}-a^{4}z^{2}-2a^{4}z^{-2}+4a^{4}+a^{3}z^{9}+3a^{3}z^{7}-17a^{3}z^{5}+18a^{3}z^{3}+z^{3}a^{-3}-12a^{3}z-2za^{-3}+5a^{3}z^{-1}+a^{-3}z^{-1}+5a^{2}z^{8}-15a^{2}z^{6}+15a^{2}z^{4}+2z^{4}a^{-2}-11a^{2}z^{2}-2z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+10a^{2}+a^{-2}+az^{9}+az^{7}+z^{7}a^{-1}-11az^{5}-3z^{5}a^{-1}+23az^{3}+11z^{3}a^{-1}-21az-11za^{-1}+9az^{-1}+5a^{-1}z^{-1}+2z^{8}-7z^{6}+14z^{4}-9z^{2}-4z^{-2}+7$ (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 7                   1 -1 5                 1   1 3               4 1   -3 1             4 1     3 -1           5 6       1 -3         4 3 1       2 -5       2 5           3 -7     4 4             0 -9   1 4               3 -11   2                 -2 -13 1                   1

Integral Khovanov Homology (db, data source)

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

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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