L11n302

L11n301

L11n303

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n302 at Knotilus!

Link Presentations

[edit Notes on L11n302's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X20,13,21,14 X22,19,11,20 X15,5,16,10 X17,9,18,8 X7,17,8,16 X9,19,10,18 X14,21,15,22 X2536 X11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -7, 6, -8, 5}, {-11, 2, 3, -9, -5, 7, -6, 8, 4, -3, 9, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-uv^{2}w+uvw-uv-uw^{2}+uw-v^{2}w^{3}+v^{2}w^{2}+vw^{4}-vw^{3}+w^{3}}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{6}-q^{5}+2q^{4}-q^{3}+2q^{2}+1+q^{-1}-q^{-2}+q^{-3}-q^{-4}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}+z^{6}-a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}+7z^{4}-4a^{2}z^{2}-17z^{2}a^{-2}+4z^{2}a^{-4}+16z^{2}-4a^{2}-16a^{-2}+5a^{-4}+15-a^{2}z^{-2}-5a^{-2}z^{-2}+2a^{-4}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$a^{2}z^{8}+z^{8}a^{-2}+z^{8}a^{-4}+z^{8}+a^{3}z^{7}+2az^{7}+2z^{7}a^{-1}+2z^{7}a^{-3}+z^{7}a^{-5}-6a^{2}z^{6}-8z^{6}a^{-2}-5z^{6}a^{-4}+z^{6}a^{-6}-8z^{6}-6a^{3}z^{5}-14az^{5}-16z^{5}a^{-1}-12z^{5}a^{-3}-4z^{5}a^{-5}+10a^{2}z^{4}+22z^{4}a^{-2}+7z^{4}a^{-4}-5z^{4}a^{-6}+20z^{4}+10a^{3}z^{3}+28az^{3}+37z^{3}a^{-1}+21z^{3}a^{-3}+2z^{3}a^{-5}-7a^{2}z^{2}-31z^{2}a^{-2}-7z^{2}a^{-4}+6z^{2}a^{-6}-25z^{2}-5a^{3}z-21az-33za^{-1}-16za^{-3}+za^{-5}+4a^{2}+20a^{-2}+6a^{-4}-2a^{-6}+17+a^{3}z^{-1}+5az^{-1}+9a^{-1}z^{-1}+5a^{-3}z^{-1}-a^{2}z^{-2}-5a^{-2}z^{-2}-2a^{-4}z^{-2}-4z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     1 1 0 9                   1     1 7               1 1 1     1 5             1 3 1       1 3           1   1         2 1         1 4 2           1 -1       1   1             2 -3       1 1               0 -5   1 1                   0 -7                         0 -9 1                       -1

Integral Khovanov Homology (db, data source)

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

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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