L11n351

L11n350

L11n352

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n351 at Knotilus!

Link Presentations

[edit Notes on L11n351's Link Presentations]

Planar diagram presentation	X6172 X3,15,4,14 X11,20,12,21 X7,18,8,19 X9,13,10,22 X21,17,22,16 X17,8,18,9 X15,11,16,10 X19,12,20,5 X2536 X13,1,14,4
Gauss code	{1, -10, -2, 11}, {10, -1, -4, 7, -5, 8, -3, 9}, {-11, 2, -8, 6, -7, 4, -9, 3, -6, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {-uv^{3}w+uv^{3}+uv^{2}w^{3}-3uv^{2}w^{2}+4uv^{2}w-2uv^{2}-uvw^{3}+4uvw^{2}-3uvw+uv-v^{2}w^{3}+3v^{2}w^{2}-4v^{2}w+v^{2}+2vw^{3}-4vw^{2}+3vw-v-w^{3}+w^{2}}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}-7q^{3}+12q^{2}-13q+15-13q^{-1}+11q^{-2}-6q^{-3}+3q^{-4}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{-4}+a^{4}z^{-2}+2a^{4}-a^{-4}+a^{2}z^{4}+2z^{4}a^{-2}-2a^{2}z^{2}+3z^{2}a^{-2}-2a^{2}z^{-2}-5a^{2}+2a^{-2}-z^{6}-2z^{4}-z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$2az^{9}+2z^{9}a^{-1}+5a^{2}z^{8}+5z^{8}a^{-2}+10z^{8}+3a^{3}z^{7}+5az^{7}+7z^{7}a^{-1}+5z^{7}a^{-3}-12a^{2}z^{6}-5z^{6}a^{-2}+3z^{6}a^{-4}-20z^{6}-3a^{3}z^{5}-13az^{5}-18z^{5}a^{-1}-7z^{5}a^{-3}+z^{5}a^{-5}+6a^{4}z^{4}+25a^{2}z^{4}-5z^{4}a^{-4}+24z^{4}+4a^{3}z^{3}+12az^{3}+12z^{3}a^{-1}+2z^{3}a^{-3}-2z^{3}a^{-5}-11a^{4}z^{2}-29a^{2}z^{2}-z^{2}a^{-2}+3z^{2}a^{-4}-22z^{2}-5a^{3}z-8az-3za^{-1}+za^{-3}+za^{-5}+6a^{4}+13a^{2}-a^{-4}+9+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 11                   1 -1 9                 2   2 7               5 1   -4 5             7 2     5 3           6 5       -1 1         9 7         2 -1       7 9           2 -3     4 6             -2 -5   2 7               5 -7 1 4                 -3 -9 3                   3

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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