L11n106

L11n105

L11n107

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n106 at Knotilus!

Link Presentations

[edit Notes on L11n106's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,16,8,17 X17,22,18,5 X13,18,14,19 X9,21,10,20 X19,14,20,15 X21,9,22,8 X15,10,16,11 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 8, -6, 9, 11, -2, -5, 7, -9, 3, -4, 5, -7, 6, -8, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{5}-2uv^{4}-uv^{3}+3uv^{2}-2uv-2v^{4}+3v^{3}-v^{2}-2v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{9/2}}}-{\frac {3}{q^{11/2}}}+{\frac {2}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4za^{9}+3a^{9}z^{-1}-z^{5}a^{7}-7z^{3}a^{7}-9za^{7}-3a^{7}z^{-1}+z^{7}a^{5}+6z^{5}a^{5}+10z^{3}a^{5}+8za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-5z^{3}a^{3}-5za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{6}-4a^{12}z^{4}+3a^{12}z^{2}-a^{12}+2a^{11}z^{7}-8a^{11}z^{5}+6a^{11}z^{3}-3a^{11}z+a^{11}z^{-1}+2a^{10}z^{8}-8a^{10}z^{6}+4a^{10}z^{4}+3a^{10}z^{2}-2a^{10}+a^{9}z^{9}-3a^{9}z^{7}-5a^{9}z^{5}+15a^{9}z^{3}-11a^{9}z+3a^{9}z^{-1}+3a^{8}z^{8}-17a^{8}z^{6}+24a^{8}z^{4}-9a^{8}z^{2}+a^{7}z^{9}-4a^{7}z^{7}-5a^{7}z^{5}+22a^{7}z^{3}-14a^{7}z+3a^{7}z^{-1}+2a^{6}z^{8}-14a^{6}z^{6}+25a^{6}z^{4}-14a^{6}z^{2}+2a^{6}+2a^{5}z^{7}-14a^{5}z^{5}+23a^{5}z^{3}-12a^{5}z+2a^{5}z^{-1}+a^{4}z^{8}-6a^{4}z^{6}+9a^{4}z^{4}-5a^{4}z^{2}+a^{3}z^{7}-6a^{3}z^{5}+10a^{3}z^{3}-6a^{3}z+a^{3}z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                       1 1 -2                         0 -4                 1 2 1   0 -6               1 1 1     1 -8             2 2 1       -1 -10           3 2 1         2 -12         2 4 1           1 -14       2 2 1             1 -16     1 2 1               0 -18   1 2                   -1 -20   1                     1 -22 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11n105 L11n107