L11n111

L11n110

L11n112

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n111 at Knotilus!

Link Presentations

[edit Notes on L11n111's Link Presentations]

Planar diagram presentation	X6172 X3,12,4,13 X7,18,8,19 X19,22,20,5 X9,21,10,20 X21,9,22,8 X16,11,17,12 X14,17,15,18 X10,15,11,16 X2536 X13,4,14,1
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 6, -5, -9, 7, 2, -11, -8, 9, -7, 8, 3, -4, 5, -6, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{5}-2t(1)t(2)^{4}-t(2)^{4}-t(1)t(2)^{3}+2t(2)^{3}+2t(1)t(2)^{2}-t(2)^{2}-t(1)t(2)-2t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {2}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {1}{q^{11/2}}}+{\frac {1}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4za^{9}+4a^{9}z^{-1}-z^{5}a^{7}-7z^{3}a^{7}-11za^{7}-6a^{7}z^{-1}+z^{7}a^{5}+6z^{5}a^{5}+11z^{3}a^{5}+11za^{5}+5a^{5}z^{-1}-z^{5}a^{3}-5z^{3}a^{3}-6za^{3}-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{6}-4a^{12}z^{4}+3a^{12}z^{2}-a^{12}+2a^{11}z^{7}-9a^{11}z^{5}+9a^{11}z^{3}-4a^{11}z+a^{11}z^{-1}+a^{10}z^{8}-3a^{10}z^{6}-4a^{10}z^{4}+9a^{10}z^{2}-3a^{10}+3a^{9}z^{7}-18a^{9}z^{5}+29a^{9}z^{3}-18a^{9}z+4a^{9}z^{-1}+a^{8}z^{8}-5a^{8}z^{6}+a^{8}z^{4}+7a^{8}z^{2}-3a^{8}+3a^{7}z^{7}-22a^{7}z^{5}+42a^{7}z^{3}-28a^{7}z+6a^{7}z^{-1}+a^{6}z^{8}-6a^{6}z^{6}+6a^{6}z^{4}+a^{6}z^{2}-a^{6}+3a^{5}z^{7}-19a^{5}z^{5}+33a^{5}z^{3}-22a^{5}z+5a^{5}z^{-1}+a^{4}z^{8}-5a^{4}z^{6}+5a^{4}z^{4}-a^{4}+a^{3}z^{7}-6a^{3}z^{5}+11a^{3}z^{3}-8a^{3}z+2a^{3}z^{-1}$ (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                   2 1   1 -6               1 1 1     1 -8             1 1 1       -1 -10           2 2 1         1 -12         2 3 1           0 -14       1 1 1             1 -16     1 2 1               0 -18   1 1                   0 -20   1                     1 -22 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11n110 L11n112