L11n13

L11n12

L11n14

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n13 at Knotilus!

Link Presentations

[edit Notes on L11n13's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,10,6,11 X3849 X11,22,12,5 X13,20,14,21 X19,14,20,15 X21,12,22,13 X9,18,10,19 X15,2,16,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -10, 4, -6, 9, -7, 8, -11, -2, 3, 10, -8, 7, -9, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

L11n12 L11n14