L11n122

L11n121

L11n123

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n122 at Knotilus!

Link Presentations

[edit Notes on L11n122's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X16,10,17,9 X11,21,12,20 X21,9,22,8 X7,19,8,18 X19,13,20,12 X10,16,11,15 X22,17,5,18 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, -6, 5, 3, -8, -4, 7, 11, -2, 8, -3, 9, 6, -7, 4, -5, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {-2uv^{2}+2uv+u+v^{3}+2v^{2}-2v}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-q^{11/2}+2q^{9/2}-2q^{7/2}+3q^{5/2}-2q^{3/2}+{\sqrt {q}}-{\frac {2}{\sqrt {q}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$za^{3}+2a^{3}z^{-1}-z^{3}a-5za-3az^{-1}+z^{3}a^{-1}+2za^{-1}+a^{-1}z^{-1}+z^{3}a^{-3}+za^{-3}-za^{-5}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-3z^{8}a^{-2}-2z^{8}a^{-4}-z^{8}+5z^{7}a^{-1}+4z^{7}a^{-3}-z^{7}a^{-5}+17z^{6}a^{-2}+11z^{6}a^{-4}+6z^{6}-a^{3}z^{5}-5z^{5}a^{-1}-z^{5}a^{-3}+5z^{5}a^{-5}-26z^{4}a^{-2}-17z^{4}a^{-4}-9z^{4}+5a^{3}z^{3}+4az^{3}+2z^{3}a^{-1}-3z^{3}a^{-3}-6z^{3}a^{-5}+3a^{2}z^{2}+13z^{2}a^{-2}+9z^{2}a^{-4}+7z^{2}-5a^{3}z-7az-3za^{-1}+za^{-5}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                     1 1 10                   1   -1 8                 1 1   0 6               2 1     -1 4           1 1 1       -1 2           1 2         1 0       1 3 1           1 -2         2             2 -4     1 1               0 -6 1                     1 -8 1                     1

Integral Khovanov Homology (db, data source)

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

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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