L11n256

L11n255

L11n257

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n256 at Knotilus!

Link Presentations

[edit Notes on L11n256's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X16,12,17,11 X18,22,19,21 X20,14,21,13 X12,20,13,19 X22,18,9,17 X15,8,16,5 X7,14,8,15 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -9, 8}, {-11, 2, 3, -6, 5, 9, -8, -3, 7, -4, 6, -5, 4, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)t(3)^{3}-2t(1)t(3)^{2}-t(1)t(2)t(3)^{2}-t(2)t(3)^{2}-t(3)^{2}+t(1)t(3)+t(1)t(2)t(3)+2t(2)t(3)+t(3)-2t(2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{7}+q^{6}-q^{5}-q^{4}+2q^{3}+q^{-3}-q^{2}-q^{-2}+4q+3q^{-1}-2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-2}+z^{4}a^{-4}-z^{4}+a^{2}z^{2}-3z^{2}a^{-2}+5z^{2}a^{-4}-z^{2}a^{-6}-3z^{2}+2a^{2}-3a^{-2}+6a^{-4}-2a^{-6}-3+a^{2}z^{-2}-2a^{-2}z^{-2}+3a^{-4}z^{-2}-a^{-6}z^{-2}-z^{-2}$ (db)
Kauffman polynomial	$z^{7}a^{-7}-6z^{5}a^{-7}+10z^{3}a^{-7}-7za^{-7}+2a^{-7}z^{-1}+z^{8}a^{-6}-6z^{6}a^{-6}+8z^{4}a^{-6}-2z^{2}a^{-6}-a^{-6}z^{-2}+a^{-6}+2z^{7}a^{-5}-16z^{5}a^{-5}+35z^{3}a^{-5}-27za^{-5}+8a^{-5}z^{-1}+z^{8}a^{-4}-7z^{6}a^{-4}+12z^{4}a^{-4}-7z^{2}a^{-4}-3a^{-4}z^{-2}+5a^{-4}+3z^{7}a^{-3}-21z^{5}a^{-3}+44z^{3}a^{-3}-34za^{-3}+10a^{-3}z^{-1}+z^{8}a^{-2}+a^{2}z^{6}-5z^{6}a^{-2}-5a^{2}z^{4}+8z^{4}a^{-2}+7a^{2}z^{2}-8z^{2}a^{-2}+a^{2}z^{-2}-2a^{-2}z^{-2}-4a^{2}+4a^{-2}+az^{7}+3z^{7}a^{-1}-3az^{5}-14z^{5}a^{-1}-az^{3}+18z^{3}a^{-1}+4az-10za^{-1}-2az^{-1}+2a^{-1}z^{-1}+z^{8}-3z^{6}-z^{4}+4z^{2}+z^{-2}-3$ (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 7 χ 15                       1 -1 13                         0 11                   1 1   0 9               3 1       -2 7             1 1 1       1 5           2 4 1         1 3         3 1 1           3 1       2 5 1             2 -1     1 1 1               1 -3     2                   2 -5 1 1                     0 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11n255 L11n257