L11n51

L11n50

L11n52

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n51 at Knotilus!

Link Presentations

[edit Notes on L11n51's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X18,14,19,13 X9,17,10,16 X17,9,18,8 X22,20,5,19 X20,15,21,16 X14,21,15,22 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 6, -5, -2, 11, -3, 4, -9, 8, 5, -6, -4, 7, -8, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{4}-2v^{3}-2v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{13/2}-4q^{11/2}+6q^{9/2}-7q^{7/2}+8q^{5/2}-8q^{3/2}+6{\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}+az^{3}-8z^{3}a^{-1}+9z^{3}a^{-3}-4z^{3}a^{-5}+3az-9za^{-1}+9za^{-3}-4za^{-5}+za^{-7}+2az^{-1}-4a^{-1}z^{-1}+3a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{2}a^{-8}+4z^{3}a^{-7}-za^{-7}+2z^{6}a^{-6}-z^{4}a^{-6}+z^{2}a^{-6}-a^{-6}+5z^{7}a^{-5}-15z^{5}a^{-5}+16z^{3}a^{-5}-6za^{-5}+a^{-5}z^{-1}+4z^{8}a^{-4}-10z^{6}a^{-4}+3z^{4}a^{-4}+4z^{2}a^{-4}-3a^{-4}+z^{9}a^{-3}+6z^{7}a^{-3}-32z^{5}a^{-3}+37z^{3}a^{-3}-17za^{-3}+3a^{-3}z^{-1}+6z^{8}a^{-2}-20z^{6}a^{-2}+12z^{4}a^{-2}+4z^{2}a^{-2}-3a^{-2}+z^{9}a^{-1}+az^{7}+2z^{7}a^{-1}-5az^{5}-22z^{5}a^{-1}+9az^{3}+34z^{3}a^{-1}-7az-19za^{-1}+2az^{-1}+4a^{-1}z^{-1}+2z^{8}-8z^{6}+8z^{4}-2$ (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 χ 14                   1 -1 12                 3   3 10               3 1   -2 8             4 3     1 6           4 3       -1 4         4 4         0 2       4 6           2 0     1 2             -1 -2   1 4               3 -4   1                 -1 -6 1                   1

Integral Khovanov Homology (db, data source)

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

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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