L11n37

L11n36

L11n38

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n37 at Knotilus!

Link Presentations

[edit Notes on L11n37's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X11,14,12,15 X8493 X5,13,6,12 X13,5,14,22 X15,19,16,18 X9,17,10,16 X17,11,18,10 X2,20,3,19
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, -9, 10, -4, 6, -7, 4, -8, 9, -10, 8, 11, -2, 3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(2v^{2}-3v+2\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{15/2}-3q^{13/2}+6q^{11/2}-8q^{9/2}+9q^{7/2}-10q^{5/2}+8q^{3/2}-7{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {1}{q^{3/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$2z^{5}a^{-3}-3z^{3}a^{-1}+7z^{3}a^{-3}-3z^{3}a^{-5}+az-5za^{-1}+9za^{-3}-6za^{-5}+za^{-7}+az^{-1}-2a^{-1}z^{-1}+3a^{-3}z^{-1}-3a^{-5}z^{-1}+a^{-7}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-3}-z^{9}a^{-5}-3z^{8}a^{-2}-6z^{8}a^{-4}-3z^{8}a^{-6}-2z^{7}a^{-1}-5z^{7}a^{-3}-6z^{7}a^{-5}-3z^{7}a^{-7}+9z^{6}a^{-2}+15z^{6}a^{-4}+5z^{6}a^{-6}-z^{6}a^{-8}+6z^{5}a^{-1}+24z^{5}a^{-3}+27z^{5}a^{-5}+9z^{5}a^{-7}-15z^{4}a^{-2}-9z^{4}a^{-4}+6z^{4}a^{-6}+3z^{4}a^{-8}-3z^{4}-az^{3}-16z^{3}a^{-1}-36z^{3}a^{-3}-28z^{3}a^{-5}-7z^{3}a^{-7}+7z^{2}a^{-2}+z^{2}a^{-4}-7z^{2}a^{-6}-3z^{2}a^{-8}+2z^{2}+2az+11za^{-1}+20za^{-3}+14za^{-5}+3za^{-7}-2a^{-2}+2a^{-6}+a^{-8}-az^{-1}-2a^{-1}z^{-1}-3a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 16                   1 -1 14                 2   2 12               4 1   -3 10             4 2     2 8           5 4       -1 6         5 4         1 4       3 5           2 2     4 5             -1 0   1 5               4 -2   2                 -2 -4 1                   1

Integral Khovanov Homology (db, data source)

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

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.

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