L10n24

L10n23

L10n25

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n24 at Knotilus!

Link Presentations

[edit Notes on L10n24's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X13,18,14,19 X9,17,10,16 X17,9,18,8 X15,20,16,5 X19,14,20,15 X2536 X4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 3, 6, -5, -2, 10, -3, -4, 8, -7, 5, -6, 4, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(v^{2}+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+2q^{5/2}-3q^{3/2}+2{\sqrt {q}}-{\frac {3}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{-3}+2za^{-3}-z^{5}a^{-1}+az^{3}-4z^{3}a^{-1}+2az-4za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-4}-4z^{4}a^{-4}+3z^{2}a^{-4}+2z^{7}a^{-3}-9z^{5}a^{-3}+10z^{3}a^{-3}-4za^{-3}+z^{8}a^{-2}-3z^{6}a^{-2}-z^{4}a^{-2}+a^{2}z^{2}+3z^{2}a^{-2}+az^{7}+3z^{7}a^{-1}-5az^{5}-14z^{5}a^{-1}+8az^{3}+18z^{3}a^{-1}-4az-8za^{-1}+az^{-1}+a^{-1}z^{-1}+z^{8}-4z^{6}+3z^{4}+z^{2}-1$ (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 χ 10               1 -1 8             1   1 6           1 1   0 4         2 1     1 2     1 1 1       1 0     3 2         1 -2   1 2           1 -4   1             -1 -6 1               1

Integral Khovanov Homology (db, data source)

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

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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