L11n428

L11n427

L11n429

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n428 at Knotilus!

Link Presentations

[edit Notes on L11n428's Link Presentations]

Planar diagram presentation	X8192 X9,20,10,21 X5,15,6,14 X12,14,7,13 X16,8,17,7 X22,18,13,17 X3,10,4,11 X18,11,19,12 X15,1,16,6 X19,4,20,5 X2,21,3,22
Gauss code	{1, -11, -7, 10, -3, 9}, {5, -1, -2, 7, 8, -4}, {4, 3, -9, -5, 6, -8, -10, 2, 11, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(w-1)(uv-1)^{2}}{uv{\sqrt {w}}}}$ (db)
Jones polynomial	$-q^{4}+q^{3}+q+1+q^{-2}+q^{-4}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{4}z^{-2}+2a^{4}-a^{2}z^{4}-z^{4}a^{-2}-6a^{2}z^{2}-4z^{2}a^{-2}-2a^{2}z^{-2}-7a^{2}-2a^{-2}+z^{6}+6z^{4}+10z^{2}+z^{-2}+7$ (db)
Kauffman polynomial	$a^{4}z^{4}-4a^{4}z^{2}-a^{4}z^{-2}+5a^{4}+z^{7}a^{-3}-6z^{5}a^{-3}+a^{3}z^{3}+9z^{3}a^{-3}-3a^{3}z-2za^{-3}+2a^{3}z^{-1}+z^{8}a^{-2}-2a^{2}z^{6}-7z^{6}a^{-2}+13a^{2}z^{4}+15z^{4}a^{-2}-24a^{2}z^{2}-12z^{2}a^{-2}-2a^{2}z^{-2}+13a^{2}+4a^{-2}+z^{7}a^{-1}-az^{5}-7z^{5}a^{-1}+5az^{3}+13z^{3}a^{-1}-7az-6za^{-1}+2az^{-1}+z^{8}-9z^{6}+27z^{4}-32z^{2}-z^{-2}+13$ (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 9                   1 -1 7                     0 5             1 1 1   1 3             1       1 1         2 1 1       2 -1       1 3 1         1 -3     1 1 1           1 -5   1 3 1             1 -7     1               1 -9 1                   1

Integral Khovanov Homology (db, data source)

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

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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