L11n367

L11n366

L11n368

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n367 at Knotilus!

Link Presentations

[edit Notes on L11n367's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X13,20,14,21 X19,9,20,8 X9,19,10,18 X15,22,16,17 X17,16,18,5 X21,14,22,15 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {-8, 6, -5, 4, -9, 7}, {10, -1, 3, 5, -6, -2, 11, -3, -4, 9, -7, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(w-1)\left(v^{2}w-1\right)}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{4}+2q^{3}+2q^{-3}-2q^{2}-q^{-2}+3q+3q^{-1}-2$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{4}z^{-2}+a^{4}-a^{2}z^{4}-z^{4}a^{-2}-4a^{2}z^{2}-3z^{2}a^{-2}-2a^{2}z^{-2}-4a^{2}-a^{-2}+z^{6}+5z^{4}+7z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$-a^{4}z^{2}-a^{4}z^{-2}+3a^{4}+z^{7}a^{-3}-5z^{5}a^{-3}+6z^{3}a^{-3}-2a^{3}z-za^{-3}+2a^{3}z^{-1}+a^{2}z^{8}+2z^{8}a^{-2}-6a^{2}z^{6}-11z^{6}a^{-2}+12a^{2}z^{4}+17z^{4}a^{-2}-13a^{2}z^{2}-9z^{2}a^{-2}-2a^{2}z^{-2}+7a^{2}+2a^{-2}+az^{9}+z^{9}a^{-1}-5az^{7}-4z^{7}a^{-1}+6az^{5}+z^{5}a^{-1}-az^{3}+5z^{3}a^{-1}-4az-3za^{-1}+2az^{-1}+3z^{8}-17z^{6}+29z^{4}-21z^{2}-z^{-2}+7$ (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 9               1 -1 7             1   1 5           1 1   0 3         2 1     1 1     1 1 1       1 -1     3 2         1 -3 1 1 2           2 -5 2 1             1 -7 2               2

Integral Khovanov Homology (db, data source)

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

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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