L11n419

L11n418

L11n420

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n419 at Knotilus!

Link Presentations

[edit Notes on L11n419's Link Presentations]

Planar diagram presentation	X8192 X16,8,17,7 X3,10,4,11 X2,18,3,17 X9,19,10,18 X20,12,21,11 X5,14,6,15 X15,13,16,22 X13,6,14,1 X19,5,20,4 X12,22,7,21
Gauss code	{1, -4, -3, 10, -7, 9}, {2, -1, -5, 3, 6, -11}, {-9, 7, -8, -2, 4, 5, -10, -6, 11, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)t(3)^{3}-t(2)t(3)^{3}+t(1)t(2)^{2}t(3)^{2}-2t(1)t(2)t(3)^{2}+t(2)t(3)^{2}-t(1)t(3)-t(1)^{2}t(2)t(3)+2t(1)t(2)t(3)+t(1)^{2}t(2)-t(1)t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{6}+3q^{5}-3q^{4}+5q^{3}-4q^{2}+4q+2q^{-1}-2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-2z^{4}a^{-2}-7z^{2}a^{-2}+3z^{2}a^{-4}+2z^{2}-8a^{-2}+5a^{-4}-a^{-6}+4-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{3}a^{-7}-za^{-7}+3z^{4}a^{-6}-4z^{2}a^{-6}+a^{-6}+z^{7}a^{-5}-3z^{5}a^{-5}+7z^{3}a^{-5}-3za^{-5}+z^{8}a^{-4}-5z^{6}a^{-4}+16z^{4}a^{-4}-18z^{2}a^{-4}-a^{-4}z^{-2}+8a^{-4}+2z^{7}a^{-3}-6z^{5}a^{-3}+11z^{3}a^{-3}-8za^{-3}+2a^{-3}z^{-1}+z^{8}a^{-2}-5z^{6}a^{-2}+16z^{4}a^{-2}-23z^{2}a^{-2}-2a^{-2}z^{-2}+12a^{-2}+z^{7}a^{-1}-3z^{5}a^{-1}+5z^{3}a^{-1}-6za^{-1}+2a^{-1}z^{-1}+3z^{4}-9z^{2}-z^{-2}+6$ (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 13               1 -1 11             2   2 9           2 2   0 7         3 1     2 5       2 3       1 3     2 2         0 1   1 3           2 -1 1 1             0 -3 2               2

Integral Khovanov Homology (db, data source)

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

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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