L11n407

L11n406

L11n408

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n407 at Knotilus!

Link Presentations

[edit Notes on L11n407's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X22,12,19,11 X10,4,11,3 X5,21,6,20 X21,5,22,18 X12,20,13,19 X14,9,15,10 X2,14,3,13 X8,15,9,16
Gauss code	{1, -10, 5, -3}, {8, 6, -7, -4}, {-6, -1, 2, -11, 9, -5, 4, -8, 10, -9, 11, -2, 3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2(t(1)-1)(t(3)-1)^{2}(t(2)t(3)+1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$3q^{5}-5q^{4}+9q^{3}-9q^{2}+11q-10+8q^{-1}-5q^{-2}+3q^{-3}-q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$a^{-6}z^{-2}+a^{-6}-z^{4}a^{-4}-3z^{2}a^{-4}-2a^{-4}z^{-2}-4a^{-4}+z^{6}a^{-2}-a^{2}z^{4}+3z^{4}a^{-2}-2a^{2}z^{2}+3z^{2}a^{-2}+a^{-2}z^{-2}+3a^{-2}+z^{6}+3z^{4}+2z^{2}$ (db)
Kauffman polynomial	$2az^{9}+2z^{9}a^{-1}+3a^{2}z^{8}+6z^{8}a^{-2}+9z^{8}+a^{3}z^{7}-3az^{7}+4z^{7}a^{-1}+8z^{7}a^{-3}-13a^{2}z^{6}-12z^{6}a^{-2}+7z^{6}a^{-4}-32z^{6}-4a^{3}z^{5}-10az^{5}-25z^{5}a^{-1}-16z^{5}a^{-3}+3z^{5}a^{-5}+17a^{2}z^{4}+4z^{4}a^{-2}-12z^{4}a^{-4}+33z^{4}+5a^{3}z^{3}+18az^{3}+20z^{3}a^{-1}+7z^{3}a^{-3}-6a^{2}z^{2}-3z^{2}a^{-2}+11z^{2}a^{-4}+6z^{2}a^{-6}-14z^{2}-2a^{3}z-6az-6za^{-1}+2za^{-3}+4za^{-5}+a^{2}-a^{-2}-6a^{-4}-4a^{-6}+3-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 11                   3 3 9                 4 2 -2 7               5 1   4 5             4 4     0 3           7 5       2 1         5 6         1 -1       3 5           -2 -3     2 5             3 -5   1 3               -2 -7   2                 2 -9 1                   -1

Integral Khovanov Homology (db, data source)

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

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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