L11a432

L11a431

L11a433

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a432 at Knotilus!

Link Presentations

[edit Notes on L11a432's Link Presentations]

Planar diagram presentation	X6172 X14,6,15,5 X8493 X2,16,3,15 X16,7,17,8 X18,14,19,13 X20,9,21,10 X12,19,5,20 X22,11,13,12 X10,21,11,22 X4,17,1,18
Gauss code	{1, -4, 3, -11}, {2, -1, 5, -3, 7, -10, 9, -8}, {6, -2, 4, -5, 11, -6, 8, -7, 10, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)(t(3)t(2)-t(2)+1)(t(2)t(3)-t(3)+1)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-8}+5q^{-7}-10q^{-6}+16q^{-5}-20q^{-4}+q^{3}+24q^{-3}-4q^{2}-22q^{-2}+8q+20q^{-1}-13$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+6a^{4}z^{4}-9a^{2}z^{4}+3z^{4}-a^{6}z^{2}+3a^{4}z^{2}-4a^{2}z^{2}+2z^{2}+a^{6}-4a^{4}+3a^{2}+a^{6}z^{-2}-2a^{4}z^{-2}+a^{2}z^{-2}$ (db)
Kauffman polynomial	$a^{9}z^{5}+5a^{8}z^{6}-5a^{8}z^{4}+a^{8}+10a^{7}z^{7}-14a^{7}z^{5}+3a^{7}z^{3}+11a^{6}z^{8}-14a^{6}z^{6}+3a^{6}z^{4}-2a^{6}z^{2}+a^{6}z^{-2}-a^{6}+7a^{5}z^{9}+a^{5}z^{7}-16a^{5}z^{5}+5a^{5}z^{3}+4a^{5}z-2a^{5}z^{-1}+2a^{4}z^{10}+17a^{4}z^{8}-44a^{4}z^{6}+36a^{4}z^{4}-9a^{4}z^{2}+2a^{4}z^{-2}-4a^{4}+13a^{3}z^{9}-22a^{3}z^{7}+7a^{3}z^{5}+3a^{3}z^{3}+4a^{3}z-2a^{3}z^{-1}+2a^{2}z^{10}+13a^{2}z^{8}-45a^{2}z^{6}+z^{6}a^{-2}+47a^{2}z^{4}-2z^{4}a^{-2}-13a^{2}z^{2}+a^{2}z^{-2}-3a^{2}+6az^{9}-9az^{7}+4z^{7}a^{-1}-2az^{5}-10z^{5}a^{-1}+6az^{3}+5z^{3}a^{-1}+7z^{8}-19z^{6}+17z^{4}-6z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     3   -3 3                   5 1   4 1                 8 3     -5 -1               12 5       7 -3             12 10         -2 -5           12 10           2 -7         8 12             4 -9       8 12               -4 -11     4 10                 6 -13   1 6                   -5 -15   4                     4 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{8}$ $r=-3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a431 L11a433