L11a533

L11a532

L11a534

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a533 at Knotilus!

Link Presentations

[edit Notes on L11a533's Link Presentations]

Planar diagram presentation	X6172 X2536 X18,12,19,11 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X22,16,13,15 X20,18,21,17 X16,22,17,21 X12,20,9,19
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, {7, -6, 8, -10, 9, -3, 11, -9, 10, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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