L11a535

L11a534

L11a536

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a535 at Knotilus!

Link Presentations

[edit Notes on L11a535's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {-2t(1)t(4)^{2}t(3)^{2}+t(1)t(2)t(4)^{2}t(3)^{2}-2t(2)t(4)^{2}t(3)^{2}+2t(4)^{2}t(3)^{2}+t(1)t(4)t(3)^{2}-t(1)t(2)t(4)t(3)^{2}+2t(2)t(4)t(3)^{2}-t(4)t(3)^{2}+2t(1)t(4)^{2}t(3)-t(1)t(2)t(4)^{2}t(3)+t(2)t(4)^{2}t(3)-t(4)^{2}t(3)+t(1)t(3)-t(1)t(2)t(3)+2t(2)t(3)-3t(1)t(4)t(3)+2t(1)t(2)t(4)t(3)-3t(2)t(4)t(3)+2t(4)t(3)-t(3)-2t(1)+2t(1)t(2)-2t(2)+2t(1)t(4)-t(1)t(2)t(4)+t(2)t(4)-t(4)+1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$-4q^{9/2}+{\frac {1}{q^{9/2}}}+7q^{7/2}-{\frac {5}{q^{7/2}}}-11q^{5/2}+{\frac {6}{q^{5/2}}}+13q^{3/2}-{\frac {12}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-15{\sqrt {q}}+{\frac {12}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+12az^{3}-6z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-11a^{3}z+16az-6za^{-1}+3a^{5}z^{-1}-10a^{3}z^{-1}+11az^{-1}-4a^{-1}z^{-1}+a^{5}z^{-3}-3a^{3}z^{-3}+3az^{-3}-a^{-1}z^{-3}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-a^{3}z^{9}-5az^{9}-4z^{9}a^{-1}-a^{4}z^{8}+a^{2}z^{8}-7z^{8}a^{-2}-5z^{8}-a^{5}z^{7}-a^{3}z^{7}+12az^{7}+4z^{7}a^{-1}-8z^{7}a^{-3}+2a^{4}z^{6}-2a^{2}z^{6}+10z^{6}a^{-2}-7z^{6}a^{-4}+13z^{6}+6a^{5}z^{5}+17a^{3}z^{5}-2az^{5}+9z^{5}a^{-3}-4z^{5}a^{-5}+6a^{4}z^{4}+19a^{2}z^{4}-4z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}+z^{4}-13a^{5}z^{3}-32a^{3}z^{3}-16az^{3}+3z^{3}a^{-5}-17a^{4}z^{2}-33a^{2}z^{2}-16z^{2}+13a^{5}z+28a^{3}z+21az+3za^{-1}-3za^{-3}+13a^{4}+24a^{2}-a^{-2}+11-6a^{5}z^{-1}-14a^{3}z^{-1}-12az^{-1}-3a^{-1}z^{-1}+a^{-3}z^{-1}-3a^{4}z^{-2}-6a^{2}z^{-2}-3z^{-2}+a^{5}z^{-3}+3a^{3}z^{-3}+3az^{-3}+a^{-1}z^{-3}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over $r$).

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   4 1   -3 6                 7 3     4 4               9 7       -2 2             6 4         2 0           7 10           3 -2         5 5             0 -4       2 8               6 -6     3 4                 -1 -8   1 5                   4 -10                         0 -12 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{5}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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