L11a536

L11a535

L11a537

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a536 at Knotilus!

Link Presentations

[edit Notes on L11a536'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 X12,22,13,21 X20,14,21,13 X14,20,9,19
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -9, 10, -11}, {7, -6, 8, -3, 11, -10, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uvw^{2}x-uvw^{2}+uvwx^{2}-3uvwx+2uvw-uvx^{2}+2uvx-2uw^{2}x+uw^{2}-2uwx^{2}+4uwx-2uw+2ux^{2}-2ux-2vw^{2}x+2vw^{2}-2vwx^{2}+4vwx-2vw+vx^{2}-2vx+2w^{2}x-w^{2}+2wx^{2}-3wx+w-x^{2}+x}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+7q^{5/2}-12q^{3/2}+14{\sqrt {q}}-{\frac {18}{\sqrt {q}}}+{\frac {15}{q^{3/2}}}-{\frac {15}{q^{5/2}}}+{\frac {9}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}+a^{5}z^{-3}-3a^{5}z-a^{5}z^{-1}+3a^{3}z^{3}-3a^{3}z^{-3}+z^{3}a^{-3}-4a^{3}z^{-1}-az^{5}-z^{5}a^{-1}+2az^{3}+3az^{-3}-z^{3}a^{-1}-a^{-1}z^{-3}+6az+7az^{-1}-3za^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+3a^{7}z-a^{7}z^{-1}+2a^{6}z^{6}-3a^{6}z^{4}+a^{6}+3a^{5}z^{7}-3a^{5}z^{5}+a^{5}z^{3}-a^{5}z^{-3}-3a^{5}z+3a^{5}z^{-1}+3a^{4}z^{8}+a^{4}z^{6}+z^{6}a^{-4}-10a^{4}z^{4}-2z^{4}a^{-4}+16a^{4}z^{2}+3a^{4}z^{-2}-11a^{4}+2a^{3}z^{9}+6a^{3}z^{7}+4z^{7}a^{-3}-19a^{3}z^{5}-11z^{5}a^{-3}+26a^{3}z^{3}+7z^{3}a^{-3}-3a^{3}z^{-3}-21a^{3}z+12a^{3}z^{-1}+a^{2}z^{10}+5a^{2}z^{8}+6z^{8}a^{-2}-3a^{2}z^{6}-16z^{6}a^{-2}-17a^{2}z^{4}+10z^{4}a^{-2}+33a^{2}z^{2}+6a^{2}z^{-2}-24a^{2}+6az^{9}+4z^{9}a^{-1}-5az^{7}-4z^{7}a^{-1}-13az^{5}-9z^{5}a^{-1}+29az^{3}+14z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-28az-13za^{-1}+14az^{-1}+6a^{-1}z^{-1}+z^{10}+8z^{8}-19z^{6}+2z^{4}+17z^{2}+3z^{-2}-13$ (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                 8 3     5 2               9 7       -2 0             9 5         4 -2           7 10           3 -4         8 8             0 -6       2 8               6 -8     4 7                 -3 -10   1 5                   4 -12   1                     -1 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a535 L11a537