L11a539

L11a538

L11a540

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a539 at Knotilus!

Link Presentations

[edit Notes on L11a539's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(w-1)(x-1)\left(uvx^{2}-uvx+uv-ux^{2}+2ux-2u-2vx^{2}+2vx-v+x^{2}-x+1\right)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$-4q^{9/2}+{\frac {2}{q^{9/2}}}+10q^{7/2}-{\frac {7}{q^{7/2}}}-16q^{5/2}+{\frac {10}{q^{5/2}}}+18q^{3/2}-{\frac {18}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-22{\sqrt {q}}+{\frac {19}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+10az^{3}-9z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-8a^{3}z+15az-11za^{-1}+3za^{-3}+2a^{5}z^{-1}-8a^{3}z^{-1}+11az^{-1}-6a^{-1}z^{-1}+a^{-3}z^{-1}+a^{5}z^{-3}-3a^{3}z^{-3}+3az^{-3}-a^{-1}z^{-3}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-5a^{5}z^{5}+4z^{5}a^{-5}+10a^{5}z^{3}-a^{5}z^{-3}-10a^{5}z+5a^{5}z^{-1}+2a^{4}z^{8}-6a^{4}z^{6}+10z^{6}a^{-4}+3a^{4}z^{4}-8z^{4}a^{-4}+7a^{4}z^{2}+4z^{2}a^{-4}+3a^{4}z^{-2}-9a^{4}-a^{-4}+2a^{3}z^{9}+a^{3}z^{7}+16z^{7}a^{-3}-23a^{3}z^{5}-25z^{5}a^{-3}+43a^{3}z^{3}+17z^{3}a^{-3}-3a^{3}z^{-3}-34a^{3}z-9za^{-3}+14a^{3}z^{-1}+2a^{-3}z^{-1}+a^{2}z^{10}+8a^{2}z^{8}+14z^{8}a^{-2}-27a^{2}z^{6}-17z^{6}a^{-2}+14a^{2}z^{4}-6z^{4}a^{-2}+19a^{2}z^{2}+15z^{2}a^{-2}+6a^{2}z^{-2}-21a^{2}-6a^{-2}+8az^{9}+6z^{9}a^{-1}-3az^{7}+13z^{7}a^{-1}-48az^{5}-59z^{5}a^{-1}+77az^{3}+61z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-50az-35za^{-1}+18az^{-1}+11a^{-1}z^{-1}+z^{10}+20z^{8}-48z^{6}+14z^{4}+23z^{2}+3z^{-2}-18$ (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 12                       1 -1 10                     3   3 8                   7 1   -6 6                 9 3     6 4               9 7       -2 2             13 9         4 0           12 15           3 -2         6 7             -1 -4       4 12               8 -6     3 6                 -3 -8   1 6                   5 -10   1                     -1 -12 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a538 L11a540