L11a530

L11a529

L11a531

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a530 at Knotilus!

Link Presentations

[edit Notes on L11a530's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X20,14,21,13 X18,10,19,9 X10,22,11,21 X14,20,7,19 X16,5,17,6 X22,18,15,17 X2738 X4,11,5,12 X6,15,1,16
Gauss code	{1, -9, 2, -10, 7, -11}, {9, -1, 4, -5, 10, -2, 3, -6}, {11, -7, 8, -4, 6, -3, 5, -8}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{3}-t(1)t(3)t(2)^{3}+2t(3)t(2)^{3}-t(2)^{3}+2t(1)^{2}t(2)^{2}+t(1)^{2}t(3)^{2}t(2)^{2}-3t(1)t(3)^{2}t(2)^{2}+3t(3)^{2}t(2)^{2}-4t(1)t(2)^{2}-3t(1)^{2}t(3)t(2)^{2}+8t(1)t(3)t(2)^{2}-5t(3)t(2)^{2}+2t(2)^{2}-3t(1)^{2}t(2)-2t(1)^{2}t(3)^{2}t(2)+4t(1)t(3)^{2}t(2)-2t(3)^{2}t(2)+3t(1)t(2)+5t(1)^{2}t(3)t(2)-8t(1)t(3)t(2)+3t(3)t(2)-t(2)+t(1)^{2}t(3)^{2}-t(1)t(3)^{2}-2t(1)^{2}t(3)+t(1)t(3)}{t(1)t(2)^{3/2}t(3)}}$ (db)
Jones polynomial	$-q^{5}+4q^{4}-9q^{3}+16q^{2}-20q+24-23q^{-1}+20q^{-2}-14q^{-3}+9q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}-3z^{2}a^{4}+a^{4}z^{-2}+3z^{4}a^{2}+z^{2}a^{2}-2a^{2}z^{-2}-3a^{2}-z^{6}-z^{4}-2z^{2}+z^{-2}+1+2z^{4}a^{-2}+z^{2}a^{-2}+a^{-2}-z^{2}a^{-4}$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+3a^{5}z^{7}-6a^{5}z^{5}+z^{5}a^{-5}+3a^{5}z^{3}-z^{3}a^{-5}+6a^{4}z^{8}-14a^{4}z^{6}+4z^{6}a^{-4}+15a^{4}z^{4}-5z^{4}a^{-4}-12a^{4}z^{2}+2z^{2}a^{-4}-a^{4}z^{-2}+5a^{4}+5a^{3}z^{9}-a^{3}z^{7}+8z^{7}a^{-3}-18a^{3}z^{5}-11z^{5}a^{-3}+21a^{3}z^{3}+5z^{3}a^{-3}-9a^{3}z+2a^{3}z^{-1}+2a^{2}z^{10}+12a^{2}z^{8}+10z^{8}a^{-2}-34a^{2}z^{6}-14z^{6}a^{-2}+32a^{2}z^{4}+8z^{4}a^{-2}-23a^{2}z^{2}-2z^{2}a^{-2}-2a^{2}z^{-2}+11a^{2}-a^{-2}+12az^{9}+7z^{9}a^{-1}-13az^{7}-z^{7}a^{-1}-11az^{5}-11z^{5}a^{-1}+18az^{3}+6z^{3}a^{-1}-9az+2az^{-1}+2z^{10}+16z^{8}-37z^{6}+27z^{4}-12z^{2}-z^{-2}+5$ (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 11                       1 -1 9                     3   3 7                   6 1   -5 5                 10 3     7 3               11 7       -4 1             13 9         4 -1           12 13           1 -3         8 11             -3 -5       6 12               6 -7     3 8                 -5 -9     6                   6 -11 1 3                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a529 L11a531