L11a64

L11a63

L11a65

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a64 at Knotilus!

Link Presentations

[edit Notes on L11a64's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{4}-2t(2)^{3}+3t(2)^{2}-2t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-12q^{9/2}+7q^{7/2}-5q^{5/2}+2q^{3/2}+q^{23/2}-3q^{21/2}+7q^{19/2}-10q^{17/2}+13q^{15/2}-14q^{13/2}+13q^{11/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-5}-z^{7}a^{-7}+z^{5}a^{-3}-4z^{5}a^{-5}-4z^{5}a^{-7}+z^{5}a^{-9}+4z^{3}a^{-3}-4z^{3}a^{-5}-6z^{3}a^{-7}+3z^{3}a^{-9}+4za^{-3}-7za^{-7}+3za^{-9}+a^{-3}z^{-1}+a^{-5}z^{-1}-4a^{-7}z^{-1}+2a^{-9}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-14}-z^{2}a^{-14}+3z^{5}a^{-13}-2z^{3}a^{-13}+6z^{6}a^{-12}-8z^{4}a^{-12}+6z^{2}a^{-12}-2a^{-12}+7z^{7}a^{-11}-9z^{5}a^{-11}+4z^{3}a^{-11}+za^{-11}+6z^{8}a^{-10}-7z^{6}a^{-10}+4z^{2}a^{-10}-a^{-10}+4z^{9}a^{-9}-6z^{7}a^{-9}+9z^{5}a^{-9}-15z^{3}a^{-9}+8za^{-9}-2a^{-9}z^{-1}+z^{10}a^{-8}+6z^{8}a^{-8}-21z^{6}a^{-8}+24z^{4}a^{-8}-19z^{2}a^{-8}+6a^{-8}+6z^{9}a^{-7}-19z^{7}a^{-7}+25z^{5}a^{-7}-24z^{3}a^{-7}+13za^{-7}-4a^{-7}z^{-1}+z^{10}a^{-6}+2z^{8}a^{-6}-16z^{6}a^{-6}+23z^{4}a^{-6}-16z^{2}a^{-6}+5a^{-6}+2z^{9}a^{-5}-5z^{7}a^{-5}-z^{5}a^{-5}+5z^{3}a^{-5}+za^{-5}-a^{-5}z^{-1}+2z^{8}a^{-4}-8z^{6}a^{-4}+8z^{4}a^{-4}-a^{-4}+z^{7}a^{-3}-5z^{5}a^{-3}+8z^{3}a^{-3}-5za^{-3}+a^{-3}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 24                       1 -1 22                     2   2 20                   5 1   -4 18                 5 2     3 16               8 5       -3 14             6 5         1 12           7 8           1 10         5 6             -1 8       2 7               5 6     3 5                 -2 4   1 4                   3 2   1                     -1 0 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a63 L11a65