L11a63

L11a62

L11a64

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a63 at Knotilus!

Link Presentations

[edit Notes on L11a63's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-3t(2)+1\right)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{17/2}-4q^{15/2}+9q^{13/2}-13q^{11/2}+17q^{9/2}-20q^{7/2}+18q^{5/2}-16q^{3/2}+11{\sqrt {q}}-{\frac {7}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+za^{-7}+a^{-7}z^{-1}-2z^{5}a^{-5}-5z^{3}a^{-5}-5za^{-5}-3a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+8z^{3}a^{-3}+8za^{-3}+3a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+2az-6za^{-1}+az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}+4z^{5}a^{-9}-z^{3}a^{-9}+9z^{6}a^{-8}-9z^{4}a^{-8}+4z^{2}a^{-8}-a^{-8}+12z^{7}a^{-7}-16z^{5}a^{-7}+9z^{3}a^{-7}-3za^{-7}+a^{-7}z^{-1}+10z^{8}a^{-6}-8z^{6}a^{-6}-7z^{4}a^{-6}+5z^{2}a^{-6}-2a^{-6}+5z^{9}a^{-5}+9z^{7}a^{-5}-39z^{5}a^{-5}+31z^{3}a^{-5}-12za^{-5}+3a^{-5}z^{-1}+z^{10}a^{-4}+17z^{8}a^{-4}-46z^{6}a^{-4}+31z^{4}a^{-4}-7z^{2}a^{-4}+8z^{9}a^{-3}-8z^{7}a^{-3}-28z^{5}a^{-3}+41z^{3}a^{-3}-18za^{-3}+3a^{-3}z^{-1}+z^{10}a^{-2}+10z^{8}a^{-2}-40z^{6}a^{-2}+41z^{4}a^{-2}-13z^{2}a^{-2}+2a^{-2}+3z^{9}a^{-1}+az^{7}-4z^{7}a^{-1}-4az^{5}-13z^{5}a^{-1}+6az^{3}+26z^{3}a^{-1}-4az-13za^{-1}+az^{-1}+2a^{-1}z^{-1}+3z^{8}-11z^{6}+13z^{4}-5z^{2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     3   3 14                   6 1   -5 12                 7 3     4 10               10 6       -4 8             10 7         3 6           8 10           2 4         8 10             -2 2       5 10               5 0     2 6                 -4 -2   1 5                   4 -4   2                     -2 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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