L11a19

L11a18

L11a20

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a19 at Knotilus!

Link Presentations

[edit Notes on L11a19's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-4t(2)^{3}+4t(2)^{2}-4t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$16q^{9/2}-18q^{7/2}+17q^{5/2}-{\frac {1}{q^{5/2}}}-16q^{3/2}+{\frac {3}{q^{3/2}}}+q^{17/2}-3q^{15/2}+7q^{13/2}-12q^{11/2}+11{\sqrt {q}}-{\frac {7}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+2za^{-7}+a^{-7}z^{-1}-2z^{5}a^{-5}-6z^{3}a^{-5}-7za^{-5}-3a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+8z^{3}a^{-3}+9za^{-3}+3a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+2az-6za^{-1}+az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-3z^{9}a^{-1}-8z^{9}a^{-3}-5z^{9}a^{-5}-10z^{8}a^{-2}-16z^{8}a^{-4}-9z^{8}a^{-6}-3z^{8}-az^{7}+4z^{7}a^{-1}+10z^{7}a^{-3}-4z^{7}a^{-5}-9z^{7}a^{-7}+41z^{6}a^{-2}+49z^{6}a^{-4}+13z^{6}a^{-6}-6z^{6}a^{-8}+11z^{6}+4az^{5}+13z^{5}a^{-1}+25z^{5}a^{-3}+33z^{5}a^{-5}+14z^{5}a^{-7}-3z^{5}a^{-9}-42z^{4}a^{-2}-40z^{4}a^{-4}-4z^{4}a^{-6}+6z^{4}a^{-8}-z^{4}a^{-10}-13z^{4}-6az^{3}-25z^{3}a^{-1}-40z^{3}a^{-3}-36z^{3}a^{-5}-13z^{3}a^{-7}+2z^{3}a^{-9}+15z^{2}a^{-2}+11z^{2}a^{-4}-4z^{2}a^{-6}-4z^{2}a^{-8}+z^{2}a^{-10}+5z^{2}+4az+13za^{-1}+20za^{-3}+16za^{-5}+5za^{-7}-2a^{-2}+2a^{-6}+a^{-8}-az^{-1}-2a^{-1}z^{-1}-3a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     2   2 14                   5 1   -4 12                 7 2     5 10               9 5       -4 8             9 7         2 6           8 9           1 4         8 9             -1 2       5 10               5 0     2 6                 -4 -2   1 5                   4 -4   2                     -2 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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