L11a20

L11a19

L11a21

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a20 at Knotilus!

Link Presentations

[edit Notes on L11a20's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)^{5}}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+14{\sqrt {q}}-{\frac {19}{\sqrt {q}}}+{\frac {20}{q^{3/2}}}-{\frac {21}{q^{5/2}}}+{\frac {17}{q^{7/2}}}-{\frac {12}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+6za^{5}+3a^{5}z^{-1}-3z^{5}a^{3}-9z^{3}a^{3}-10za^{3}-3a^{3}z^{-1}+z^{7}a+4z^{5}a+8z^{3}a+7za+2az^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-3a^{5}z^{9}-8a^{3}z^{9}-5az^{9}-4a^{6}z^{8}-12a^{4}z^{8}-17a^{2}z^{8}-9z^{8}-3a^{7}z^{7}-4a^{5}z^{7}+a^{3}z^{7}-6az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+6a^{6}z^{6}+30a^{4}z^{6}+40a^{2}z^{6}-4z^{6}a^{-2}+13z^{6}+8a^{7}z^{5}+24a^{5}z^{5}+32a^{3}z^{5}+30az^{5}+13z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}+2a^{6}z^{4}-22a^{4}z^{4}-32a^{2}z^{4}+5z^{4}a^{-2}-6z^{4}-7a^{7}z^{3}-27a^{5}z^{3}-44a^{3}z^{3}-33az^{3}-8z^{3}a^{-1}+z^{3}a^{-3}-3a^{8}z^{2}-5a^{6}z^{2}+5a^{4}z^{2}+10a^{2}z^{2}-z^{2}a^{-2}+2z^{2}+3a^{7}z+14a^{5}z+22a^{3}z+15az+4za^{-1}+a^{8}+2a^{6}-2a^{2}-a^{7}z^{-1}-3a^{5}z^{-1}-3a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 8 3     -5 0               11 6       5 -2             11 10         -1 -4           10 9           1 -6         7 11             4 -8       5 10               -5 -10     2 7                 5 -12   1 5                   -4 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a19 L11a21