L11a80

L11a79

L11a81

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a80 at Knotilus!

Link Presentations

[edit Notes on L11a80's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{5}-2t(2)^{5}-4t(1)t(2)^{4}+8t(2)^{4}+9t(1)t(2)^{3}-11t(2)^{3}-11t(1)t(2)^{2}+9t(2)^{2}+8t(1)t(2)-4t(2)-2t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+10q^{7/2}-16q^{5/2}+20q^{3/2}-23{\sqrt {q}}+{\frac {22}{\sqrt {q}}}-{\frac {19}{q^{3/2}}}+{\frac {13}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+8az^{3}-9z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-5a^{3}z+8az-9za^{-1}+3za^{-3}+a^{5}z^{-1}-2a^{3}z^{-1}+3az^{-1}-3a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-4a^{5}z^{5}+4z^{5}a^{-5}+6a^{5}z^{3}-4a^{5}z+a^{5}z^{-1}+3a^{4}z^{8}-10a^{4}z^{6}+10z^{6}a^{-4}+11a^{4}z^{4}-8z^{4}a^{-4}-4a^{4}z^{2}+4z^{2}a^{-4}-a^{-4}+4a^{3}z^{9}-8a^{3}z^{7}+16z^{7}a^{-3}-4a^{3}z^{5}-23z^{5}a^{-3}+16a^{3}z^{3}+13z^{3}a^{-3}-9a^{3}z-4za^{-3}+2a^{3}z^{-1}+a^{-3}z^{-1}+2a^{2}z^{10}+8a^{2}z^{8}+16z^{8}a^{-2}-39a^{2}z^{6}-24z^{6}a^{-2}+42a^{2}z^{4}+6z^{4}a^{-2}-15a^{2}z^{2}+4z^{2}a^{-2}+2a^{2}-2a^{-2}+13az^{9}+9z^{9}a^{-1}-25az^{7}-6az^{5}-33z^{5}a^{-1}+27az^{3}+30z^{3}a^{-1}-15az-14za^{-1}+3az^{-1}+3a^{-1}z^{-1}+2z^{10}+21z^{8}-63z^{6}+46z^{4}-11z^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   7 1   -6 6                 9 3     6 4               11 7       -4 2             12 9         3 0           11 12           1 -2         8 11             -3 -4       5 11               6 -6     3 8                 -5 -8   1 6                   5 -10   2                     -2 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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.

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