L11n98

L11n97

L11n99

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n98 at Knotilus!

Link Presentations

[edit Notes on L11n98's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,21,14,20 X7,17,8,16 X9,19,10,18 X17,9,18,8 X19,11,20,10 X15,5,16,22 X21,15,22,14 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 6, -5, 7, 11, -2, -3, 9, -8, 4, -6, 5, -7, 3, -9, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(2)^{7}-t(1)t(2)^{5}+t(2)^{5}+2t(1)t(2)^{4}-2t(2)^{4}-2t(1)t(2)^{3}+2t(2)^{3}+t(1)t(2)^{2}-t(2)^{2}-t(1)}{{\sqrt {t(1)}}t(2)^{7/2}}}$ (db)
Jones polynomial	$q^{17/2}-2q^{15/2}+4q^{13/2}-4q^{11/2}+4q^{9/2}-4q^{7/2}+2q^{5/2}-q^{3/2}-{\sqrt {q}}-{\frac {1}{q^{3/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-3}+z^{5}a^{-1}-8z^{5}a^{-3}+6z^{3}a^{-1}-20z^{3}a^{-3}+3z^{3}a^{-5}+z^{3}a^{-7}+10za^{-1}-20za^{-3}+7za^{-5}+za^{-7}+4a^{-1}z^{-1}-7a^{-3}z^{-1}+3a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-z^{8}a^{-2}-z^{8}a^{-4}+9z^{7}a^{-1}+10z^{7}a^{-3}-z^{7}a^{-7}+9z^{6}a^{-2}+9z^{6}a^{-4}-2z^{6}a^{-6}-2z^{6}a^{-8}-28z^{5}a^{-1}-33z^{5}a^{-3}-3z^{5}a^{-5}-2z^{5}a^{-9}-24z^{4}a^{-2}-26z^{4}a^{-4}+z^{4}a^{-6}+2z^{4}a^{-8}-z^{4}a^{-10}+37z^{3}a^{-1}+46z^{3}a^{-3}+6z^{3}a^{-5}+3z^{3}a^{-9}+23z^{2}a^{-2}+24z^{2}a^{-4}+z^{2}a^{-8}+2z^{2}a^{-10}-20za^{-1}-29za^{-3}-8za^{-5}+za^{-7}-7a^{-2}-7a^{-4}-a^{-10}+4a^{-1}z^{-1}+7a^{-3}z^{-1}+3a^{-5}z^{-1}$ (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                     1   1 14                   3 1   -2 12                 1 1     0 10               3 3       0 8           1 2 1         0 6           1 3           2 4       1 2 2             -1 2         2               2 0     1                   1 -2 1                       1 -4 1                       1

Integral Khovanov Homology (db, data source)

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

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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