L10n101

L10n100

L10n102

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n101 at Knotilus!

Link Presentations

[edit Notes on L10n101's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X18,12,19,11 X20,16,17,15 X16,20,9,19 X12,18,13,17 X2536 X9,1,10,4
Gauss code	{1, -9, -2, 10}, {9, -1, -3, 4}, {8, -5, 7, -6}, {-10, 2, 5, -8, -4, 3, 6, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(4)-1)\left(-t(4)^{2}+t(1)t(4)+t(2)t(4)-t(1)t(3)t(4)+t(1)t(2)t(3)t(4)-t(2)t(3)t(4)-t(4)+t(1)t(2)t(3)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}t(4)^{3/2}}}$ (db)
Jones polynomial	$2q^{9/2}-4q^{7/2}+2q^{5/2}-4q^{3/2}-q^{17/2}-q^{13/2}-3q^{11/2}+2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-a^{-9}z^{-3}-a^{-9}z^{-1}+3a^{-7}z^{-3}+2za^{-7}+5a^{-7}z^{-1}-3a^{-5}z^{-3}-3za^{-5}-6a^{-5}z^{-1}-z^{5}a^{-3}-3z^{3}a^{-3}+a^{-3}z^{-3}-za^{-3}+a^{-3}z^{-1}+z^{3}a^{-1}+2za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{7}a^{-9}-7z^{5}a^{-9}+15z^{3}a^{-9}-a^{-9}z^{-3}-14za^{-9}+6a^{-9}z^{-1}+z^{6}a^{-8}-9z^{4}a^{-8}+19z^{2}a^{-8}+3a^{-8}z^{-2}-13a^{-8}+z^{7}a^{-7}-10z^{5}a^{-7}+28z^{3}a^{-7}-3a^{-7}z^{-3}-27za^{-7}+14a^{-7}z^{-1}+2z^{6}a^{-6}-14z^{4}a^{-6}+33z^{2}a^{-6}+6a^{-6}z^{-2}-24a^{-6}+z^{7}a^{-5}-4z^{5}a^{-5}+10z^{3}a^{-5}-3a^{-5}z^{-3}-16za^{-5}+12a^{-5}z^{-1}+3z^{6}a^{-4}-10z^{4}a^{-4}+14z^{2}a^{-4}+3a^{-4}z^{-2}-11a^{-4}+z^{7}a^{-3}-6z^{3}a^{-3}-a^{-3}z^{-3}+3a^{-3}z^{-1}+2z^{6}a^{-2}-5z^{4}a^{-2}+a^{-2}+z^{5}a^{-1}-3z^{3}a^{-1}+3za^{-1}-a^{-1}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 18                     1 1 16                     1 1 14                 1     1 12             4         4 10           3 5 1       1 8         2             2 6         3 1           2 4     4 2               2 2   1 3                 2 0   1                   -1 -2 1                     1

Integral Khovanov Homology (db, data source)

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

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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