L10n112

L10n111

L10n113

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n112 at Knotilus!

Link Presentations

[edit Notes on L10n112's Link Presentations]

Planar diagram presentation	X6172 X2536 X11,19,12,18 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X15,17,16,20 X19,13,20,16 X17,9,18,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 10}, {-7, 6, -8, 9}, {-10, 3, -9, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(2)+t(1)t(3)t(2)-t(3)t(2)+t(1)t(4)t(2)-t(1)t(3)t(4)t(2)+t(3)t(4)t(2)+2t(1)t(5)t(2)-t(1)t(3)t(5)t(2)+t(3)t(5)t(2)-t(1)t(4)t(5)t(2)-t(5)t(2)+t(3)-t(1)t(4)+t(1)t(3)t(4)-2t(3)t(4)+t(4)-t(1)t(5)-t(3)t(5)+t(1)t(4)t(5)+t(3)t(4)t(5)-t(4)t(5)+t(5)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}{\sqrt {t(4)}}{\sqrt {t(5)}}}}$ (db)
Jones polynomial	$q^{9}-q^{8}+6q^{7}-5q^{6}+11q^{5}-5q^{4}+10q^{3}-5q^{2}+4q$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$a^{-10}z^{-4}+a^{-10}z^{-2}-4a^{-8}z^{-4}-7a^{-8}z^{-2}-4a^{-8}+6a^{-6}z^{-4}+6z^{2}a^{-6}+15a^{-6}z^{-2}+14a^{-6}-3z^{4}a^{-4}-4a^{-4}z^{-4}-10z^{2}a^{-4}-13a^{-4}z^{-2}-16a^{-4}+a^{-2}z^{-4}+4z^{2}a^{-2}+4a^{-2}z^{-2}+6a^{-2}$ (db)
Kauffman polynomial	$z^{8}a^{-6}+z^{8}a^{-8}+5z^{7}a^{-5}+6z^{7}a^{-7}+z^{7}a^{-9}+10z^{6}a^{-4}+12z^{6}a^{-6}+3z^{6}a^{-8}+z^{6}a^{-10}+6z^{5}a^{-3}-6z^{5}a^{-7}-25z^{4}a^{-4}-39z^{4}a^{-6}-19z^{4}a^{-8}-5z^{4}a^{-10}-10z^{3}a^{-3}-30z^{3}a^{-5}-30z^{3}a^{-7}-10z^{3}a^{-9}+10z^{2}a^{-2}+30z^{2}a^{-4}+40z^{2}a^{-6}+30z^{2}a^{-8}+10z^{2}a^{-10}+20za^{-3}+55za^{-5}+55za^{-7}+20za^{-9}-10a^{-2}-25a^{-4}-31a^{-6}-25a^{-8}-10a^{-10}-15a^{-3}z^{-1}-41a^{-5}z^{-1}-41a^{-7}z^{-1}-15a^{-9}z^{-1}+5a^{-2}z^{-2}+14a^{-4}z^{-2}+18a^{-6}z^{-2}+14a^{-8}z^{-2}+5a^{-10}z^{-2}+4a^{-3}z^{-3}+12a^{-5}z^{-3}+12a^{-7}z^{-3}+4a^{-9}z^{-3}-a^{-2}z^{-4}-4a^{-4}z^{-4}-6a^{-6}z^{-4}-4a^{-8}z^{-4}-a^{-10}z^{-4}$ (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   \ 0 1 2 3 4 5 6 7 8 χ 19                 1 1 17               1 1 0 15             5     5 13             1     1 11         11 5       6 9       4 10         6 7     6 1           5 5 1   4             5 3 5 6               -1 1 4                 4

Integral Khovanov Homology (db, data source)

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

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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