L11n301

L11n300

L11n302

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n301 at Knotilus!

Link Presentations

[edit Notes on L11n301's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X13,21,14,20 X19,11,20,22 X15,5,16,10 X17,9,18,8 X7,17,8,16 X9,19,10,18 X21,15,22,14 X2536 X11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -7, 6, -8, 5}, {-11, 2, -3, 9, -5, 7, -6, 8, -4, 3, -9, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)t(3)^{4}-2t(1)t(2)^{2}t(3)^{3}+t(2)^{2}t(3)^{3}+2t(1)t(2)t(3)^{3}-3t(2)t(3)^{3}+t(3)^{3}+2t(1)t(2)^{2}t(3)^{2}-t(2)^{2}t(3)^{2}+t(1)t(3)^{2}-4t(1)t(2)t(3)^{2}+4t(2)t(3)^{2}-2t(3)^{2}-t(1)t(2)^{2}t(3)-t(1)t(3)+3t(1)t(2)t(3)-2t(2)t(3)+2t(3)-t(1)t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{11}+3q^{10}-6q^{9}+9q^{8}-11q^{7}+12q^{6}-10q^{5}+9q^{4}-4q^{3}+3q^{2}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{2}a^{-10}-a^{-10}z^{-2}-2a^{-10}+3z^{4}a^{-8}+10z^{2}a^{-8}+4a^{-8}z^{-2}+11a^{-8}-2z^{6}a^{-6}-10z^{4}a^{-6}-20z^{2}a^{-6}-5a^{-6}z^{-2}-18a^{-6}+3z^{4}a^{-4}+10z^{2}a^{-4}+2a^{-4}z^{-2}+9a^{-4}$ (db)
Kauffman polynomial	$z^{5}a^{-13}-2z^{3}a^{-13}+za^{-13}+3z^{6}a^{-12}-6z^{4}a^{-12}+3z^{2}a^{-12}-a^{-12}+4z^{7}a^{-11}-6z^{5}a^{-11}+z^{3}a^{-11}-2za^{-11}+a^{-11}z^{-1}+3z^{8}a^{-10}-9z^{4}a^{-10}+6z^{2}a^{-10}-a^{-10}z^{-2}+z^{9}a^{-9}+8z^{7}a^{-9}-26z^{5}a^{-9}+33z^{3}a^{-9}-19za^{-9}+5a^{-9}z^{-1}+7z^{8}a^{-8}-17z^{6}a^{-8}+23z^{4}a^{-8}-19z^{2}a^{-8}-4a^{-8}z^{-2}+13a^{-8}+z^{9}a^{-7}+7z^{7}a^{-7}-28z^{5}a^{-7}+48z^{3}a^{-7}-35za^{-7}+9a^{-7}z^{-1}+4z^{8}a^{-6}-14z^{6}a^{-6}+32z^{4}a^{-6}-38z^{2}a^{-6}-5a^{-6}z^{-2}+22a^{-6}+3z^{7}a^{-5}-9z^{5}a^{-5}+18z^{3}a^{-5}-19za^{-5}+5a^{-5}z^{-1}+6z^{4}a^{-4}-16z^{2}a^{-4}-2a^{-4}z^{-2}+11a^{-4}$ (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 9 χ 23                   1 -1 21                 2   2 19               4 1   -3 17             5 2     3 15           7 5       -2 13         5 4         1 11       5 7           2 9     4 5             -1 7   1 6               5 5 2 3                 -1 3 3                   3

Integral Khovanov Homology (db, data source)

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

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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