L11n107

L11n106

L11n108

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n107 at Knotilus!

Link Presentations

[edit Notes on L11n107's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(3t(2)^{2}-4t(2)+3\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$12q^{9/2}-11q^{7/2}+6q^{5/2}-3q^{3/2}+q^{21/2}-3q^{19/2}+7q^{17/2}-10q^{15/2}+13q^{13/2}-14q^{11/2}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-2z^{5}a^{-5}-z^{5}a^{-7}+3z^{3}a^{-3}-5z^{3}a^{-5}-z^{3}a^{-7}+z^{3}a^{-9}+5za^{-3}-5za^{-5}-za^{-7}+za^{-9}+2a^{-3}z^{-1}-2a^{-5}z^{-1}-a^{-7}z^{-1}+a^{-9}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-12}-3z^{4}a^{-12}+3z^{2}a^{-12}-a^{-12}+3z^{7}a^{-11}-8z^{5}a^{-11}+6z^{3}a^{-11}-za^{-11}+4z^{8}a^{-10}-8z^{6}a^{-10}+z^{4}a^{-10}+2z^{2}a^{-10}+2z^{9}a^{-9}+4z^{7}a^{-9}-19z^{5}a^{-9}+12z^{3}a^{-9}-2za^{-9}-a^{-9}z^{-1}+10z^{8}a^{-8}-21z^{6}a^{-8}+11z^{4}a^{-8}-6z^{2}a^{-8}+3a^{-8}+2z^{9}a^{-7}+8z^{7}a^{-7}-25z^{5}a^{-7}+19z^{3}a^{-7}-4za^{-7}-a^{-7}z^{-1}+6z^{8}a^{-6}-9z^{6}a^{-6}+7z^{4}a^{-6}-2z^{2}a^{-6}+7z^{7}a^{-5}-14z^{5}a^{-5}+19z^{3}a^{-5}-10za^{-5}+2a^{-5}z^{-1}+3z^{6}a^{-4}+3z^{2}a^{-4}-3a^{-4}+6z^{3}a^{-3}-7za^{-3}+2a^{-3}z^{-1}$ (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   \ 0 1 2 3 4 5 6 7 8 9 χ 22                   1 -1 20                 2   2 18               5 1   -4 16             5 2     3 14           8 5       -3 12         6 5         1 10       6 8           2 8     5 6             -1 6   1 6               5 4 2 5                 -3 2 3                   3

Integral Khovanov Homology (db, data source)

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

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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