L11a439

L11a438

L11a440

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a439 at Knotilus!

Link Presentations

[edit Notes on L11a439's Link Presentations]

Planar diagram presentation	X6172 X14,6,15,5 X10,4,11,3 X2,16,3,15 X20,18,21,17 X16,9,17,10 X8,20,9,19 X18,8,19,7 X22,12,13,11 X12,14,5,13 X4,21,1,22
Gauss code	{1, -4, 3, -11}, {2, -1, 8, -7, 6, -3, 9, -10}, {10, -2, 4, -6, 5, -8, 7, -5, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)^{2}(t(3)-1)^{2}(t(2)t(3)+1)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{10}-4q^{9}+9q^{8}-14q^{7}+18q^{6}-20q^{5}+21q^{4}-16q^{3}+13q^{2}-7q+4-q^{-1}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{4}a^{-8}+2z^{2}a^{-8}+a^{-8}-2z^{6}a^{-6}-7z^{4}a^{-6}-7z^{2}a^{-6}+a^{-6}z^{-2}-2a^{-6}+z^{8}a^{-4}+5z^{6}a^{-4}+9z^{4}a^{-4}+6z^{2}a^{-4}-2a^{-4}z^{-2}-a^{-4}-z^{6}a^{-2}-3z^{4}a^{-2}-z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-12}+4z^{5}a^{-11}-z^{3}a^{-11}+9z^{6}a^{-10}-8z^{4}a^{-10}+3z^{2}a^{-10}-a^{-10}+13z^{7}a^{-9}-18z^{5}a^{-9}+9z^{3}a^{-9}-2za^{-9}+12z^{8}a^{-8}-15z^{6}a^{-8}+4z^{2}a^{-8}-a^{-8}+7z^{9}a^{-7}+z^{7}a^{-7}-27z^{5}a^{-7}+21z^{3}a^{-7}-6za^{-7}+2z^{10}a^{-6}+15z^{8}a^{-6}-49z^{6}a^{-6}+38z^{4}a^{-6}-9z^{2}a^{-6}+a^{-6}z^{-2}-a^{-6}+12z^{9}a^{-5}-28z^{7}a^{-5}+8z^{5}a^{-5}+10z^{3}a^{-5}-2za^{-5}-2a^{-5}z^{-1}+2z^{10}a^{-4}+7z^{8}a^{-4}-40z^{6}a^{-4}+46z^{4}a^{-4}-15z^{2}a^{-4}+2a^{-4}z^{-2}-2a^{-4}+5z^{9}a^{-3}-15z^{7}a^{-3}+10z^{5}a^{-3}+z^{3}a^{-3}+2za^{-3}-2a^{-3}z^{-1}+4z^{8}a^{-2}-15z^{6}a^{-2}+17z^{4}a^{-2}-5z^{2}a^{-2}+a^{-2}z^{-2}-2a^{-2}+z^{7}a^{-1}-3z^{5}a^{-1}+2z^{3}a^{-1}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 21                       1 1 19                     3   -3 17                   6 1   5 15                 8 3     -5 13               10 6       4 11             12 10         -2 9           9 8           1 7         7 12             5 5       6 9               -3 3     3 9                 6 1   1 4                   -3 -1   3                     3 -3 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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