L11a506

L11a505

L11a507

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a506 at Knotilus!

Link Presentations

[edit Notes on L11a506's Link Presentations]

Planar diagram presentation	X8192 X16,8,17,7 X14,6,15,5 X10,4,11,3 X4,14,5,13 X2,18,3,17 X18,9,19,10 X12,21,7,22 X22,11,13,12 X20,16,21,15 X6,19,1,20
Gauss code	{1, -6, 4, -5, 3, -11}, {2, -1, 7, -4, 9, -8}, {5, -3, 10, -2, 6, -7, 11, -10, 8, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(3)-1)\left(t(2)^{2}t(1)^{2}+t(2)^{2}t(3)^{2}t(1)^{2}-2t(2)t(3)^{2}t(1)^{2}+t(3)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(3)t(1)^{2}+2t(2)t(3)t(1)^{2}-t(3)t(1)^{2}-2t(2)^{2}t(1)-t(2)^{2}t(3)^{2}t(1)+3t(2)t(3)^{2}t(1)-2t(3)^{2}t(1)+3t(2)t(1)+2t(2)^{2}t(3)t(1)-3t(2)t(3)t(1)+2t(3)t(1)-t(1)+t(2)^{2}-t(2)t(3)^{2}+t(3)^{2}-2t(2)-t(2)^{2}t(3)+2t(2)t(3)-t(3)+1\right)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{7}-4q^{6}+9q^{5}-16q^{4}+22q^{3}-24q^{2}+26q-21+17q^{-1}-10q^{-2}+5q^{-3}-q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}+3z^{4}a^{-4}+3z^{2}a^{-4}-z^{8}a^{-2}-5z^{6}a^{-2}-a^{2}z^{4}-10z^{4}a^{-2}-a^{2}z^{2}-7z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+a^{2}+a^{-2}+2z^{6}+6z^{4}+4z^{2}-2z^{-2}-2$ (db)
Kauffman polynomial	$z^{4}a^{-8}+4z^{5}a^{-7}-z^{3}a^{-7}+9z^{6}a^{-6}-6z^{4}a^{-6}+z^{2}a^{-6}+15z^{7}a^{-5}-20z^{5}a^{-5}+9z^{3}a^{-5}+18z^{8}a^{-4}-34z^{6}a^{-4}+24z^{4}a^{-4}-8z^{2}a^{-4}+13z^{9}a^{-3}+a^{3}z^{7}-17z^{7}a^{-3}-2a^{3}z^{5}-8z^{5}a^{-3}+a^{3}z^{3}+10z^{3}a^{-3}+4z^{10}a^{-2}+5a^{2}z^{8}+19z^{8}a^{-2}-15a^{2}z^{6}-77z^{6}a^{-2}+13a^{2}z^{4}+71z^{4}a^{-2}-2a^{2}z^{2}-21z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{2}-2a^{-2}+8az^{9}+21z^{9}a^{-1}-23az^{7}-56z^{7}a^{-1}+16az^{5}+34z^{5}a^{-1}-az^{3}-2z^{3}a^{-1}+2az+2za^{-1}-2az^{-1}-2a^{-1}z^{-1}+4z^{10}+6z^{8}-49z^{6}+53z^{4}-14z^{2}+2z^{-2}-3$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     3   -3 11                   6 1   5 9                 10 3     -7 7               12 6       6 5             13 11         -2 3           13 11           2 1         10 15             5 -1       7 11               -4 -3     4 11                 7 -5   1 6                   -5 -7   4                     4 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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