L11a266

L11a265

L11a267

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a266 at Knotilus!

Link Presentations

[edit Notes on L11a266's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X22,12,9,11 X2,9,3,10 X4,22,5,21 X14,6,15,5 X16,20,17,19 X18,8,19,7 X6,18,7,17 X20,16,21,15 X8,14,1,13
Gauss code	{1, -4, 2, -5, 6, -9, 8, -11}, {4, -1, 3, -2, 11, -6, 10, -7, 9, -8, 7, -10, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(2)^{2}t(1)^{3}-3t(2)t(1)^{3}+t(1)^{3}+2t(2)^{3}t(1)^{2}-9t(2)^{2}t(1)^{2}+9t(2)t(1)^{2}-3t(1)^{2}-3t(2)^{3}t(1)+9t(2)^{2}t(1)-9t(2)t(1)+2t(1)+t(2)^{3}-3t(2)^{2}+2t(2)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$18q^{9/2}-16q^{7/2}+11q^{5/2}-7q^{3/2}+q^{21/2}-4q^{19/2}+8q^{17/2}-12q^{15/2}+16q^{13/2}-19q^{11/2}+3{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}-z^{5}a^{-7}+za^{-7}-2z^{5}a^{-5}-3z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}-z^{5}a^{-3}+2za^{-3}+a^{-3}z^{-1}+z^{3}a^{-1}+za^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-12}-2z^{4}a^{-12}+z^{2}a^{-12}+4z^{7}a^{-11}-10z^{5}a^{-11}+6z^{3}a^{-11}-za^{-11}+6z^{8}a^{-10}-14z^{6}a^{-10}+7z^{4}a^{-10}+4z^{9}a^{-9}-z^{7}a^{-9}-15z^{5}a^{-9}+13z^{3}a^{-9}-3za^{-9}+z^{10}a^{-8}+11z^{8}a^{-8}-28z^{6}a^{-8}+17z^{4}a^{-8}-2z^{2}a^{-8}+7z^{9}a^{-7}-4z^{7}a^{-7}-16z^{5}a^{-7}+19z^{3}a^{-7}-5za^{-7}+z^{10}a^{-6}+10z^{8}a^{-6}-20z^{6}a^{-6}+12z^{4}a^{-6}-z^{2}a^{-6}+3z^{9}a^{-5}+6z^{7}a^{-5}-20z^{5}a^{-5}+20z^{3}a^{-5}-8za^{-5}+a^{-5}z^{-1}+5z^{8}a^{-4}-4z^{6}a^{-4}-z^{4}a^{-4}+2z^{2}a^{-4}-a^{-4}+5z^{7}a^{-3}-8z^{5}a^{-3}+6z^{3}a^{-3}-4za^{-3}+a^{-3}z^{-1}+3z^{6}a^{-2}-5z^{4}a^{-2}+2z^{2}a^{-2}+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 22                       1 -1 20                     3   3 18                   5 1   -4 16                 7 3     4 14               9 5       -4 12             10 7         3 10           9 10           1 8         7 9             -2 6       4 9               5 4     3 7                 -4 2   1 5                   4 0   2                     -2 -2 1                       1

Integral Khovanov Homology (db, data source)

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

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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