L11a22

L11a21

L11a23

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a22 at Knotilus!

Link Presentations

[edit Notes on L11a22's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X18,10,19,9 X22,14,5,13 X20,16,21,15 X16,20,17,19 X14,22,15,21 X8,18,9,17 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 4, -2, 11, -3, 5, -8, 6, -7, 9, -4, 7, -6, 8, -5}

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	$12q^{9/2}-11q^{7/2}+7q^{5/2}-5q^{3/2}+q^{21/2}-3q^{19/2}+6q^{17/2}-8q^{15/2}+11q^{13/2}-13q^{11/2}+2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-3}-z^{5}a^{-5}-z^{5}a^{-7}+z^{3}a^{-1}-2z^{3}a^{-3}-2z^{3}a^{-7}+z^{3}a^{-9}+2za^{-1}-2za^{-3}+2za^{-5}-3za^{-7}+za^{-9}+a^{-1}z^{-1}-a^{-3}z^{-1}+a^{-5}z^{-1}-2a^{-7}z^{-1}+a^{-9}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-2z^{9}a^{-5}-6z^{9}a^{-7}-4z^{9}a^{-9}-2z^{8}a^{-4}-2z^{8}a^{-6}-5z^{8}a^{-8}-5z^{8}a^{-10}-2z^{7}a^{-3}+3z^{7}a^{-5}+21z^{7}a^{-7}+13z^{7}a^{-9}-3z^{7}a^{-11}-2z^{6}a^{-2}+11z^{6}a^{-6}+28z^{6}a^{-8}+18z^{6}a^{-10}-z^{6}a^{-12}-z^{5}a^{-1}-8z^{5}a^{-5}-32z^{5}a^{-7}-14z^{5}a^{-9}+9z^{5}a^{-11}+4z^{4}a^{-2}+3z^{4}a^{-4}-23z^{4}a^{-6}-44z^{4}a^{-8}-19z^{4}a^{-10}+3z^{4}a^{-12}+3z^{3}a^{-1}+7z^{3}a^{-3}+12z^{3}a^{-5}+20z^{3}a^{-7}+8z^{3}a^{-9}-4z^{3}a^{-11}-z^{2}a^{-2}+17z^{2}a^{-6}+27z^{2}a^{-8}+10z^{2}a^{-10}-z^{2}a^{-12}-3za^{-1}-5za^{-3}-7za^{-5}-8za^{-7}-3za^{-9}-a^{-2}-4a^{-6}-7a^{-8}-3a^{-10}+a^{-1}z^{-1}+a^{-3}z^{-1}+a^{-5}z^{-1}+2a^{-7}z^{-1}+a^{-9}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 22                       1 -1 20                     2   2 18                   4 1   -3 16                 4 2     2 14               7 4       -3 12             6 4         2 10           6 7           1 8         5 6             -1 6       2 6               4 4     3 5                 -2 2   1 4                   3 0   1                     -1 -2 1                       1

Integral Khovanov Homology (db, data source)

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

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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