L11a21

L11a20

L11a22

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a21 at Knotilus!

Link Presentations

[edit Notes on L11a21's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}+1\right)\left(2t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{23/2}-4q^{21/2}+9q^{19/2}-13q^{17/2}+17q^{15/2}-18q^{13/2}+17q^{11/2}-15q^{9/2}+9q^{7/2}-6q^{5/2}+2q^{3/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-5}-z^{7}a^{-7}+z^{5}a^{-3}-4z^{5}a^{-5}-3z^{5}a^{-7}+z^{5}a^{-9}+4z^{3}a^{-3}-6z^{3}a^{-5}-2z^{3}a^{-7}+2z^{3}a^{-9}+5za^{-3}-5za^{-5}-za^{-7}+za^{-9}+2a^{-3}z^{-1}-2a^{-5}z^{-1}-a^{-7}z^{-1}+a^{-9}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-14}+4z^{5}a^{-13}-z^{3}a^{-13}+9z^{6}a^{-12}-9z^{4}a^{-12}+4z^{2}a^{-12}-a^{-12}+12z^{7}a^{-11}-16z^{5}a^{-11}+7z^{3}a^{-11}-za^{-11}+10z^{8}a^{-10}-10z^{6}a^{-10}-4z^{4}a^{-10}+4z^{2}a^{-10}+5z^{9}a^{-9}+4z^{7}a^{-9}-22z^{5}a^{-9}+12z^{3}a^{-9}-2za^{-9}-a^{-9}z^{-1}+z^{10}a^{-8}+13z^{8}a^{-8}-33z^{6}a^{-8}+19z^{4}a^{-8}-6z^{2}a^{-8}+3a^{-8}+7z^{9}a^{-7}-11z^{7}a^{-7}-8z^{5}a^{-7}+15z^{3}a^{-7}-4za^{-7}-a^{-7}z^{-1}+z^{10}a^{-6}+5z^{8}a^{-6}-21z^{6}a^{-6}+19z^{4}a^{-6}-4z^{2}a^{-6}+2z^{9}a^{-5}-2z^{7}a^{-5}-11z^{5}a^{-5}+20z^{3}a^{-5}-10za^{-5}+2a^{-5}z^{-1}+2z^{8}a^{-4}-7z^{6}a^{-4}+6z^{4}a^{-4}+2z^{2}a^{-4}-3a^{-4}+z^{7}a^{-3}-5z^{5}a^{-3}+9z^{3}a^{-3}-7za^{-3}+2a^{-3}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 24                       1 -1 22                     3   3 20                   6 1   -5 18                 7 3     4 16               10 6       -4 14             8 7         1 12           9 10           1 10         6 8             -2 8       3 9               6 6     3 6                 -3 4   1 5                   4 2   1                     -1 0 1                       1

Integral Khovanov Homology (db, data source)

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

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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