L11a83

L11a82

L11a84

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a83 at Knotilus!

Link Presentations

[edit Notes on L11a83's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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