L11a5

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a5 at Knotilus!

Link Presentations

[edit Notes on L11a5's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X10,6,11,5 X8493 X22,12,5,11 X20,14,21,13 X14,20,15,19 X12,22,13,21 X18,10,19,9 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 10, -4, 6, -9, 7, -8, 11, -2, 3, -10, 8, -7, 9, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(3v^{2}-5v+3\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -11 q^{9/2}+13 q^{7/2}-14 q^{5/2}+\frac{1}{q^{5/2}}+12 q^{3/2}-\frac{4}{q^{3/2}}-q^{17/2}+3 q^{15/2}-5 q^{13/2}+8 q^{11/2}-10 \sqrt{q}+\frac{6}{\sqrt{q}}} (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}+z^{5}a^{-3}+z^{5}a^{-5}-az^{3}+z^{3}a^{-1}+2z^{3}a^{-5}-z^{3}a^{-7}-za^{-3}+2za^{-5}-za^{-7}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-4}-2z^{10}a^{-6}-5z^{9}a^{-3}-9z^{9}a^{-5}-4z^{9}a^{-7}-6z^{8}a^{-2}-z^{8}a^{-4}+2z^{8}a^{-6}-3z^{8}a^{-8}-6z^{7}a^{-1}+12z^{7}a^{-3}+36z^{7}a^{-5}+17z^{7}a^{-7}-z^{7}a^{-9}+9z^{6}a^{-2}+18z^{6}a^{-4}+16z^{6}a^{-6}+13z^{6}a^{-8}-6z^{6}-4az^{5}+4z^{5}a^{-1}-13z^{5}a^{-3}-46z^{5}a^{-5}-21z^{5}a^{-7}+4z^{5}a^{-9}-a^{2}z^{4}-4z^{4}a^{-2}-23z^{4}a^{-4}-27z^{4}a^{-6}-15z^{4}a^{-8}+6z^{4}+4az^{3}+4z^{3}a^{-1}+8z^{3}a^{-3}+22z^{3}a^{-5}+11z^{3}a^{-7}-3z^{3}a^{-9}+z^{2}a^{-2}+6z^{2}a^{-4}+11z^{2}a^{-6}+6z^{2}a^{-8}-2za^{-3}-4za^{-5}-2za^{-7}+1-az^{-1}-a^{-1}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ).

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 18                       1 1 16                     2   -2 14                   3 1   2 12                 5 2     -3 10               6 3       3 8             7 5         -2 6           7 6           1 4         5 7             2 2       5 7               -2 0     3 7                 4 -2   1 3                   -2 -4   3                     3 -6 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{5}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{7}} $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}} ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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