L11a183

L11a182

L11a184

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a183 at Knotilus!

Link Presentations

[edit Notes on L11a183's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X18,7,19,8 X20,10,21,9 X22,17,7,18 X14,6,15,5 X16,12,17,11 X12,16,13,15 X4,14,5,13 X6,20,1,19 X2,21,3,22
Gauss code	{1, -11, 2, -9, 6, -10}, {3, -1, 4, -2, 7, -8, 9, -6, 8, -7, 5, -3, 10, -4, 11, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}+t(2)^{4}-5t(1)^{2}t(2)^{3}+9t(1)t(2)^{3}-4t(2)^{3}+6t(1)^{2}t(2)^{2}-13t(1)t(2)^{2}+6t(2)^{2}-4t(1)^{2}t(2)+9t(1)t(2)-5t(2)+t(1)^{2}-3t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-15q^{9/2}+20q^{7/2}-{\frac {1}{q^{7/2}}}-24q^{5/2}+{\frac {4}{q^{5/2}}}+23q^{3/2}-{\frac {9}{q^{3/2}}}+q^{15/2}-4q^{13/2}+9q^{11/2}-21{\sqrt {q}}+{\frac {15}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-5}+2z^{3}a^{-5}+za^{-5}-z^{7}a^{-3}-3z^{5}a^{-3}-3z^{3}a^{-3}-za^{-3}-a^{-3}z^{-1}-z^{7}a^{-1}+az^{5}-3z^{5}a^{-1}+2az^{3}-3z^{3}a^{-1}+az+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-2z^{4}a^{-8}+z^{2}a^{-8}+4z^{7}a^{-7}-9z^{5}a^{-7}+6z^{3}a^{-7}-za^{-7}+7z^{8}a^{-6}-15z^{6}a^{-6}+9z^{4}a^{-6}-z^{2}a^{-6}+6z^{9}a^{-5}-5z^{7}a^{-5}-9z^{5}a^{-5}+8z^{3}a^{-5}-za^{-5}+2z^{10}a^{-4}+13z^{8}a^{-4}-33z^{6}a^{-4}+19z^{4}a^{-4}-3z^{2}a^{-4}+12z^{9}a^{-3}-12z^{7}a^{-3}+a^{3}z^{5}-10z^{5}a^{-3}-a^{3}z^{3}+9z^{3}a^{-3}-a^{-3}z^{-1}+2z^{10}a^{-2}+15z^{8}a^{-2}+4a^{2}z^{6}-30z^{6}a^{-2}-5a^{2}z^{4}+13z^{4}a^{-2}+2a^{2}z^{2}-z^{2}a^{-2}+a^{-2}+6z^{9}a^{-1}+8az^{7}+5z^{7}a^{-1}-12az^{5}-23z^{5}a^{-1}+8az^{3}+16z^{3}a^{-1}-2az-2za^{-1}-a^{-1}z^{-1}+9z^{8}-9z^{6}+2z^{2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     3   3 12                   6 1   -5 10                 9 3     6 8               11 6       -5 6             13 9         4 4           11 12           1 2         10 12             -2 0       6 12               6 -2     3 9                 -6 -4   1 6                   5 -6   3                     -3 -8 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ 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}^{3}\oplus{\mathbb Z}_2} ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ 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}^{12}\oplus{\mathbb Z}_2^{12}} ${\mathbb {Z} }^{13}$ $r=3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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