L11a184

L11a183

L11a185

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a184 at Knotilus!

Link Presentations

[edit Notes on L11a184's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {\left(t(1)t(2)^{2}-t(2)^{2}-2t(1)t(2)+2t(2)+t(1)-2\right)\left(2t(1)t(2)^{2}-t(2)^{2}-2t(1)t(2)+2t(2)+t(1)-1\right)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+16{\sqrt {q}}-{\frac {23}{\sqrt {q}}}+{\frac {25}{q^{3/2}}}-{\frac {27}{q^{5/2}}}+{\frac {23}{q^{7/2}}}-{\frac {17}{q^{9/2}}}+{\frac {11}{q^{11/2}}}-{\frac {5}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z^{5}-a^{5}z^{3}+a^{5}z+a^{3}z^{7}+2a^{3}z^{5}-a^{3}z^{3}-3a^{3}z+a^{3}z^{-1}+az^{7}+3az^{5}-z^{5}a^{-1}+4az^{3}-2z^{3}a^{-1}+2az-za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-a^{8}z^{4}+5a^{7}z^{7}-9a^{7}z^{5}+3a^{7}z^{3}+10a^{6}z^{8}-22a^{6}z^{6}+12a^{6}z^{4}+9a^{5}z^{9}-12a^{5}z^{7}-5a^{5}z^{5}+7a^{5}z^{3}-2a^{5}z+3a^{4}z^{10}+16a^{4}z^{8}-47a^{4}z^{6}+29a^{4}z^{4}-2a^{4}z^{2}+17a^{3}z^{9}-26a^{3}z^{7}+z^{5}a^{-3}+12a^{3}z^{3}-z^{3}a^{-3}-5a^{3}z-a^{3}z^{-1}+3a^{2}z^{10}+16a^{2}z^{8}-39a^{2}z^{6}+4z^{6}a^{-2}+23a^{2}z^{4}-5z^{4}a^{-2}-2a^{2}z^{2}+2z^{2}a^{-2}+a^{2}+8az^{9}-az^{7}+8z^{7}a^{-1}-16az^{5}-11z^{5}a^{-1}+16az^{3}+7z^{3}a^{-1}-5az-2za^{-1}-az^{-1}+10z^{8}-11z^{6}+2z^{4}+2z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 10 3     -7 0               13 6       7 -2             13 11         -2 -4           14 12           2 -6         10 14             4 -8       7 13               -6 -10     4 10                 6 -12   1 7                   -6 -14   4                     4 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-3$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{14}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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