L11a89

L11a88

L11a90

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a89 at Knotilus!

Link Presentations

[edit Notes on L11a89's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uv^{4}-8uv^{3}+10uv^{2}-6uv+u+v^{5}-6v^{4}+10v^{3}-8v^{2}+2v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {11}{q^{5/2}}}-{\frac {15}{q^{7/2}}}+{\frac {17}{q^{9/2}}}-{\frac {18}{q^{11/2}}}+{\frac {14}{q^{13/2}}}-{\frac {11}{q^{15/2}}}+{\frac {7}{q^{17/2}}}-{\frac {3}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4za^{9}+3a^{9}z^{-1}-5z^{3}a^{7}-8za^{7}-3a^{7}z^{-1}+2z^{5}a^{5}+4z^{3}a^{5}+5za^{5}+2a^{5}z^{-1}+z^{5}a^{3}-2za^{3}-a^{3}z^{-1}-z^{3}a-za$ (db)
Kauffman polynomial	$-z^{6}a^{12}+3z^{4}a^{12}-3z^{2}a^{12}+a^{12}-3z^{7}a^{11}+8z^{5}a^{11}-6z^{3}a^{11}+3za^{11}-a^{11}z^{-1}-4z^{8}a^{10}+7z^{6}a^{10}+2z^{4}a^{10}-4z^{2}a^{10}+2a^{10}-3z^{9}a^{9}-z^{7}a^{9}+19z^{5}a^{9}-22z^{3}a^{9}+12za^{9}-3a^{9}z^{-1}-z^{10}a^{8}-9z^{8}a^{8}+28z^{6}a^{8}-26z^{4}a^{8}+9z^{2}a^{8}-7z^{9}a^{7}+8z^{7}a^{7}+14z^{5}a^{7}-32z^{3}a^{7}+16za^{7}-3a^{7}z^{-1}-z^{10}a^{6}-11z^{8}a^{6}+34z^{6}a^{6}-40z^{4}a^{6}+16z^{2}a^{6}-2a^{6}-4z^{9}a^{5}+z^{7}a^{5}+12z^{5}a^{5}-22z^{3}a^{5}+12za^{5}-2a^{5}z^{-1}-6z^{8}a^{4}+11z^{6}a^{4}-10z^{4}a^{4}+5z^{2}a^{4}-5z^{7}a^{3}+8z^{5}a^{3}-4z^{3}a^{3}+4za^{3}-a^{3}z^{-1}-3z^{6}a^{2}+5z^{4}a^{2}-z^{2}a^{2}-z^{5}a+2z^{3}a-za$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     2   -2 -2                   5 1   4 -4                 7 3     -4 -6               8 4       4 -8             9 7         -2 -10           9 8           1 -12         6 10             4 -14       5 8               -3 -16     2 6                 4 -18   1 5                   -4 -20   2                     2 -22 1                       -1

Integral Khovanov Homology (db, data source)

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

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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