L11a85

L11a84

L11a86

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a85 at Knotilus!

Link Presentations

[edit Notes on L11a85's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{5}-4uv^{4}+5uv^{3}-5uv^{2}+4uv-2u-2v^{5}+4v^{4}-5v^{3}+5v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {12}{q^{9/2}}}+{\frac {9}{q^{7/2}}}-{\frac {6}{q^{5/2}}}+{\frac {3}{q^{3/2}}}+{\frac {1}{q^{23/2}}}-{\frac {2}{q^{21/2}}}+{\frac {5}{q^{19/2}}}-{\frac {8}{q^{17/2}}}+{\frac {11}{q^{15/2}}}-{\frac {14}{q^{13/2}}}+{\frac {12}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{11}-2a^{11}z^{-1}+3z^{3}a^{9}+9za^{9}+5a^{9}z^{-1}-3z^{5}a^{7}-11z^{3}a^{7}-11za^{7}-3a^{7}z^{-1}+z^{7}a^{5}+4z^{5}a^{5}+5z^{3}a^{5}+3za^{5}-z^{5}a^{3}-3z^{3}a^{3}-2za^{3}$ (db)
Kauffman polynomial	$a^{14}z^{4}-2a^{14}z^{2}+a^{14}+2a^{13}z^{5}-2a^{13}z^{3}+3a^{12}z^{6}-2a^{12}z^{4}+4a^{11}z^{7}-5a^{11}z^{5}+6a^{11}z^{3}-5a^{11}z+2a^{11}z^{-1}+4a^{10}z^{8}-4a^{10}z^{6}-3a^{10}z^{4}+10a^{10}z^{2}-5a^{10}+3a^{9}z^{9}-a^{9}z^{7}-12a^{9}z^{5}+20a^{9}z^{3}-14a^{9}z+5a^{9}z^{-1}+a^{8}z^{10}+6a^{8}z^{8}-22a^{8}z^{6}+15a^{8}z^{4}+3a^{8}z^{2}-5a^{8}+6a^{7}z^{9}-14a^{7}z^{7}-a^{7}z^{5}+13a^{7}z^{3}-8a^{7}z+3a^{7}z^{-1}+a^{6}z^{10}+5a^{6}z^{8}-27a^{6}z^{6}+29a^{6}z^{4}-10a^{6}z^{2}+3a^{5}z^{9}-8a^{5}z^{7}+6a^{5}z^{3}-a^{5}z+3a^{4}z^{8}-12a^{4}z^{6}+14a^{4}z^{4}-5a^{4}z^{2}+a^{3}z^{7}-4a^{3}z^{5}+5a^{3}z^{3}-2a^{3}z$ (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 0                       1 1 -2                     2   -2 -4                   4 1   3 -6                 6 3     -3 -8               6 3       3 -10             6 6         0 -12           8 6           2 -14         4 7             3 -16       4 7               -3 -18     1 4                 3 -20   1 4                   -3 -22   1                     1 -24 1                       -1

Integral Khovanov Homology (db, data source)

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

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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