L11n55

L11n54

L11n56

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n55 at Knotilus!

Link Presentations

[edit Notes on L11n55's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,11,17,12 X7,15,8,14 X15,9,16,8 X20,13,21,14 X22,17,5,18 X18,21,19,22 X12,19,13,20 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, 3, -9, 6, 4, -5, -3, 7, -8, 9, -6, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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