L11n104

L11n103

L11n105

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n104 at Knotilus!

Link Presentations

[edit Notes on L11n104's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,16,8,17 X22,18,5,17 X18,14,19,13 X21,14,22,15 X9,20,10,21 X15,8,16,9 X19,10,20,11 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 8, -7, 9, 11, -2, 5, 6, -8, 3, 4, -5, -9, 7, -6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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