L10a55

L10a54

L10a56

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a55 at Knotilus!

Link Presentations

[edit Notes on L10a55's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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