L10a76

L10a75

L10a77

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a76 at Knotilus!

Link Presentations

[edit Notes on L10a76's Link Presentations]

Planar diagram presentation	X8192 X14,9,15,10 X6718 X20,15,7,16 X16,6,17,5 X4,20,5,19 X10,4,11,3 X12,17,13,18 X18,11,19,12 X2,14,3,13
Gauss code	{1, -10, 7, -6, 5, -3}, {3, -1, 2, -7, 9, -8, 10, -2, 4, -5, 8, -9, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{4}-4u^{2}v^{3}+5u^{2}v^{2}-2u^{2}v-uv^{4}+7uv^{3}-11uv^{2}+7uv-u-2v^{3}+5v^{2}-4v+1}{uv^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+12{\sqrt {q}}-{\frac {17}{\sqrt {q}}}+{\frac {17}{q^{3/2}}}-{\frac {16}{q^{5/2}}}+{\frac {13}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+4az^{5}-z^{5}a^{-1}+a^{5}z^{3}-5a^{3}z^{3}+7az^{3}-2z^{3}a^{-1}+a^{5}z-4a^{3}z+6az-2za^{-1}-a^{3}z^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-3a^{3}z^{9}-3az^{9}-7a^{4}z^{8}-15a^{2}z^{8}-8z^{8}-7a^{5}z^{7}-9a^{3}z^{7}-10az^{7}-8z^{7}a^{-1}-4a^{6}z^{6}+8a^{4}z^{6}+29a^{2}z^{6}-4z^{6}a^{-2}+13z^{6}-a^{7}z^{5}+11a^{5}z^{5}+28a^{3}z^{5}+32az^{5}+15z^{5}a^{-1}-z^{5}a^{-3}+6a^{6}z^{4}+a^{4}z^{4}-17a^{2}z^{4}+5z^{4}a^{-2}-7z^{4}+a^{7}z^{3}-5a^{5}z^{3}-23a^{3}z^{3}-28az^{3}-10z^{3}a^{-1}+z^{3}a^{-3}-2a^{6}z^{2}-3a^{4}z^{2}-a^{2}z^{2}+2a^{5}z+7a^{3}z+10az+5za^{-1}+a^{4}+3a^{2}+3-a^{3}z^{-1}-3az^{-1}-2a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                   3   -3 4                 6 1   5 2               7 4     -3 0             10 5       5 -2           8 8         0 -4         8 9           -1 -6       5 8             3 -8     3 8               -5 -10   1 5                 4 -12   3                   -3 -14 1                     1

Integral Khovanov Homology (db, data source)

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

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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