L10a34

L10a33

L10a35

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a34 at Knotilus!

Link Presentations

[edit Notes on L10a34's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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