L10a105

L10a104

L10a106

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a105 at Knotilus!

Link Presentations

[edit Notes on L10a105's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X16,10,17,9 X6,13,7,14 X14,7,15,8 X8,15,1,16 X20,18,9,17 X4,19,5,20 X18,5,19,6
Gauss code	{1, -2, 3, -9, 10, -5, 6, -7}, {4, -1, 2, -3, 5, -6, 7, -4, 8, -10, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{3}t(1)^{3}-2t(2)^{2}t(1)^{3}-2t(2)^{3}t(1)^{2}+4t(2)^{2}t(1)^{2}-3t(2)t(1)^{2}-3t(2)^{2}t(1)+4t(2)t(1)-2t(1)-2t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {4}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {8}{q^{13/2}}}+{\frac {6}{q^{15/2}}}-{\frac {4}{q^{17/2}}}+{\frac {2}{q^{19/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z^{3}a^{9}+3za^{9}+a^{9}z^{-1}-2z^{5}a^{7}-8z^{3}a^{7}-8za^{7}-a^{7}z^{-1}+z^{7}a^{5}+5z^{5}a^{5}+8z^{3}a^{5}+5za^{5}-z^{5}a^{3}-4z^{3}a^{3}-4za^{3}$ (db)
Kauffman polynomial	$a^{13}z^{3}-a^{13}z+2a^{12}z^{4}-a^{12}z^{2}+3a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+4a^{10}z^{6}-5a^{10}z^{4}+2a^{10}z^{2}+5a^{9}z^{7}-13a^{9}z^{5}+14a^{9}z^{3}-7a^{9}z+a^{9}z^{-1}+3a^{8}z^{8}-4a^{8}z^{6}-6a^{8}z^{4}+6a^{8}z^{2}-a^{8}+a^{7}z^{9}+4a^{7}z^{7}-23a^{7}z^{5}+25a^{7}z^{3}-9a^{7}z+a^{7}z^{-1}+5a^{6}z^{8}-17a^{6}z^{6}+13a^{6}z^{4}-2a^{6}z^{2}+a^{5}z^{9}-12a^{5}z^{5}+16a^{5}z^{3}-4a^{5}z+2a^{4}z^{8}-9a^{4}z^{6}+12a^{4}z^{4}-5a^{4}z^{2}+a^{3}z^{7}-5a^{3}z^{5}+8a^{3}z^{3}-4a^{3}z$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                     1 1 -2                   1   -1 -4                 3 1   2 -6               4 2     -2 -8             3 2       1 -10           4 4         0 -12         4 3           1 -14       2 4             2 -16     2 4               -2 -18     2                 2 -20 1 2                   -1 -22 1                     1

Integral Khovanov Homology (db, data source)

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

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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