L10a102

L10a101

L10a103

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a102 at Knotilus!

Link Presentations

[edit Notes on L10a102's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)t(2)+1)\left(2t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(1)+2t(2)t(1)-t(1)-t(2)+2\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{9/2}}}-{\frac {3}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {6}{q^{17/2}}}-{\frac {6}{q^{19/2}}}+{\frac {6}{q^{21/2}}}-{\frac {4}{q^{23/2}}}+{\frac {2}{q^{25/2}}}-{\frac {1}{q^{27/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$z^{5}a^{11}+4z^{3}a^{11}+4za^{11}-z^{7}a^{9}-5z^{5}a^{9}-7z^{3}a^{9}-2za^{9}+a^{9}z^{-1}-z^{7}a^{7}-6z^{5}a^{7}-11z^{3}a^{7}-6za^{7}-a^{7}z^{-1}$ (db)
Kauffman polynomial	$a^{17}z^{3}-a^{17}z+2a^{16}z^{4}-a^{16}z^{2}+3a^{15}z^{5}-2a^{15}z^{3}+a^{15}z+4a^{14}z^{6}-7a^{14}z^{4}+6a^{14}z^{2}+3a^{13}z^{7}-4a^{13}z^{5}+a^{13}z+2a^{12}z^{8}-3a^{12}z^{6}-a^{12}z^{2}+a^{11}z^{9}-2a^{11}z^{7}+4a^{11}z^{5}-11a^{11}z^{3}+5a^{11}z+3a^{10}z^{8}-11a^{10}z^{6}+12a^{10}z^{4}-6a^{10}z^{2}+a^{9}z^{9}-4a^{9}z^{7}+5a^{9}z^{5}-3a^{9}z^{3}+a^{9}z^{-1}+a^{8}z^{8}-4a^{8}z^{6}+3a^{8}z^{4}+2a^{8}z^{2}-a^{8}+a^{7}z^{7}-6a^{7}z^{5}+11a^{7}z^{3}-6a^{7}z+a^{7}z^{-1}$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                     1 1 -8                   1 1 0 -10                 2     2 -12               2 1     -1 -14             4 2       2 -16           2 2         0 -18         4 4           0 -20       2 2             0 -22     2 4               -2 -24   1 3                 2 -26   1                   -1 -28 1                     1

Integral Khovanov Homology (db, data source)

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

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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