L10a38

L10a37

L10a39

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a38 at Knotilus!

Link Presentations

[edit Notes on L10a38's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {\left(v^{2}-v+1\right)\left(3uv^{2}-3uv+u+v^{3}-3v^{2}+3v\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{25/2}}}+{\frac {3}{q^{23/2}}}-{\frac {6}{q^{21/2}}}+{\frac {11}{q^{19/2}}}-{\frac {13}{q^{17/2}}}+{\frac {14}{q^{15/2}}}-{\frac {14}{q^{13/2}}}+{\frac {10}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}z^{-1}-4za^{11}-5a^{11}z^{-1}+6z^{3}a^{9}+14za^{9}+8a^{9}z^{-1}-3z^{5}a^{7}-10z^{3}a^{7}-11za^{7}-4a^{7}z^{-1}-z^{5}a^{5}-2z^{3}a^{5}-za^{5}$ (db)
Kauffman polynomial	$a^{15}z^{5}-2a^{15}z^{3}+a^{15}z+3a^{14}z^{6}-6a^{14}z^{4}+5a^{14}z^{2}-2a^{14}+4a^{13}z^{7}-4a^{13}z^{5}-a^{13}z^{3}+a^{13}z+a^{13}z^{-1}+3a^{12}z^{8}+5a^{12}z^{6}-21a^{12}z^{4}+22a^{12}z^{2}-9a^{12}+a^{11}z^{9}+11a^{11}z^{7}-23a^{11}z^{5}+16a^{11}z^{3}-9a^{11}z+5a^{11}z^{-1}+7a^{10}z^{8}+a^{10}z^{6}-29a^{10}z^{4}+35a^{10}z^{2}-14a^{10}+a^{9}z^{9}+13a^{9}z^{7}-32a^{9}z^{5}+32a^{9}z^{3}-22a^{9}z+8a^{9}z^{-1}+4a^{8}z^{8}+2a^{8}z^{6}-18a^{8}z^{4}+19a^{8}z^{2}-8a^{8}+6a^{7}z^{7}-13a^{7}z^{5}+15a^{7}z^{3}-12a^{7}z+4a^{7}z^{-1}+3a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{5}-2a^{5}z^{3}+a^{5}z$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                     1 1 -6                   3 1 -2 -8                 5     5 -10               5 3     -2 -12             9 5       4 -14           6 6         0 -16         7 8           -1 -18       4 6             2 -20     2 7               -5 -22   1 4                 3 -24   2                   -2 -26 1                     1

Integral Khovanov Homology (db, data source)

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

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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