L10a25

L10a24

L10a26

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a25 at Knotilus!

Link Presentations

[edit Notes on L10a25's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uv^{4}-6uv^{3}+6uv^{2}-4uv+u+v^{5}-4v^{4}+6v^{3}-6v^{2}+2v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	${\frac {13}{q^{9/2}}}-{\frac {12}{q^{7/2}}}+{\frac {9}{q^{5/2}}}-{\frac {7}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {6}{q^{15/2}}}+{\frac {9}{q^{13/2}}}-{\frac {12}{q^{11/2}}}-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}-3z^{3}a^{7}-6za^{7}-3a^{7}z^{-1}+2z^{5}a^{5}+6z^{3}a^{5}+8za^{5}+4a^{5}z^{-1}+z^{5}a^{3}-4za^{3}-2a^{3}z^{-1}-z^{3}a-za$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+6z^{4}a^{10}-3z^{2}a^{10}+a^{10}-4z^{7}a^{9}+6z^{5}a^{9}-2z^{3}a^{9}+2za^{9}-a^{9}z^{-1}-3z^{8}a^{8}-z^{6}a^{8}+11z^{4}a^{8}-10z^{2}a^{8}+3a^{8}-z^{9}a^{7}-9z^{7}a^{7}+25z^{5}a^{7}-27z^{3}a^{7}+13za^{7}-3a^{7}z^{-1}-7z^{8}a^{6}+9z^{6}a^{6}+2z^{4}a^{6}-10z^{2}a^{6}+3a^{6}-z^{9}a^{5}-10z^{7}a^{5}+29z^{5}a^{5}-33z^{3}a^{5}+18za^{5}-4a^{5}z^{-1}-4z^{8}a^{4}+4z^{6}a^{4}+2z^{4}a^{4}-4z^{2}a^{4}+2a^{4}-5z^{7}a^{3}+10z^{5}a^{3}-8z^{3}a^{3}+7za^{3}-2a^{3}z^{-1}-3z^{6}a^{2}+5z^{4}a^{2}-z^{2}a^{2}-z^{5}a+2z^{3}a-za$ (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 2                     1 1 0                   2   -2 -2                 5 1   4 -4               5 3     -2 -6             7 4       3 -8           6 5         -1 -10         6 7           -1 -12       4 7             3 -14     2 5               -3 -16   1 4                 3 -18   2                   -2 -20 1                     1

Integral Khovanov Homology (db, data source)

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

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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