L10a165

L10a164

L10a166

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a165 at Knotilus!

Link Presentations

[edit Notes on L10a165's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uvwx-2uvw-2uvx+2uv-2uwx+2uw+3ux-2u-2vwx+3vw+2vx-2v+2wx-2w-2x+1}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}{\sqrt {x}}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {11}{q^{7/2}}}+{\frac {10}{q^{9/2}}}-{\frac {11}{q^{11/2}}}+{\frac {5}{q^{13/2}}}-{\frac {5}{q^{15/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{11}z^{-3}-4a^{9}z^{-1}-3a^{9}z^{-3}+6za^{7}+8a^{7}z^{-1}+3a^{7}z^{-3}-4z^{3}a^{5}-6za^{5}-4a^{5}z^{-1}-a^{5}z^{-3}+z^{5}a^{3}+z^{3}a^{3}-z^{3}a$ (db)
Kauffman polynomial	$-z^{5}a^{11}+4z^{3}a^{11}-6za^{11}+4a^{11}z^{-1}-a^{11}z^{-3}-z^{6}a^{10}+6z^{2}a^{10}+3a^{10}z^{-2}-8a^{10}-z^{7}a^{9}-3z^{5}a^{9}+12z^{3}a^{9}-14za^{9}+9a^{9}z^{-1}-3a^{9}z^{-3}-z^{8}a^{8}-2z^{6}a^{8}+12z^{2}a^{8}+6a^{8}z^{-2}-15a^{8}-z^{9}a^{7}-6z^{5}a^{7}+13z^{3}a^{7}-14za^{7}+9a^{7}z^{-1}-3a^{7}z^{-3}-5z^{8}a^{6}+9z^{6}a^{6}-9z^{4}a^{6}+6z^{2}a^{6}+3a^{6}z^{-2}-8a^{6}-z^{9}a^{5}-5z^{7}a^{5}+10z^{5}a^{5}-z^{3}a^{5}-6za^{5}+4a^{5}z^{-1}-a^{5}z^{-3}-4z^{8}a^{4}+6z^{6}a^{4}-2z^{4}a^{4}-6z^{7}a^{3}+13z^{5}a^{3}-5z^{3}a^{3}-4z^{6}a^{2}+7z^{4}a^{2}-z^{5}a+z^{3}a$ (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                   3   -3 -2                 4 1   3 -4               5 4     -1 -6             6 3       3 -8           4 5         1 -10         7 6           1 -12       4 10             6 -14     1 1               0 -16     4                 4 -18 1 1                   0 -20 1                     1

Integral Khovanov Homology (db, data source)

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

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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