L10a154

L10a153

L10a155

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a154 at Knotilus!

Link Presentations

[edit Notes on L10a154's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(w-1)\left(2uvw-3uv+uw^{2}-3uw+u+vw^{2}-3vw+v-3w^{2}+2w\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q-3+7q^{-1}-11q^{-2}+13q^{-3}-13q^{-4}+13q^{-5}-9q^{-6}+7q^{-7}-2q^{-8}+q^{-9}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}-2a^{8}z^{-2}-4a^{8}+6a^{6}z^{2}+a^{6}z^{-2}+6a^{6}-3a^{4}z^{4}-5a^{4}z^{2}-4a^{4}-a^{2}z^{4}+2a^{2}z^{2}+2a^{2}+z^{2}$ (db)
Kauffman polynomial	$z^{6}a^{10}-4z^{4}a^{10}+6z^{2}a^{10}+a^{10}z^{-2}-4a^{10}+2z^{7}a^{9}-4z^{5}a^{9}+4za^{9}-2a^{9}z^{-1}+3z^{8}a^{8}-5z^{6}a^{8}-z^{4}a^{8}+5z^{2}a^{8}+2a^{8}z^{-2}-5a^{8}+z^{9}a^{7}+9z^{7}a^{7}-29z^{5}a^{7}+22z^{3}a^{7}-4za^{7}-2a^{7}z^{-1}+8z^{8}a^{6}-11z^{6}a^{6}-5z^{4}a^{6}+6z^{2}a^{6}+a^{6}z^{-2}-2a^{6}+z^{9}a^{5}+15z^{7}a^{5}-41z^{5}a^{5}+36z^{3}a^{5}-12za^{5}+5z^{8}a^{4}+z^{6}a^{4}-16z^{4}a^{4}+14z^{2}a^{4}-2a^{4}+8z^{7}a^{3}-13z^{5}a^{3}+12z^{3}a^{3}-4za^{3}+6z^{6}a^{2}-7z^{4}a^{2}+6z^{2}a^{2}-2a^{2}+3z^{5}a-2z^{3}a+z^{4}-z^{2}$ (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 3                     1 1 1                   2   -2 -1                 5 1   4 -3               7 3     -4 -5             6 4       2 -7           7 7         0 -9         6 6           0 -11       5 9             4 -13     2 4               -2 -15     5                 5 -17 1 2                   -1 -19 1                     1

Integral Khovanov Homology (db, data source)

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

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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