L10a125

L10a124

L10a126

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a125 at Knotilus!

Link Presentations

[edit Notes on L10a125's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uvw^{3}-2uvw^{2}+2uvw-2uv-2uw^{3}+3uw^{2}-3uw+2u-2vw^{3}+3vw^{2}-3vw+2v+2w^{3}-2w^{2}+2w-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q^{4}-4q^{3}+7q^{2}-10q+11-11q^{-1}+11q^{-2}-6q^{-3}+5q^{-4}-q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$2a^{6}z^{-2}+a^{6}-3z^{2}a^{4}-5a^{4}z^{-2}-8a^{4}+3z^{4}a^{2}+9z^{2}a^{2}+4a^{2}z^{-2}+10a^{2}-z^{6}-3z^{4}-4z^{2}-z^{-2}-3+z^{4}a^{-2}+z^{2}a^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-5a^{6}z^{4}+9a^{6}z^{2}+2a^{6}z^{-2}-7a^{6}+a^{5}z^{7}-a^{5}z^{5}-6a^{5}z^{3}+11a^{5}z-5a^{5}z^{-1}+a^{4}z^{8}+2a^{4}z^{6}-14a^{4}z^{4}+z^{4}a^{-4}+20a^{4}z^{2}+5a^{4}z^{-2}-14a^{4}+a^{3}z^{9}+3a^{3}z^{5}+4z^{5}a^{-3}-17a^{3}z^{3}-3z^{3}a^{-3}+21a^{3}z-9a^{3}z^{-1}+5a^{2}z^{8}-7a^{2}z^{6}+7z^{6}a^{-2}-3a^{2}z^{4}-8z^{4}a^{-2}+11a^{2}z^{2}+z^{2}a^{-2}+4a^{2}z^{-2}-10a^{2}+az^{9}+6az^{7}+7z^{7}a^{-1}-8az^{5}-8z^{5}a^{-1}-8az^{3}+13az+3za^{-1}-5az^{-1}-a^{-1}z^{-1}+4z^{8}-z^{6}-3z^{4}+z^{2}+z^{-2}-2$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                     1 1 7                   3   -3 5                 4 1   3 3               6 3     -3 1             5 4       1 -1           7 7         0 -3         4 4           0 -5       2 7             5 -7     3 4               -1 -9   1 5                 4 -11                       0 -13 1                     1

Integral Khovanov Homology (db, data source)

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

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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