L10a127

L10a126

L10a128

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a127 at Knotilus!

Link Presentations

[edit Notes on L10a127's Link Presentations]

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X18,10,19,9 X16,12,17,11 X20,18,11,17 X4,15,1,16 X10,20,5,19
Gauss code	{1, -4, 3, -9}, {2, -1, 5, -3, 6, -10}, {7, -2, 4, -5, 9, -7, 8, -6, 10, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)(w-1)(2vw-v-w+2)}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-8q^{6}+12q^{5}-15q^{4}+17q^{3}-14q^{2}+13q-7+4q^{-1}-q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}+2z^{4}a^{-4}-z^{4}a^{-6}-z^{4}+z^{2}a^{-2}+z^{2}a^{-4}-z^{2}a^{-6}-z^{2}-a^{-2}+1-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$2z^{9}a^{-3}+2z^{9}a^{-5}+5z^{8}a^{-2}+11z^{8}a^{-4}+6z^{8}a^{-6}+6z^{7}a^{-1}+10z^{7}a^{-3}+11z^{7}a^{-5}+7z^{7}a^{-7}-13z^{6}a^{-4}-5z^{6}a^{-6}+4z^{6}a^{-8}+4z^{6}+az^{5}-8z^{5}a^{-1}-20z^{5}a^{-3}-24z^{5}a^{-5}-12z^{5}a^{-7}+z^{5}a^{-9}-11z^{4}a^{-2}-z^{4}a^{-4}-3z^{4}a^{-6}-6z^{4}a^{-8}-7z^{4}-az^{3}+2z^{3}a^{-1}+7z^{3}a^{-3}+11z^{3}a^{-5}+6z^{3}a^{-7}-z^{3}a^{-9}+6z^{2}a^{-2}+2z^{2}a^{-4}+2z^{2}a^{-6}+2z^{2}a^{-8}+4z^{2}-za^{-1}-za^{-3}+a^{-2}+a^{-4}+1+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                     1 -1 15                   3   3 13                 5 1   -4 11               7 3     4 9             8 5       -3 7           9 7         2 5         7 10           3 3       6 7             -1 1     3 9               6 -1   1 4                 -3 -3   3                   3 -5 1                     -1

Integral Khovanov Homology (db, data source)

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

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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