L10a56

L10a55

L10a57

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a56 at Knotilus!

Link Presentations

[edit Notes on L10a56's Link Presentations]

Planar diagram presentation	X8192 X16,7,17,8 X10,4,11,3 X12,5,13,6 X20,12,7,11 X18,13,19,14 X2,15,3,16 X4,19,5,20 X14,10,15,9 X6,18,1,17
Gauss code	{1, -7, 3, -8, 4, -10}, {2, -1, 9, -3, 5, -4, 6, -9, 7, -2, 10, -6, 8, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{4}-4u^{2}v^{3}+6u^{2}v^{2}-4u^{2}v+u^{2}-2uv^{4}+8uv^{3}-13uv^{2}+8uv-2u+v^{4}-4v^{3}+6v^{2}-4v+1}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {11}{q^{9/2}}}-q^{7/2}+{\frac {16}{q^{7/2}}}+5q^{5/2}-{\frac {21}{q^{5/2}}}-11q^{3/2}+{\frac {22}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {5}{q^{11/2}}}+16{\sqrt {q}}-{\frac {21}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+3az^{5}-z^{5}a^{-1}+a^{5}z^{3}-3a^{3}z^{3}+3az^{3}-z^{3}a^{-1}-az+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-5a^{3}z^{9}-5az^{9}-12a^{4}z^{8}-24a^{2}z^{8}-12z^{8}-11a^{5}z^{7}-14a^{3}z^{7}-14az^{7}-11z^{7}a^{-1}-5a^{6}z^{6}+16a^{4}z^{6}+42a^{2}z^{6}-5z^{6}a^{-2}+16z^{6}-a^{7}z^{5}+16a^{5}z^{5}+39a^{3}z^{5}+39az^{5}+16z^{5}a^{-1}-z^{5}a^{-3}+4a^{6}z^{4}-4a^{4}z^{4}-16a^{2}z^{4}+4z^{4}a^{-2}-4z^{4}-6a^{5}z^{3}-18a^{3}z^{3}-18az^{3}-6z^{3}a^{-1}-a^{3}z-az-a^{2}+a^{3}z^{-1}+az^{-1}$ (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 8                     1 1 6                   4   -4 4                 7 1   6 2               9 4     -5 0             12 7       5 -2           11 10         -1 -4         10 11           -1 -6       7 12             5 -8     4 9               -5 -10   1 7                 6 -12   4                   -4 -14 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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