L10a30

L10a29

L10a31

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a30 at Knotilus!

Link Presentations

[edit Notes on L10a30's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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