L10a130

L10a129

L10a131

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a130 at Knotilus!

Link Presentations

[edit Notes on L10a130's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(3)^{2}t(2)^{2}-t(1)t(2)^{2}+2t(1)t(3)t(2)^{2}-3t(3)t(2)^{2}+t(2)^{2}+2t(1)t(3)^{2}t(2)-3t(3)^{2}t(2)+3t(1)t(2)-5t(1)t(3)t(2)+5t(3)t(2)-2t(2)-t(1)t(3)^{2}+t(3)^{2}-2t(1)+3t(1)t(3)-2t(3)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{-2}-3q^{-3}+7q^{-4}-9q^{-5}+13q^{-6}-12q^{-7}+12q^{-8}-9q^{-9}+6q^{-10}-3q^{-11}+q^{-12}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{12}-3a^{10}z^{2}+a^{10}z^{-2}-2a^{10}+2a^{8}z^{4}+a^{8}z^{2}-2a^{8}z^{-2}-3a^{8}+3a^{6}z^{4}+6a^{6}z^{2}+a^{6}z^{-2}+4a^{6}+a^{4}z^{4}+a^{4}z^{2}$ (db)
Kauffman polynomial	$z^{6}a^{14}-3z^{4}a^{14}+3z^{2}a^{14}-a^{14}+3z^{7}a^{13}-9z^{5}a^{13}+8z^{3}a^{13}-2za^{13}+3z^{8}a^{12}-4z^{6}a^{12}-6z^{4}a^{12}+8z^{2}a^{12}-a^{12}+z^{9}a^{11}+8z^{7}a^{11}-27z^{5}a^{11}+22z^{3}a^{11}-6za^{11}+7z^{8}a^{10}-9z^{6}a^{10}-6z^{4}a^{10}+7z^{2}a^{10}+a^{10}z^{-2}-3a^{10}+z^{9}a^{9}+11z^{7}a^{9}-26z^{5}a^{9}+15z^{3}a^{9}-2a^{9}z^{-1}+4z^{8}a^{8}+2z^{6}a^{8}-13z^{4}a^{8}+12z^{2}a^{8}+2a^{8}z^{-2}-6a^{8}+6z^{7}a^{7}-5z^{5}a^{7}-z^{3}a^{7}+4za^{7}-2a^{7}z^{-1}+6z^{6}a^{6}-9z^{4}a^{6}+9z^{2}a^{6}+a^{6}z^{-2}-4a^{6}+3z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-z^{2}a^{4}$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                     1 1 -5                   3 1 -2 -7                 4     4 -9               5 3     -2 -11             8 4       4 -13           5 6         1 -15         7 7           0 -17       4 7             3 -19     2 5               -3 -21   1 4                 3 -23   2                   -2 -25 1                     1

Integral Khovanov Homology (db, data source)

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

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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