L10a37

L10a36

L10a38

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a37 at Knotilus!

Link Presentations

[edit Notes on L10a37's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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