L10a71

L10a70

L10a72

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a71 at Knotilus!

Link Presentations

[edit Notes on L10a71's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{4}-4u^{2}v^{3}+5u^{2}v^{2}-3u^{2}v+u^{2}-2uv^{4}+7uv^{3}-11uv^{2}+7uv-2u+v^{4}-3v^{3}+5v^{2}-4v+1}{uv^{2}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-9q^{9/2}+14q^{7/2}-18q^{5/2}+19q^{3/2}-19{\sqrt {q}}+{\frac {14}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-3z^{5}a^{-1}+2z^{5}a^{-3}+az^{3}-2z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-az+2za^{-1}+za^{-3}-za^{-5}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-3z^{9}a^{-1}-3z^{9}a^{-3}-17z^{8}a^{-2}-8z^{8}a^{-4}-9z^{8}-10az^{7}-17z^{7}a^{-1}-15z^{7}a^{-3}-8z^{7}a^{-5}-5a^{2}z^{6}+25z^{6}a^{-2}+8z^{6}a^{-4}-4z^{6}a^{-6}+8z^{6}-a^{3}z^{5}+16az^{5}+42z^{5}a^{-1}+39z^{5}a^{-3}+13z^{5}a^{-5}-z^{5}a^{-7}+5a^{2}z^{4}-4z^{4}a^{-2}+z^{4}a^{-4}+5z^{4}a^{-6}+5z^{4}-6az^{3}-22z^{3}a^{-1}-26z^{3}a^{-3}-9z^{3}a^{-5}+z^{3}a^{-7}-2z^{2}a^{-2}-2z^{2}a^{-4}-2z^{2}a^{-6}-2z^{2}-az+4za^{-3}+3za^{-5}-a^{-2}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                     1 1 12                   3   -3 10                 6 1   5 8               8 3     -5 6             10 6       4 4           10 9         -1 2         9 9           0 0       6 11             5 -2     4 8               -4 -4   1 6                 5 -6   4                   -4 -8 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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