L10a75

L10a74

L10a76

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a75 at Knotilus!

Link Presentations

[edit Notes on L10a75's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X16,5,17,6 X20,11,7,12 X18,13,19,14 X14,17,15,18 X12,19,13,20 X6718 X4,15,5,16
Gauss code	{1, -2, 3, -10, 4, -9}, {9, -1, 2, -3, 5, -8, 6, -7, 10, -4, 7, -6, 8, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {3u^{2}v^{2}-2u^{2}v-2uv^{2}+5uv-2u-2v+3}{uv}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {6}{q^{13/2}}}+{\frac {6}{q^{15/2}}}-{\frac {6}{q^{17/2}}}+{\frac {5}{q^{19/2}}}-{\frac {3}{q^{21/2}}}+{\frac {2}{q^{23/2}}}-{\frac {1}{q^{25/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z^{3}a^{11}+2za^{11}-z^{5}a^{9}-3z^{3}a^{9}-2za^{9}-z^{5}a^{7}-2z^{3}a^{7}+za^{7}+a^{7}z^{-1}-z^{5}a^{5}-4z^{3}a^{5}-4za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$a^{15}z^{5}-3a^{15}z^{3}+a^{15}z+2a^{14}z^{6}-6a^{14}z^{4}+3a^{14}z^{2}+2a^{13}z^{7}-5a^{13}z^{5}+2a^{13}z^{3}+2a^{12}z^{8}-7a^{12}z^{6}+10a^{12}z^{4}-5a^{12}z^{2}+a^{11}z^{9}-3a^{11}z^{7}+5a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{8}-12a^{10}z^{6}+20a^{10}z^{4}-8a^{10}z^{2}+a^{9}z^{9}-4a^{9}z^{7}+9a^{9}z^{5}-6a^{9}z^{3}+a^{9}z+a^{8}z^{8}-2a^{8}z^{6}+2a^{8}z^{4}-a^{8}z^{2}+a^{7}z^{7}-a^{7}z^{5}-3a^{7}z^{3}+3a^{7}z-a^{7}z^{-1}+a^{6}z^{6}-2a^{6}z^{4}-a^{6}z^{2}+a^{6}+a^{5}z^{5}-4a^{5}z^{3}+4a^{5}z-a^{5}z^{-1}$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                     1 1 -6                   1 1 0 -8                 2     2 -10               2 1     -1 -12             4 2       2 -14           2 2         0 -16         4 4           0 -18       2 3             1 -20     1 3               -2 -22   1 2                 1 -24   1                   -1 -26 1                     1

Integral Khovanov Homology (db, data source)

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

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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