L10a17

L10a16

L10a18

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a17 at Knotilus!

Link Presentations

[edit Notes on L10a17's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(3v^{2}-4v+3\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$13q^{9/2}-13q^{7/2}+9q^{5/2}-6q^{3/2}-q^{19/2}+4q^{17/2}-7q^{15/2}+11q^{13/2}-13q^{11/2}+2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-za^{-9}+3z^{3}a^{-7}+4za^{-7}+a^{-7}z^{-1}-2z^{5}a^{-5}-5z^{3}a^{-5}-5za^{-5}-2a^{-5}z^{-1}-z^{5}a^{-3}-z^{3}a^{-3}+z^{3}a^{-1}+2za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-11}-z^{3}a^{-11}+4z^{6}a^{-10}-7z^{4}a^{-10}+3z^{2}a^{-10}+6z^{7}a^{-9}-10z^{5}a^{-9}+4z^{3}a^{-9}-2za^{-9}+4z^{8}a^{-8}+2z^{6}a^{-8}-17z^{4}a^{-8}+12z^{2}a^{-8}-2a^{-8}+z^{9}a^{-7}+12z^{7}a^{-7}-29z^{5}a^{-7}+23z^{3}a^{-7}-8za^{-7}+a^{-7}z^{-1}+7z^{8}a^{-6}-5z^{6}a^{-6}-10z^{4}a^{-6}+14z^{2}a^{-6}-5a^{-6}+z^{9}a^{-5}+9z^{7}a^{-5}-22z^{5}a^{-5}+21z^{3}a^{-5}-9za^{-5}+2a^{-5}z^{-1}+3z^{8}a^{-4}-z^{6}a^{-4}-3z^{4}a^{-4}+5z^{2}a^{-4}-3a^{-4}+3z^{7}a^{-3}-3z^{5}a^{-3}+2z^{6}a^{-2}-3z^{4}a^{-2}+a^{-2}+z^{5}a^{-1}-3z^{3}a^{-1}+3za^{-1}-a^{-1}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 20                     1 1 18                   3   -3 16                 4 1   3 14               7 3     -4 12             6 4       2 10           7 7         0 8         6 6           0 6       3 7             4 4     3 6               -3 2   1 5                 4 0   1                   -1 -2 1                     1

Integral Khovanov Homology (db, data source)

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

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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