L11a517

L11a516

L11a518

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a517 at Knotilus!

Link Presentations

[edit Notes on L11a517's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X20,16,21,15 X16,5,17,6 X18,11,19,12 X10,17,11,18 X4,19,5,20
Gauss code	{1, -4, 2, -11, 8, -6}, {4, -1, 5, -10, 9, -3}, {3, -2, 7, -8, 10, -9, 11, -7, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(3)-1)\left(t(1)t(3)^{2}t(2)^{2}-t(3)^{2}t(2)^{2}+2t(3)t(2)^{2}-t(2)^{2}-2t(1)t(3)^{2}t(2)+t(3)^{2}t(2)-t(1)t(2)+3t(1)t(3)t(2)-3t(3)t(2)+2t(2)+t(1)t(3)^{2}+t(1)-2t(1)t(3)-1\right)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-5q^{-5}+4q^{4}+12q^{-4}-10q^{3}-18q^{-3}+18q^{2}+26q^{-2}-24q-28q^{-1}+29$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-6a^{2}z^{4}-3z^{4}a^{-2}+11z^{4}+a^{4}z^{2}-7a^{2}z^{2}-4z^{2}a^{-2}+10z^{2}+a^{4}-4a^{2}-a^{-2}+4+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$4a^{2}z^{10}+4z^{10}+11a^{3}z^{9}+22az^{9}+11z^{9}a^{-1}+11a^{4}z^{8}+17a^{2}z^{8}+13z^{8}a^{-2}+19z^{8}+5a^{5}z^{7}-18a^{3}z^{7}-42az^{7}-10z^{7}a^{-1}+9z^{7}a^{-3}+a^{6}z^{6}-24a^{4}z^{6}-58a^{2}z^{6}-21z^{6}a^{-2}+4z^{6}a^{-4}-58z^{6}-8a^{5}z^{5}+3a^{3}z^{5}+20az^{5}-4z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-a^{6}z^{4}+15a^{4}z^{4}+50a^{2}z^{4}+18z^{4}a^{-2}-4z^{4}a^{-4}+56z^{4}+2a^{5}z^{3}+2a^{3}z^{3}+8z^{3}a^{-1}+7z^{3}a^{-3}-z^{3}a^{-5}-4a^{4}z^{2}-21a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}-27z^{2}-2a^{3}z-4az-3za^{-1}-za^{-3}+3a^{4}+7a^{2}+2a^{-2}+7+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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 5 χ 11                       1 -1 9                     3   3 7                   7 1   -6 5                 11 3     8 3               13 7       -6 1             16 11         5 -1           14 15           1 -3         12 14             -2 -5       8 16               8 -7     4 10                 -6 -9   1 8                   7 -11   4                     -4 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=0$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{16}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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