K12a1019

Knotscape

Three-fold symmetrical representation

Knotscape

Three-fold symmetrical representation

K12a1019 as a geodesic line of the oblate spheroid

Knot presentations

Dowker-Thistlethwaite code

6 12 10 16 14 20 18 24 22 4 2 8

Notes on presentations of K12a1019

Polynomial invariants

Jones polynomial	$q^{6}-6q^{5}+16q^{4}-30q^{3}+44q^{2}-54q+59-54q^{-1}+44q^{-2}-30q^{-3}+16q^{-4}-6q^{-5}+q^{-6}$
Alexander polynomial	$t^{4}-10t^{3}+39t^{2}-80t+101-80t^{-1}+39t^{-2}-10t^{-3}+t^{-4}$
Conway polynomial	$z^{8}-2z^{6}-z^{4}+2z^{2}+1$
Determinant	361
Signature	0
HOMFLY-PT polynomial	$z^{8}-2a^{2}z^{6}-2z^{6}a^{-2}+2z^{6}+a^{4}z^{4}-2a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}+z^{4}+2a^{2}z^{2}+2z^{2}a^{-2}-2z^{2}-a^{4}+2a^{2}+2a^{-2}-a^{-4}-1$
Kauffman polynomial	$8az^{11}+8z^{11}a^{-1}+22a^{2}z^{10}+22z^{10}a^{-2}+44z^{10}+25a^{3}z^{9}+37az^{9}+37z^{9}a^{-1}+25z^{9}a^{-3}+16a^{4}z^{8}-23a^{2}z^{8}-23z^{8}a^{-2}+16z^{8}a^{-4}-78z^{8}+6a^{5}z^{7}-42a^{3}z^{7}-108az^{7}-108z^{7}a^{-1}-42z^{7}a^{-3}+6z^{7}a^{-5}+a^{6}z^{6}-22a^{4}z^{6}-8a^{2}z^{6}-8z^{6}a^{-2}-22z^{6}a^{-4}+z^{6}a^{-6}+30z^{6}-5a^{5}z^{5}+23a^{3}z^{5}+78az^{5}+78z^{5}a^{-1}+23z^{5}a^{-3}-5z^{5}a^{-5}+7a^{4}z^{4}+5a^{2}z^{4}+5z^{4}a^{-2}+7z^{4}a^{-4}-4z^{4}-7a^{3}z^{3}-17az^{3}-17z^{3}a^{-1}-7z^{3}a^{-3}+a^{4}z^{2}+5a^{2}z^{2}+5z^{2}a^{-2}+z^{2}a^{-4}+8z^{2}+a^{5}z+a^{3}z+za^{-3}+za^{-5}-a^{4}-2a^{2}-2a^{-2}-a^{-4}-1$

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