K12a541

Knotscape

Old German army general's shoulder epaulette braiding

Knot presentations

Dowker-Thistlethwaite code

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

Notes on presentations of K12a541

Polynomial invariants

Jones polynomial	$q^{10}-4q^{9}+9q^{8}-14q^{7}+19q^{6}-23q^{5}+23q^{4}-21q^{3}+17q^{2}-11q+7-3q^{-1}+q^{-2}$
Alexander polynomial	$t^{4}-7t^{3}+19t^{2}-31t+37-31t^{-1}+19t^{-2}-7t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+z^{6}-3z^{4}-2z^{2}+1$
Determinant	153
Signature	4
HOMFLY-PT polynomial	$z^{8}a^{-4}-2z^{6}a^{-2}+5z^{6}a^{-4}-2z^{6}a^{-6}-8z^{4}a^{-2}+10z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+z^{4}-9z^{2}a^{-2}+9z^{2}a^{-4}-7z^{2}a^{-6}+2z^{2}a^{-8}+3z^{2}-2a^{-2}+2a^{-4}-2a^{-6}+a^{-8}+2$
Kauffman polynomial	$z^{11}a^{-3}+z^{11}a^{-5}+3z^{10}a^{-2}+7z^{10}a^{-4}+4z^{10}a^{-6}+3z^{9}a^{-1}+7z^{9}a^{-3}+13z^{9}a^{-5}+9z^{9}a^{-7}-8z^{8}a^{-2}-11z^{8}a^{-4}+11z^{8}a^{-6}+13z^{8}a^{-8}+z^{8}-14z^{7}a^{-1}-37z^{7}a^{-3}-40z^{7}a^{-5}-4z^{7}a^{-7}+13z^{7}a^{-9}-5z^{6}a^{-2}-24z^{6}a^{-4}-50z^{6}a^{-6}-17z^{6}a^{-8}+9z^{6}a^{-10}-5z^{6}+21z^{5}a^{-1}+44z^{5}a^{-3}+23z^{5}a^{-5}-21z^{5}a^{-7}-17z^{5}a^{-9}+4z^{5}a^{-11}+26z^{4}a^{-2}+46z^{4}a^{-4}+43z^{4}a^{-6}+5z^{4}a^{-8}-8z^{4}a^{-10}+z^{4}a^{-12}+9z^{4}-10z^{3}a^{-1}-12z^{3}a^{-3}+z^{3}a^{-5}+13z^{3}a^{-7}+9z^{3}a^{-9}-z^{3}a^{-11}-17z^{2}a^{-2}-19z^{2}a^{-4}-16z^{2}a^{-6}-4z^{2}a^{-8}+3z^{2}a^{-10}-7z^{2}-2za^{-7}-2za^{-9}+2a^{-2}+2a^{-4}+2a^{-6}+a^{-8}+2$

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