K12a975

Knotscape

Four-fold symmetrical depiction as decorative knot

Square depiction as two doubly-interlinked 4_1 knots on a closed loop.

Simple symmetrical squared depiction.

Circular cross decorative knot.

(alternate).

Dowker-Thistlethwaite code

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

Jones polynomial	$q^{4}-4q^{3}+10q^{2}-19q+28-33q^{-1}+36q^{-2}-33q^{-3}+27q^{-4}-19q^{-5}+10q^{-6}-4q^{-7}+q^{-8}$
Alexander polynomial	$-4t^{3}+22t^{2}-52t+69-52t^{-1}+22t^{-2}-4t^{-3}$
Conway polynomial	$-4z^{6}-2z^{4}+1$
Determinant	225
Signature	0
HOMFLY-PT polynomial	$a^{8}-4z^{2}a^{6}-4a^{6}+6z^{4}a^{4}+12z^{2}a^{4}+6a^{4}-3z^{6}a^{2}-9z^{4}a^{2}-11z^{2}a^{2}-4a^{2}-z^{6}+2z^{2}+2+z^{4}a^{-2}+z^{2}a^{-2}$
Kauffman polynomial	$3a^{5}z^{11}+3a^{3}z^{11}+6a^{6}z^{10}+20a^{4}z^{10}+14a^{2}z^{10}+4a^{7}z^{9}+11a^{5}z^{9}+33a^{3}z^{9}+26az^{9}+a^{8}z^{8}-17a^{6}z^{8}-50a^{4}z^{8}-5a^{2}z^{8}+27z^{8}-16a^{7}z^{7}-73a^{5}z^{7}-131a^{3}z^{7}-55az^{7}+19z^{7}a^{-1}-4a^{8}z^{6}+5a^{6}z^{6}-72a^{2}z^{6}+10z^{6}a^{-2}-53z^{6}+24a^{7}z^{5}+105a^{5}z^{5}+139a^{3}z^{5}+30az^{5}-24z^{5}a^{-1}+4z^{5}a^{-3}+6a^{8}z^{4}+21a^{6}z^{4}+61a^{4}z^{4}+88a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}+36z^{4}-16a^{7}z^{3}-56a^{5}z^{3}-60a^{3}z^{3}-10az^{3}+10z^{3}a^{-1}-4a^{8}z^{2}-18a^{6}z^{2}-38a^{4}z^{2}-37a^{2}z^{2}+z^{2}a^{-2}-12z^{2}+4a^{7}z+12a^{5}z+12a^{3}z+4az+a^{8}+4a^{6}+6a^{4}+4a^{2}+2$