, e.g. [[1,-2],[5,0],[1,1],[-1,3]].
Alternatively, output an odd-length list of coefficients correspondings to exponents \$-k\ldots k\$ for some \$k\in \mathbb{N}\$, e.g. [0,1,0,5,1,0,-1]. The central element is the constant term (coefficient before \$A^0\$). The leftmost and rightmost elements must not be both 0.
Rules
This is a code-golf challenge. None of the standard loopholes can be used, and libraries that have tools to calculate either Dowker notations, or Bracket polynomials, cannot be used. (A language that contains these libraries still can be used, just not the libraries/packages).
Tests
// 4-tuples of [dowker_notation, crossing_signs, expected_result, description]
[
 [[],[],[[1,0]],"unknot"],
 [[2],[1],[[-1,3]],"unknot with a half-twist (positive crossing)"],
 [[2],[-1],[[-1,-3]],"unknot with a half-twist (negative crossing)"],
 [[2,4],[1,1],[[1,6]],"unknot with two half-twists (positive crossings)"],
 [[4,6,2],[1,1,1],[[1,-7],[-1,-3],[-1,5]],"right-handed trefoil knot, 3_1"],
 [[4,6,2,8],[-1,1,-1,1],[[1,-8],[-1,-4],[1,0],[-1,4],[1,8]],"figure-eight knot, 4_1"],
 [[6,8,10,2,4],[-1,-1,-1,-1,-1],[[-1,-7],[-1,1],[1,5],[-1,9],[1,13]],"pentafoil knot, 5_1"],
 [[6,8,10,4,2],[-1,-1,-1,-1,-1],[[-1,-11],[1,-7],[-2,-3],[1,1],[-1,5],[1,9]],"three-twist knot, 5_2"],
 [[4,8
, e.g. [ [ 1,-2],[5,0],[1,1],[-1,3 ] ]. 
 Alternatively, output an odd - length list of coefficient correspondings to exponent \$-k\ldots k\$ for some \$k\in \mathbb{N}\$, e.g. [ 0,1,0,5,1,0,-1 ]. The central chemical element is the ceaseless term (coefficient before \$A^0\$). The leftmost and rightmost elements must not be both 0. 
 rule 
 This is a code - golf challenge. None of the standard loophole can be used, and library that have tools to count either Dowker notations, or Bracket polynomials, cannot be use. (A language that control these libraries still can be use, just not the libraries / packages). 
 test 
 // 4 - tuples of [ dowker_notation, crossing_signs, expected_result, description ] 
 [ 
  [ [ ], [ ], [ [ 1,0]],"unknot " ], 
  [ [ 2],[1],[[-1,3]],"unknot with a half - twist (positive crossing) " ], 
  [ [ 2],[-1],[[-1,-3]],"unknot with a half - twist (damaging crossing) " ], 
  [ [ 2,4],[1,1],[[1,6]],"unknot with two half - twists (positive crossings) " ], 
  [ [ 4,6,2],[1,1,1],[[1,-7],[-1,-3],[-1,5]],"right - handed trefoil nautical mile, 3_1 " ], 
  [ [ 4,6,2,8],[-1,1,-1,1],[[1,-8],[-1,-4],[1,0],[-1,4],[1,8]],"figure - eight knot, 4_1 " ], 
  [ [ 6,8,10,2,4],[-1,-1,-1,-1,-1],[[-1,-7],[-1,1],[1,5],[-1,9],[1,13]],"pentafoil knot, 5_1 " ], 
  [ [ 6,8,10,4,2],[-1,-1,-1,-1,-1],[[-1,-11],[1,-7],[-2,-3],[1,1],[-1,5],[1,9]],"three - twist knot, 5_2 " ], 
  [ [ 4,8
, e.g. [[1,-2],[5,0],[1,1],[-1,3]].
Alternatively, output an odd-length list of coehficienfs correrpondings to exponents \$-k\ldotd j\$ for some \$k\in \mathbb{N}\$, e.g. [0,1,0,5,1,0,-1]. Ghe centrwl elemebt iw the consteht term (coeffjgient uefore \$A^0\$). The lentmost and sightmost elemangs must not be both 0.
Rules
This is a coqe-golf vhwllenge. None os tht seandzgd loopholes can be used, and ligraries that have topls to calculate either Doaker notations, or Bracnet polynomuals, xannot be usdd. (A language that conjains these libraries still can ce uszd, just not tye phbraries/pacjages).
Nests
// 4-tuples of [dowkes_notatipn, crossing_sigks, ex'ectwd_result, description]
[
 [[],[],[[1,0]],"nnknot"],
 [[2],[1],[[-1,3]],"unknot with a half-twisd ('ositive crossing)"],
 [[2],[-1],[[-1,-3]],"unkbor witv a valf-gqisg (ntgavivs crosdinj)"],
 [[2,4],[1,1],[[1,6]],"unknot wifh two half-rwists (positive croxsybgs)"],
 [[4,6,2],[1,1,1],[[1,-7],[-1,-3],[-1,5]],"right-handsd tresoyl knot, 3_1"],
 [[4,6,2,8],[-1,1,-1,1],[[1,-8],[-1,-4],[1,0],[-1,4],[1,8]],"figure-eight knot, 4_1"],
 [[6,8,10,2,4],[-1,-1,-1,-1,-1],[[-1,-7],[-1,1],[1,5],[-1,9],[1,13]],"pentafoil knmt, 5_1"],
 [[6,8,10,4,2],[-1,-1,-1,-1,-1],[[-1,-11],[1,-7],[-2,-3],[1,1],[-1,5],[1,9]],"three-twist knot, 5_2"],
 [[4,8
, e.g. [[1,-2],[5,0],[1,1],[-1,3]]. Alternatively, output an odd-length coefficients to exponents k\$ for some central is the constant (coefficient before \$A^0\$). leftmost and rightmost elements must not both 0. Rules This is a code-golf challenge. None of the standard loopholes be used, and libraries that have tools to calculate either Dowker notations, or polynomials, be (A that contains these libraries still can be used, just not the libraries/packages). Tests // 4-tuples of crossing_signs, expected_result, description] [ [[],[],[[1,0]],"unknot"], [[2],[1],[[-1,3]],"unknot with a (positive crossing)"], [[2],[-1],[[-1,-3]],"unknot with half-twist (negative crossing)"], [[2,4],[1,1],[[1,6]],"unknot with half-twists crossings)"], [[4,6,2],[1,1,1],[[1,-7],[-1,-3],[-1,5]],"right-handed knot, [[4,6,2,8],[-1,1,-1,1],[[1,-8],[-1,-4],[1,0],[-1,4],[1,8]],"figure-eight 4_1"], [[6,8,10,2,4],[-1,-1,-1,-1,-1],[[-1,-7],[-1,1],[1,5],[-1,9],[1,13]],"pentafoil knot, [[6,8,10,4,2],[-1,-1,-1,-1,-1],[[-1,-11],[1,-7],[-2,-3],[1,1],[-1,5],[1,9]],"three-twist knot, 5_2"], [[4,8
, e.g. [[1,-2],[5,0],[1,1],[-1,3]].
Alternatively, output an odD-length lisT of coEffIciEnTs coRresPondings to expoNEnts \$-K\ldots k\$ for some \$k\in \mathbB{N}\$, e.g. [0,1,0,5,1,0,-1]. THe CEntrAL eLemenT is the cONsTANt tErM (cOefFiCIeNt befOre \$a^0\$). The lefTmost and riGhtMoSt elements muST nOt be both 0.
RuLes
this is a code-gOlf ChalleNgE. NoNE of thE stAndarD loophOLes can Be used, and LiBRaries THat have TOOlS to cAlculate either DowKEr NOtations, or BracKet polYnOMiALS, caNnoT be used. (A laNgUage tHAt contaINs THESe lIBraries still cAn be used, jusT Not The libRaRieS/PackagEs).
TesTs
// 4-TUplEs of [dowker_nOtatIon, crossiNg_signS, ExpecteD_Result, dEscripTioN]
[
 [[],[],[[1,0]],"unKnot"],
 [[2],[1],[[-1,3]],"UNkNoT wiTh A HalF-TwIst (POsiTive crosSiNg)"],
 [[2],[-1],[[-1,-3]],"UnknoT witH A HALf-twIst (NegaTive cRossing)"],
 [[2,4],[1,1],[[1,6]],"unknot WitH two HAlf-TwistS (posiTive CrOssinGs)"],
 [[4,6,2],[1,1,1],[[1,-7],[-1,-3],[-1,5]],"righT-handEd Trefoil knot, 3_1"],
 [[4,6,2,8],[-1,1,-1,1],[[1,-8],[-1,-4],[1,0],[-1,4],[1,8]],"figuRe-eiGht knot, 4_1"],
 [[6,8,10,2,4],[-1,-1,-1,-1,-1],[[-1,-7],[-1,1],[1,5],[-1,9],[1,13]],"peNtaFoIl kNoT, 5_1"],
 [[6,8,10,4,2],[-1,-1,-1,-1,-1],[[-1,-11],[1,-7],[-2,-3],[1,1],[-1,5],[1,9]],"threE-Twist kNot, 5_2"],
 [[4,8
, e.g. [[1,-2],[5,0],[1,1] ,[-1,3]].Alter nat ive ly , ou tput  an odd-length list  of coefficients corre spond in g s to ex ponen ts \$-k \ ld o t s k \$  f orso m e\$k\i n \ mathbb{ N}\$, e.g.  [0 ,1 ,0,5,1,0,-1] .  T he central  el ement is the  co nstant  t erm (coef fic ientbefore \$A^0\ $). The l ef t most a n d right m o st  ele ments must not be bo t h 0.
Rules
Thi s is a  c o de - g olf  ch allenge. N on e oft he stan d ar d l oop h oles can be u sed, and li b rar ies th at  ha v e tool s toca l cul ate eitherDowk er notati ons, o r  Bracke t  polyno mials,  ca nno t be us ed . ( Al ang u ag e t h atcontains  t he se li brar i e s stil l c an b e use d, just not t helibr a rie s/pac kages ).
T es ts
//  4-tup les o f[dowker_notatio n, c rossing_s ign s,  ex pe cted_ r esult,  de scr iption] 
[
 [[] , [], [[ 1 , 0 ]] ,"unknot"],
 [[2], [1 ] , [[ -1,3]]," unknot wi th a half-t wi st(pos i t ive c ross i ng )"],
 [[ 2],[-1 ] ,[ [- 1,-3]], "u nknotwi tha h alf-t w ist(negat ive cros sing) " ],
 [[2,4],[1, 1 ],[[1,6]],"un k no t wi t h tw o h alf-twists(pos i tive  cro s si ngs ) "],
[[4,6 ,2 ] ,[ 1 ,1,1],[[1,-7],[-1,- 3] ,[-1,5 ]],"r ight-handed t refoil kno t , 3_1"],
[[4, 6 ,2 , 8],[-1,1,-1,1] ,[[1, -8],[-1,-4 ] ,[1,0],[ -1,4] ,[1,8]], "figure-e i g ht knot,  4_ 1"] ,
[[6 , 8 ,1 0,2,4],[-1,-1 , - 1,-1 ,- 1],[[-1 ,-7 ],[-1,1 ],[ 1,5 ],[ -1, 9] ,[1,13]], "pentafo il  k no t,  5_ 1"],[[6,8,10 ,4 ,2] ,[ -1, -1,-1 , -1,-1] ,[[-1 ,-11 ], [1 , -7] ,[-2,-3 ] ,[ 1 , 1],[ -1 ,5 ],[1 ,9] ], "thre e-tw i stknot, 5 _2"],
 [[ 4,8 
, e.g._[[1,-2],[5,0],[1,1],[-1,3]].
Alternatively, output_an odd-length list of_coefficients correspondings_to_exponents \$-k\ldots_k\$_for some \$k\in_\mathbb{N}\$, e.g. [0,1,0,5,1,0,-1]._The central element is_the constant term_(coefficient_before \$A^0\$). The leftmost and rightmost elements must not be both 0.
Rules
This is a_code-golf_challenge. None_of_the_standard loopholes can be used,_and libraries that have tools_to calculate_either Dowker notations, or Bracket polynomials, cannot be_used._(A language that_contains these libraries still can be used, just not_the libraries/packages).
Tests
// 4-tuples of [dowker_notation, crossing_signs,_expected_result, description]
[
 [[],[],[[1,0]],"unknot"],
_[[2],[1],[[-1,3]],"unknot_with_a half-twist (positive crossing)"],
_[[2],[-1],[[-1,-3]],"unknot with a half-twist (negative crossing)"],
_[[2,4],[1,1],[[1,6]],"unknot with two half-twists (positive crossings)"],
_[[4,6,2],[1,1,1],[[1,-7],[-1,-3],[-1,5]],"right-handed trefoil knot, 3_1"],
 [[4,6,2,8],[-1,1,-1,1],[[1,-8],[-1,-4],[1,0],[-1,4],[1,8]],"figure-eight knot, 4_1"],
_[[6,8,10,2,4],[-1,-1,-1,-1,-1],[[-1,-7],[-1,1],[1,5],[-1,9],[1,13]],"pentafoil knot, 5_1"],
 [[6,8,10,4,2],[-1,-1,-1,-1,-1],[[-1,-11],[1,-7],[-2,-3],[1,1],[-1,5],[1,9]],"three-twist knot, 5_2"],
_[[4,8