A Knotty situation

Brain-Flak, 1316 bytes

(({})<({()<(({}<>))><>}){(({})[()()]<{([{}]({})<>({}<>))}{}(([({}<>)]<<>({}<>)<>((({})<<>{({}<>)<>}<>>))>)){({}<>)<>}<>{}(({}<{}(({}<{({}<>)<>}>))>))<>{({}<>)<>}>)}<>>){(({}){}()<({}<>)>)<>{}(({}){}<>)<>}<>{}{}(()){(<({}<({}<>)>)>)<>((){[()](<(({})<>){({}[({})]<>({}<>))}{}({}<>({}<{}<>{({}<>)<>}>)[()])<>({}({})[()])(([()]{()(<({}[({})]())>)}{})<{(<{}{}>)}{}><>{()((<({}()[({}<>)])<>>))}{}<{}{}>)((){[()]<({}()<({}<({}<<>({()<({}<>)<>>}<>){({}[()]<(({})<({()<({}<>)<>>})<>>)<>{({}[()]<<>({}<>)>)}{}>)}<>>)<>>)>)((){[()](<{}(({})<<>(({})<(<<>({}<<>({}<(()()){({}[()]<([{}]()<>)<>({}<<>{({}({})<>[({}<>)])}{}{}>){({}<>)<>}<>>)}{}>{})>)>)<>{}{({}<>)<>}<>([({}<>)]<((()))>)(())<>({}<>)<>{}({}[()]){<>({}<<>(()()){({}[()]<({}<<>{({}<>)<>}>){({}[({})]<>({}<>))}{}(({})<<>({}<>)<>([{}])>)>)}{}{}>)<>({}<(({})())>[()]<>)}{}({}<<>{}([{}]()<{({}<>)<>}>){({}({})<>[({}<>)])}{}{}>){({}<>)<>}<>{}{}{}>{})>)>)}{}){(<{}(({})<<>(({}{})<<>(<({}<>)>)<>{}{({}<>)<>}<>>(({}){}){})>)>)}>}{}){(<{}([{}]<({}<<>([{}]()<>)<>({}<<>{({}({})<>[({}<>)])}{}{}>){({}<>)<>}<>>({})({}){})>)>)}{}>)}{}){{}(([{}]){}<>{}{}<<>({}<>{}){([{}]({}()()<{}({}<>)(())<>>))}{}{}{}>{})(())<>{{}({}<>)(())<>}(<>)<>}{}}{}{}<>{}{}({}<{{}({}<>)(())<>}<>{{}{((<(())>))}{}}{}{{}({}<>)(())<>}>)<>{{}({}<(<()>)<>([]){{}({}<>)(())<>([])}{}>)<>{{}({}<>)<>}{}{}({}<>)<>}<>

Try it online!

I regret nothing. Input is a flattened list of pairs.

# Part 1: extract edges
(({})<

({()<(({}<>))><>}){

(({})[()()]<

{([{}]({})<>({}<>))}{}(([({}<>)]<<>({}<>)<>((({})<<>{({}<>)<>}<>>))>)){({}<>)<>}

<>{}(({}<{}(({}<{({}<>)<>}>))>))<>{({}<>)<>}

>)}

<>>){(({}){}()<({}<>)>)<>{}(({}){}<>)<>}<>

{}{}(())

# Part 2: Compute bracket polynomial
{

  # Move degree/sign to other stack
  (<({}<({}<>)>)>)<>

  # If current shape has crossings:
  ((){[()](<

    # Consider first currently listed edge in set
    # Find the other edge leaving same crossing
    (({})<>){({}[({})]<>({}<>))}{}

    # Move to top of other stack
    # Also check for twist
    ({}<>({}<{}<>{({}<>)<>}>)[()])

    # Check for twist in current edge
    <>({}({})[()])

    (

      # Remove current edge if twist
      ([()]{()(<({}[({})]())>)}{})<{(<{}{}>)}{}>

      # Remove matching edge if twist
      <>{()((<({}()[({}<>)])<>>))}{}<{}{}>

    # Push 1 minus number of twists from current vertex.
    )

    # If number of twists is not 1:
    ((){[()]<

      # While testing whether number of twists is 2:
      ({}()<

        # Keep sign/degree on third stack:
        ({}<({}<

          # Duplicate current configuration
          <>({()<({}<>)<>>}<>){({}[()]<(({})<({()<({}<>)<>>})<>>)<>{({}[()]<<>({}<>)>)}{}>)}

        # Push sign and degree on separate stacks
        <>>)<>>)

      # If number of twists is not 2: (i.e., no twists)
      >)((){[()](<{}

        # Make first copy of sign/degree
        (({})<<>(({})<

          # Make second copy of sign/degree
          (<<>({}<<>({}<

            # Do twice:
            (()()){({}[()]<

              # Prepare search for vertex leading into crossing on other side
              ([{}]()<>)

              # While keeping destination on third stack:
              <>({}<

                # Search for matching edge
                <>{({}({})<>[({}<>)])}{}

              # Replace old destination
              {}>)

              # Move back to original stack
              {({}<>)<>}<>

            >)}{}

          # Add orientation to degree
          >{})>)>)

          # Move duplicate to left stack
          <>{}{({}<>)<>}<>

          # Create "fake" edges from current crossing as termination conditions
          ([({}<>)]<((()))>)(())<>

          # Create representation of "top" new edge
          ({}<>)<>{}({}[()])

          # While didn't reach initial crossing again:
          {

            # Keep destination of new edge on third stack
            <>({}<<>

              # Do twice:
              (()()){({}[()]<

                # Search for crossing
                ({}<<>{({}<>)<>}>){({}[({})]<>({}<>))}{}

                # Reverse orientation of crossing
                (({})<<>({}<>)<>([{}])>)

              >)}{}

              # Remove extraneous search term
              {}

            # Push new destination for edge
            >)

            # Set up next edge
            <>({}<(({})())>[()]<>)

          }

          # Get destination of last edge to link up
          {}({}<

            # Find edge headed toward original crossing
            <>{}([{}]()<{({}<>)<>}>){({}({})<>[({}<>)])}

          # Replace destination
          {}{}>)

          # Move everything to left stack
          {({}<>)<>}

          # Clean up temporary data
          <>{}{}{}

        # Push new sign/degree of negatively smoothed knot
        >{})>)

      # Else (two twists)
      # i.e., crossing is the twist in unknot with one half-twist
      >)}{}){(<{}

        # Copy sign and degree+orientation
        (({})<<>(({}{})<

          # Move sign to left stack
          <>(<({}<>)>)

          # Move copy of configuration to left stack
          <>{}{({}<>)<>}

        # Add an additional 4*orientation to degree
        <>>(({}){}){})>)

      >)}

    # Else (one twist)
    >}{}){(<

      # Invert sign and get degree
      {}([{}]<({}<

        # Search term for other edge leading to this crossing
        <>([{}]()<>)

        # With destination on third stack:
        <>({}<

          # Find matching edge
          <>{({}({})<>[({}<>)])}{}

        # Replace destination
        {}>)

        # Move stuff back to left stack
        {({}<>)<>}<>

      # Add 3*orientation to degree
      >({})({}){})>)

    >)}{}

  # Else (no crossings)
  >)}{}){{}

    # If this came from the 2-twist case, undo splitting.
    # If this came from an initial empty input, use implicit zeros to not join anything
    # New sign = sign - 2 * next entry sign
    (([{}]){}<>{}{}<

      # New degree = average of both degrees
      <>({}<>{})

      # Find coefficient corresponding to degree
      {([{}]({}()()<{}({}<>)(())<>>))}{}{}

    # Add sign to coefficient
    {}>{})

    # Move rest of polynomial back to right stack
    (())<>{{}({}<>)(())<>}

    # Set up next configuration
    (<>)<>

  }{}

}{}{}<>{}

# Step 3: Put polynomial in correct form

# Keeping constant term:
{}({}<

  # Move to other stack to get access to terms of highest absolute degree
  {{}({}<>)(())<>}<>

  # Remove outer zeros
  {{}{((<(())>))}{}}

  # Move back to right stack to get access to lower order terms
  {}{{}({}<>)(())<>}

>)<>

# While terms remain:
{

  # Move term with positive coefficient
  {}({}<(<()>)<>([]){{}({}<>)(())<>([])}{}>)<>{{}({}<>)<>}{}

  # Move term with negative coefficient
  {}({}<>)<>

}<>

K (ngn/k), 196 193 185 177 172 bytes

-2 bytes thanks to @coltim

{N::2*n:#x;+/(-m-|/m:#'b)(|0,)/'b:(+/1-2*s){,/(&0|2*x;(#1_?{y[x]&:y@|x;y}[+,/y]/!N){-x+|x,:&4}/1;&0|-2*x)}'N!({(x,'1+|x;x+/:!2)}'(0<x)|:/'+(2*!n;-1+x|-x))@'/:+(0<y)=s:!n#2}

Try it online!

ungolfed:

f:{                                      /args: x:dowker notation, y:crossing signs
 N::2*n:#x                               /n:number of crossings, N:twice n
 s:1-2*!n#2                              /s:all possible sets of smoothing signs for the crossings (n*(2^n) matrix of -1 1)
 S:{(x,'1+|x;0 1+\:x:(0<x)|:/y,-1+x|-x)} /S:smoothe the crossing numbered x,y in dowker notation (x can be negative)
 a:N!(x S'2*!n)@'/:+~y=s                 /generate all possible smoothed knots (results consist of neighbour pairs that form sets of disjoint unknots)
 u::{#1_?{y[x]&:|y x;y}[+,/x]/!N}        /count number of disjoint unknots in a smoothed knot, minus one
 /       ^^^^^^^^^^^^^^^^^^^^^           /  transitive closure
 g::{-x+|x,:&4}                          /if x=L(A) is a laurent polynomial represented as list of coefficients centred on A^0 (the constant term)
 /                                       /  then g[x] represents (-A^2-A^(-2))*L(A)
 b:(+/s){,/(0<x)|:/(u[y]g/1;&2*x|-x)}'a  /build a laurent polynomials for each set of smoothing signs
 +/(-m-|/m:#'b)(|0,)/'b}                 /centre them and pad them to form a matrix and sum them to get the bracket polynomial