# A Sample KnotTheory` Session

### Setup

The first step is to load KnotTheory` as in the Setup section:

In[1]:=` << KnotTheory``

Let us now introduce the four star knots that will accompany us throughout this session:

 `In[2]:=` ```K = Knot[8, 17]; K11 = Knot[11, Alternating, 231]; L = Link[8, NonAlternating, 6]; TK = TorusKnot[7,5];```

### Presentations, Graphical Output and Tube Plots

#### Planar Diagrams

 `In[3]:=` `PD[K]` `Out[3]=` ```PD[X[6, 2, 7, 1], X[14, 8, 15, 7], X[8, 3, 9, 4], X[2, 13, 3, 14], X[12, 5, 13, 6], X[4, 9, 5, 10], X[16, 12, 1, 11], X[10, 16, 11, 15]]```

#### Gauss Codes

 `In[4]:=` `{GaussCode[K], GaussCode[L]}` `Out[4]=` ```{GaussCode[1, -4, 3, -6, 5, -1, 2, -3, 6, -8, 7, -5, 4, -2, 8, -7], GaussCode[{1, -7, 2, -8}, {-5, 4, -6, 3}, {7, -1, -4, 5, 8, -2, -3, 6}]}```

#### DT (Dowker-Thistlethwaite) Codes

 `In[5]:=` `DTCode[K]` `Out[5]=` `DTCode[6, 8, 12, 14, 4, 16, 2, 10]`

#### Braid Representatives

 `In[6]:=` `br = BR[K]` `Out[6]=` `BR[3, {-1, -1, 2, -1, 2, -1, 2, 2}]`
 `In[7]:=` `{First[br], Crossings[br], BraidIndex[K]}` `Out[7]=` `{3, 8, 3}`

#### Drawing Braids

 `In[8]:=` `Show[BraidPlot[br]]` `Out[8]=` `-Graphics-`

 `In[9]:=` `Show[DrawMorseLink[K]]` `Out[9]=` `-Graphics-`
 `In[10]:=` `Show[DrawMorseLink[L]]` `Out[10]=` `-Graphics-`

#### Drawing with TubePlot

 `In[11]:=` `Show[TubePlot[TK]]` `Out[11]=` `-Graphics3D-`

### Three Dimensional Invariants

 `In[12]:=` ```(#[K]&) /@ { SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex }``` `Out[12]=` `{NegativeAmphicheiral, 1, 3, 3, 4, 1}`

### Polynomial Invariants

#### The Alexander-Conway Polynomial

 `In[13]:=` `alex = Alexander[K11][t]` `Out[13]=` ``` -4 5 12 20 2 3 4 -23 - t + -- - -- + -- + 20 t - 12 t + 5 t - t 3 2 t t t```
 `In[14]:=` `Conway[K11][t]` `Out[14]=` ``` 2 4 6 8 1 + t - 2 t - 3 t - t```
##### "Similar" Knots (within the Atlas)
 `In[15]:=` `Select[AllKnots[], (alex === Alexander[#][t])&]` `Out[15]=` ```{Knot[11, Alternating, 57], Knot[11, Alternating, 108], Knot[11, Alternating, 139], Knot[11, Alternating, 231]}```
##### The Determinant and the Signature
 `In[16]:=` `{KnotDet[K], KnotSignature[K]}` `Out[16]=` `{37, 0}`

#### The Jones Polynomial

 `In[17]:=` `J=Jones[K11][q]` `Out[17]=` ``` -7 3 7 12 14 16 16 2 3 4 -12 + q - -- + -- - -- + -- - -- + -- + 10 q - 5 q + 2 q - q 6 5 4 3 2 q q q q q q```
##### "Similar" Knots (within the Atlas)
 `In[18]:=` `Select[AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q])&]` `Out[18]=` `{Knot[11, Alternating, 57], Knot[11, Alternating, 231]}`

#### The A2 Invariant

 `In[19]:=` `A2Invariant[L][q]` `Out[19]=` ``` -30 3 4 5 4 3 2 -16 -14 -10 -8 q + --- + --- + --- + --- + --- + --- + q + q + q + q + 28 26 24 22 20 18 q q q q q q -6 q```

#### The HOMFLY-PT Polynomial

 `In[20]:=` `HOMFLYPT[L][a, z]` `Out[20]=` ``` 6 8 10 4 8 a 2 a a 4 2 4 4 2 a - 2 a + -- - ---- + --- + 4 a z + a z 2 2 2 z z z```

#### The Kauffman Polynomial

 `In[21]:=` `Kauffman[L][a, z]` `Out[21]=` ``` 6 8 10 7 9 4 6 8 10 a 2 a a 2 a 2 a 7 2 a - 2 a - 9 a - 6 a + -- + ---- + --- - ---- - ---- + 8 a z + 2 2 2 z z z z z 9 4 2 8 2 10 2 7 3 9 3 4 4 8 a z - 4 a z + 14 a z + 10 a z - 6 a z - 6 a z + a z - 8 4 10 4 7 5 9 5 8 6 10 6 7 a z - 6 a z + a z + a z + a z + a z```

### Finite Type (Vassiliev) Invariants

 `In[22]:=` `{Vassiliev[2][K], Vassiliev[3][K]}` `Out[22]=` `{-1, 0}`

### Khovanov Homology

 `In[23]:=` `Kh[TK][q, t]` `Out[23]=` ``` 23 25 27 2 31 3 29 4 31 4 33 5 35 5 q + q + q t + q t + q t + q t + q t + q t + 31 6 33 6 35 7 37 7 33 8 35 8 37 9 q t + q t + q t + q t + q t + 2 q t + q t + 39 9 37 10 41 11 39 12 41 12 45 12 2 q t + 2 q t + 3 q t + q t + 2 q t + q t + 43 13 45 13 43 14 47 14 47 15 47 16 2 q t + 2 q t + q t + q t + 2 q t + q t + 51 16 51 17 q t + q t```

### The Coloured Jones Polynomials

 `In[24]:=` `ColouredJones[K, #][q]& /@ {2, 3, 4, 5, 6, 7}` `Out[24]=` ``` -12 3 -10 9 14 3 28 25 14 47 29 25 {55 + q - --- + q + -- - -- - -- + -- - -- - -- + -- - -- - -- - 11 9 8 7 6 5 4 3 2 q q q q q q q q q q 2 3 4 5 6 7 8 9 25 q - 29 q + 47 q - 14 q - 25 q + 28 q - 3 q - 14 q + 9 q + 10 11 12 -24 3 -22 5 -20 14 q - 3 q + q , 225 + q - --- + q + --- + q - --- - 23 21 19 q q q 6 29 17 43 40 55 73 64 108 61 146 --- + --- + --- - --- - --- + --- + --- - --- - --- + -- + --- - 18 17 16 15 14 13 12 11 10 9 8 q q q q q q q q q q q 53 177 38 205 26 216 6 2 3 4 -- - --- + -- + --- - -- - --- + - + 6 q - 216 q - 26 q + 205 q + 7 6 5 4 3 2 q q q q q q q 5 6 7 8 9 10 11 38 q - 177 q - 53 q + 146 q + 61 q - 108 q - 64 q + 12 13 14 15 16 17 18 73 q + 55 q - 40 q - 43 q + 17 q + 29 q - 6 q - 19 20 21 22 23 24 14 q + q + 5 q + q - 3 q + q , -40 3 -38 5 3 -35 17 6 31 1233 + q - --- + q + --- - --- + q - --- + --- + --- + 39 37 36 34 33 32 q q q q q q -31 82 16 96 69 52 216 146 120 216 260 q - --- - --- + --- + --- + --- - --- - --- + --- + --- + --- - 29 28 27 26 25 24 23 22 21 20 q q q q q q q q q q 323 393 7 340 605 292 631 265 347 945 149 --- - --- - --- + --- + --- - --- - --- - --- + --- + --- - --- - 19 18 17 16 15 14 13 12 11 10 9 q q q q q q q q q q q 759 522 261 1161 11 771 694 144 2 --- - --- + --- + ---- + -- - --- - --- + --- + 144 q - 694 q - 8 7 6 5 4 3 2 q q q q q q q q 3 4 5 6 7 8 9 771 q + 11 q + 1161 q + 261 q - 522 q - 759 q - 149 q + 10 11 12 13 14 15 945 q + 347 q - 265 q - 631 q - 292 q + 605 q + 16 17 18 19 20 21 22 340 q - 7 q - 393 q - 323 q + 260 q + 216 q + 120 q - 23 24 25 26 27 28 29 146 q - 216 q + 52 q + 69 q + 96 q - 16 q - 82 q + 31 32 33 34 35 36 37 38 39 q + 31 q + 6 q - 17 q + q - 3 q + 5 q + q - 3 q + 40 -60 3 -58 5 3 3 2 5 8 q , 4841 + q - --- + q + --- - --- - --- - --- - --- + --- + 59 57 56 55 54 53 52 q q q q q q q 26 4 30 43 34 35 112 107 31 197 237 --- + --- - --- - --- - --- + --- + --- + --- - --- - --- - --- - 51 50 49 48 47 46 45 44 43 42 41 q q q q q q q q q q q 60 270 462 264 285 728 603 141 976 1094 186 --- + --- + --- + --- - --- - --- - --- + --- + --- + ---- + --- - 40 39 38 37 36 35 34 33 32 31 30 q q q q q q q q q q q 1134 1650 699 1099 2200 1387 888 2662 2125 494 ---- - ---- - --- + ---- + ---- + ---- - --- - ---- - ---- + --- + 29 28 27 26 25 24 23 22 21 20 q q q q q q q q q q 2955 2877 9 3114 3506 568 3121 4033 1086 3040 ---- + ---- + --- - ---- - ---- - --- + ---- + ---- + ---- - ---- - 19 18 17 16 15 14 13 12 11 10 q q q q q q q q q q 4387 1560 2881 4660 1920 2707 4762 2247 2479 ---- - ---- + ---- + ---- + ---- - ---- - ---- - ---- + ---- + 9 8 7 6 5 4 3 2 q q q q q q q q q 2 3 4 5 6 7 2479 q - 2247 q - 4762 q - 2707 q + 1920 q + 4660 q + 2881 q - 8 9 10 11 12 13 1560 q - 4387 q - 3040 q + 1086 q + 4033 q + 3121 q - 14 15 16 17 18 19 568 q - 3506 q - 3114 q + 9 q + 2877 q + 2955 q + 20 21 22 23 24 25 494 q - 2125 q - 2662 q - 888 q + 1387 q + 2200 q + 26 27 28 29 30 31 1099 q - 699 q - 1650 q - 1134 q + 186 q + 1094 q + 32 33 34 35 36 37 976 q + 141 q - 603 q - 728 q - 285 q + 264 q + 38 39 40 41 42 43 44 462 q + 270 q - 60 q - 237 q - 197 q - 31 q + 107 q + 45 46 47 48 49 50 51 112 q + 35 q - 34 q - 43 q - 30 q + 4 q + 26 q + 52 53 54 55 56 57 58 59 60 8 q - 5 q - 2 q - 3 q - 3 q + 5 q + q - 3 q + q , -84 3 -82 5 3 3 6 10 3 3 26111 + q - --- + q + --- - --- - --- - --- + --- - --- + --- + 83 81 80 79 78 77 76 75 q q q q q q q q 29 15 31 49 14 16 61 153 14 117 273 --- - --- - --- - --- + --- + --- + --- + --- + --- - --- - --- - 74 73 72 71 70 69 68 67 66 65 64 q q q q q q q q q q q 149 92 203 641 463 57 691 870 1005 189 1343 --- - --- + --- + --- + --- + --- - --- - --- - ---- - --- + ---- + 63 62 61 60 59 58 57 56 55 54 53 q q q q q q q q q q q 1882 1543 247 1725 3355 2544 658 3470 4894 2812 ---- + ---- - --- - ---- - ---- - ---- + --- + ---- + ---- + ---- - 52 51 50 49 48 47 46 45 44 43 q q q q q q q q q q 590 5842 7188 3328 2689 8288 8456 4312 5674 --- - ---- - ---- - ---- + ---- + ---- + ---- + ---- - ---- - 42 41 40 39 38 37 36 35 34 q q q q q q q q q 11801 10070 1954 8878 14013 11845 1776 13643 16608 ----- - ----- - ---- + ---- + ----- + ----- - ---- - ----- - ----- - 33 32 31 30 29 28 27 26 25 q q q q q q q q q 8872 6032 16953 18906 4000 12400 20628 15121 1645 ---- + ---- + ----- + ----- + ---- - ----- - ----- - ----- + ---- + 24 23 22 21 20 19 18 17 16 q q q q q q q q q 17146 23466 9075 9763 22071 19122 2210 15962 25565 ----- + ----- + ---- - ---- - ----- - ----- - ---- + ----- + ----- + 15 14 13 12 11 10 9 8 7 q q q q q q q q q 12389 7226 22014 21134 4957 14414 2 ----- - ---- - ----- - ----- - ---- + ----- + 14414 q - 4957 q - 6 5 4 3 2 q q q q q q 3 4 5 6 7 8 21134 q - 22014 q - 7226 q + 12389 q + 25565 q + 15962 q - 9 10 11 12 13 14 2210 q - 19122 q - 22071 q - 9763 q + 9075 q + 23466 q + 15 16 17 18 19 17146 q + 1645 q - 15121 q - 20628 q - 12400 q + 20 21 22 23 24 25 4000 q + 18906 q + 16953 q + 6032 q - 8872 q - 16608 q - 26 27 28 29 30 31 13643 q - 1776 q + 11845 q + 14013 q + 8878 q - 1954 q - 32 33 34 35 36 37 10070 q - 11801 q - 5674 q + 4312 q + 8456 q + 8288 q + 38 39 40 41 42 43 2689 q - 3328 q - 7188 q - 5842 q - 590 q + 2812 q + 44 45 46 47 48 49 4894 q + 3470 q + 658 q - 2544 q - 3355 q - 1725 q - 50 51 52 53 54 55 247 q + 1543 q + 1882 q + 1343 q - 189 q - 1005 q - 56 57 58 59 60 61 62 870 q - 691 q + 57 q + 463 q + 641 q + 203 q - 92 q - 63 64 65 66 67 68 69 149 q - 273 q - 117 q + 14 q + 153 q + 61 q + 16 q + 70 71 72 73 74 75 76 14 q - 49 q - 31 q - 15 q + 29 q + 3 q - 3 q + 77 78 79 80 81 82 83 84 10 q - 6 q - 3 q - 3 q + 5 q + q - 3 q + q , -112 3 -110 5 3 3 6 6 127145 + q - ---- + q + ---- - ---- - ---- - ---- + ---- + 111 109 108 107 106 105 q q q q q q 12 8 6 10 16 26 41 7 74 44 ---- - ---- + ---- + ---- - ---- - --- - --- + --- + --- + --- + 104 103 102 101 100 99 98 97 96 95 q q q q q q q q q q 71 43 78 159 283 154 143 317 550 516 93 --- + --- - --- - --- - --- - --- + --- + --- + --- + --- + --- - 94 93 92 91 90 89 88 87 86 85 84 q q q q q q q q q q q 417 1159 1332 683 256 1725 2573 2216 836 1934 --- - ---- - ---- - --- + --- + ---- + ---- + ---- + --- - ---- - 83 82 81 80 79 78 77 76 75 74 q q q q q q q q q q 4278 4774 3301 913 5542 8189 7815 2389 5364 ---- - ---- - ---- + --- + ---- + ---- + ---- + ---- - ---- - 73 72 71 70 69 68 67 66 65 q q q q q q q q q 11792 14143 8598 2327 13890 21402 18063 4775 12979 ----- - ----- - ---- + ---- + ----- + ----- + ----- + ---- - ----- - 64 63 62 61 60 59 58 57 56 q q q q q q q q q 28137 29695 16138 7462 32099 41960 31319 3221 ----- - ----- - ----- + ---- + ----- + ----- + ----- + ---- - 55 54 53 52 51 50 49 48 q q q q q q q q 31814 52797 48414 18546 26269 60101 65456 37209 ----- - ----- - ----- - ----- + ----- + ----- + ----- + ----- - 47 46 45 44 43 42 41 40 q q q q q q q q 15734 62982 80416 56819 1416 61028 91744 75657 ----- - ----- - ----- - ----- + ---- + ----- + ----- + ----- + 39 38 37 36 35 34 33 32 q q q q q q q q 14955 55290 98994 91866 31357 46837 102372 104758 ----- - ----- - ----- - ----- - ----- + ----- + ------ + ------ + 31 30 29 28 27 26 25 24 q q q q q q q q 46339 37376 102713 113992 58991 28020 101063 120209 ----- - ----- - ------ - ------ - ----- + ----- + ------ + ------ + 23 22 21 20 19 18 17 16 q q q q q q q q 68876 19792 98300 123826 76313 12722 95254 125943 ----- - ----- - ----- - ------ - ----- + ----- + ----- + ------ + 15 14 13 12 11 10 9 8 q q q q q q q q 81699 7156 92151 126720 85773 2177 89089 ----- - ---- - ----- - ------ - ----- + ---- + ----- + 89089 q + 7 6 5 4 3 2 q q q q q q q 2 3 4 5 6 7 2177 q - 85773 q - 126720 q - 92151 q - 7156 q + 81699 q + 8 9 10 11 12 125943 q + 95254 q + 12722 q - 76313 q - 123826 q - 13 14 15 16 17 98300 q - 19792 q + 68876 q + 120209 q + 101063 q + 18 19 20 21 22 28020 q - 58991 q - 113992 q - 102713 q - 37376 q + 23 24 25 26 27 46339 q + 104758 q + 102372 q + 46837 q - 31357 q - 28 29 30 31 32 91866 q - 98994 q - 55290 q + 14955 q + 75657 q + 33 34 35 36 37 91744 q + 61028 q + 1416 q - 56819 q - 80416 q - 38 39 40 41 42 62982 q - 15734 q + 37209 q + 65456 q + 60101 q + 43 44 45 46 47 26269 q - 18546 q - 48414 q - 52797 q - 31814 q + 48 49 50 51 52 3221 q + 31319 q + 41960 q + 32099 q + 7462 q - 53 54 55 56 57 16138 q - 29695 q - 28137 q - 12979 q + 4775 q + 58 59 60 61 62 18063 q + 21402 q + 13890 q + 2327 q - 8598 q - 63 64 65 66 67 68 14143 q - 11792 q - 5364 q + 2389 q + 7815 q + 8189 q + 69 70 71 72 73 74 5542 q + 913 q - 3301 q - 4774 q - 4278 q - 1934 q + 75 76 77 78 79 80 836 q + 2216 q + 2573 q + 1725 q + 256 q - 683 q - 81 82 83 84 85 86 1332 q - 1159 q - 417 q + 93 q + 516 q + 550 q + 87 88 89 90 91 92 93 317 q + 143 q - 154 q - 283 q - 159 q - 78 q + 43 q + 94 95 96 97 98 99 100 71 q + 44 q + 74 q + 7 q - 41 q - 26 q - 16 q + 101 102 103 104 105 106 107 10 q + 6 q - 8 q + 12 q + 6 q - 6 q - 3 q - 108 109 110 111 112 3 q + 5 q + q - 3 q + q }```