A Sample KnotTheory` Session

KnotTheory`/Navigation  The Mathematica Package KnotTheory` Acknowledgement Setup Naming and Enumeration Presentations Planar Diagrams Some further details Gauss Codes DT (Dowker-Thistlethwaite) Codes Braid Representatives MorseLink Presentations Arc Presentations Conway Notation Graphical Input Graphical Output Drawing Planar Diagrams How does it work? Drawing MorseLink Presentations Drawing Braids Structure and Operations Invariants Invariants from Braid Theory Three Dimensional Invariants The Alexander-Conway Polynomial The Multivariable Alexander Polynomial The Determinant and the Signature The Jones Polynomial How is the Jones polynomial computed? The Coloured Jones Polynomials The A2 Invariant Quantum knot invariants The HOMFLY-PT Polynomial The Kauffman Polynomial Finite Type (Vassiliev) Invariants Khovanov Homology Heegaard Floer Knot Homology R-Matrix Invariants Extras Included with KnotTheory` Drawing with TubePlot Standalone_TubePlot Using the LinKnot package WikiLink QuantumGroups` Lightly Documented Features A Sample KnotTheory` Session Further Usage Example Prime Links with a Non-Prime Component "Rubberband" Brunnian Links Identifying Knots within a List Cabling Burau's Theorem Further Knot Theory Software KnotPlot Knotscape Xknots Extending/Modifying KnotTheory` Bugs How to Edit this Manual...

Setup

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

In[1]:= << KnotTheory`

Loading KnotTheory` (version of September 14, 2005, 13:37:36)...

8_17

K11a231

L8n6

T(7,5)

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-

Drawing MorseLink Presentations

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]}

K11a57

K11a108

K11a139

K11a231

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 }