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1050 1052

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KnotPlot

PD Presentation:

X1425 X3849 X9,17,10,16 X5,15,6,14 X15,7,16,6 X13,1,14,20 X19,11,20,10 X11,19,12,18 X17,13,18,12 X7283

Gauss Code:

{-1, 10, -2, 1, -4, 5, -10, 2, -3, 7, -8, 9, -6, 4, -5, 3, -9, 8, -7, 6}

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 16 18 20 6 12 10

Minimum Braid Representative: Length is 11, width is 4 Braid index is 4

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index Reversible 2--3 3 3 / NotAvailable 1

Alexander Polynomial:

2t-3 - 7t-2 + 15t-1 - 19 + 15t - 7t2 + 2t3

Conway Polynomial:

1 + 5z2 + 5z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{67, 2}

Jones Polynomial:

- q-2 + 3q-1 - 6 + 9q - 10q2 + 12q3 - 10q4 + 8q5 - 5q6 + 2q7 - q8

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

- q-6 + q-4 - q-2 - 1 + 2q2 - 2q4 + 3q6 + q8 + 2q10 + 3q12 - q14 + 2q16 - 2q18 - 2q20 - q24

HOMFLY-PT Polynomial:

- 3a-6 - 3a-6z2 - a-6z4 + 4a-4 + 7a-4z2 + 4a-4z4 + a-4z6 + a-2 + 3a-2z2 + 3a-2z4 + a-2z6 - 1 - 2z2 - z4

Kauffman Polynomial:

2a-9z - 3a-9z3 + a-9z5 + a-8z2 - 4a-8z4 + 2a-8z6 - 3a-7z + 5a-7z3 - 6a-7z5 + 3a-7z7 + 3a-6 - 8a-6z2 + 9a-6z4 - 6a-6z6 + 3a-6z8 - 9a-5z + 21a-5z3 - 16a-5z5 + 5a-5z7 + a-5z9 + 4a-4 - 8a-4z2 + 13a-4z4 - 12a-4z6 + 6a-4z8 - 5a-3z + 15a-3z3 - 16a-3z5 + 6a-3z7 + a-3z9 - a-2 + 4a-2z2 - 6a-2z4 - a-2z6 + 3a-2z8 - 6a-1z5 + 4a-1z7 - 1 + 3z2 - 6z4 + 3z6 + az - 2az3 + az5

V2 and V3, the type 2 and 3 Vassiliev invariants:

{5, 8}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=2 is the signature of 1051. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                     1 j = 15                   1   j = 13                 4 1   j = 11               4 1     j = 9             6 4       j = 7           6 4         j = 5         4 6           j = 3       5 6             j = 1     2 5               j = -1   1 4                 j = -3   2                   j = -5 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-7 - 3q-6 + q-5 + 9q-4 - 16q-3 - q-2 + 34q-1 - 37 - 16q + 73q2 - 52q3 - 43q4 + 107q5 - 52q6 - 67q7 + 117q8 - 39q9 - 73q10 + 95q11 - 17q12 - 59q13 + 55q14 - q15 - 33q16 + 21q17 + 3q18 - 11q19 + 5q20 + q21 - 2q22 + q23
3	- q-15 + 3q-14 - q-13 - 4q-12 - 2q-11 + 13q-10 + 4q-9 - 26q-8 - 12q-7 + 45q-6 + 30q-5 - 67q-4 - 67q-3 + 94q-2 + 117q-1 - 109 - 187q + 107q2 + 274q3 - 93q4 - 354q5 + 49q6 + 440q7 - 5q8 - 490q9 - 67q10 + 545q11 + 111q12 - 544q13 - 181q14 + 550q15 + 209q16 - 496q17 - 258q18 + 450q19 + 268q20 - 365q21 - 278q22 + 280q23 + 261q24 - 191q25 - 232q26 + 114q27 + 188q28 - 57q29 - 134q30 + 13q31 + 94q32 - 2q33 - 48q34 - 10q35 + 29q36 + 3q37 - 10q38 - 4q39 + 7q40 - q41 - q42 - q43 + 2q44 - q45
4	q-26 - 3q-25 + q-24 + 4q-23 - 3q-22 + 5q-21 - 16q-20 + 4q-19 + 22q-18 - 9q-17 + 15q-16 - 64q-15 + 2q-14 + 80q-13 + 5q-12 + 46q-11 - 195q-10 - 58q-9 + 179q-8 + 113q-7 + 194q-6 - 433q-5 - 308q-4 + 201q-3 + 360q-2 + 648q-1 - 626 - 817q - 113q2 + 582q3 + 1498q4 - 478q5 - 1397q6 - 871q7 + 490q8 + 2494q9 + 115q10 - 1706q11 - 1824q12 + q13 + 3246q14 + 904q15 - 1620q16 - 2585q17 - 663q18 + 3546q19 + 1567q20 - 1252q21 - 2958q22 - 1264q23 + 3402q24 + 1967q25 - 726q26 - 2921q27 - 1715q28 + 2841q29 + 2090q30 - 85q31 - 2487q32 - 1974q33 + 1935q34 + 1882q35 + 544q36 - 1695q37 - 1914q38 + 906q39 + 1327q40 + 905q41 - 779q42 - 1464q43 + 137q44 + 621q45 + 825q46 - 107q47 - 810q48 - 153q49 + 102q50 + 472q51 + 131q52 - 297q53 - 107q54 - 77q55 + 165q56 + 97q57 - 71q58 - 16q59 - 58q60 + 36q61 + 30q62 - 18q63 + 10q64 - 18q65 + 6q66 + 6q67 - 7q68 + 5q69 - 3q70 + q71 + q72 - 2q73 + q74
5	- q-40 + 3q-39 - q-38 - 4q-37 + 3q-36 - 2q-34 + 8q-33 - 17q-31 + 2q-30 + 11q-29 + 5q-28 + 16q-27 - 8q-26 - 47q-25 - 28q-24 + 38q-23 + 75q-22 + 63q-21 - 25q-20 - 164q-19 - 166q-18 + 16q-17 + 270q-16 + 351q-15 + 94q-14 - 384q-13 - 664q-12 - 373q-11 + 419q-10 + 1117q-9 + 913q-8 - 280q-7 - 1585q-6 - 1786q-5 - 270q-4 + 1991q-3 + 2979q-2 + 1315q-1 - 2072 - 4327q - 2983q2 + 1575q3 + 5671q4 + 5202q5 - 406q6 - 6686q7 - 7706q8 - 1574q9 + 7118q10 + 10368q11 + 4095q12 - 6908q13 - 12643q14 - 7036q15 + 5928q16 + 14594q17 + 9954q18 - 4535q19 - 15723q20 - 12725q21 + 2703q22 + 16472q23 + 14948q24 - 911q25 - 16419q26 - 16783q27 - 967q28 + 16248q29 + 17958q30 + 2515q31 - 15406q32 - 18820q33 - 4122q34 + 14640q35 + 19124q36 + 5408q37 - 13224q38 - 19175q39 - 6840q40 + 11780q41 + 18729q42 + 8016q43 - 9708q44 - 17880q45 - 9233q46 + 7479q47 + 16448q48 + 10089q49 - 4862q50 - 14488q51 - 10599q52 + 2272q53 + 11995q54 + 10485q55 + 196q56 - 9193q57 - 9753q58 - 2136q59 + 6285q60 + 8403q61 + 3408q62 - 3589q63 - 6642q64 - 3942q65 + 1447q66 + 4722q67 + 3721q68 + 94q69 - 2911q70 - 3137q71 - 863q72 + 1521q73 + 2191q74 + 1138q75 - 510q76 - 1426q77 - 992q78 + 35q79 + 704q80 + 727q81 + 207q82 - 338q83 - 425q84 - 189q85 + 73q86 + 229q87 + 155q88 - 26q89 - 85q90 - 70q91 - 34q92 + 41q93 + 46q94 - q95 - 6q96 - 3q97 - 20q98 + 4q99 + 12q100 - 4q101 - q102 + 5q103 - 5q104 - q105 + 3q106 - q107 - q108 + 2q109 - q110

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 51]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[9, 17, 10, 16], X[5, 15, 6, 14], > X[15, 7, 16, 6], X[13, 1, 14, 20], X[19, 11, 20, 10], X[11, 19, 12, 18], > X[17, 13, 18, 12], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 51]]
Out[3]=	GaussCode[-1, 10, -2, 1, -4, 5, -10, 2, -3, 7, -8, 9, -6, 4, -5, 3, -9, 8, -7, > 6]
In[4]:=	DTCode[Knot[10, 51]]
Out[4]=	DTCode[4, 8, 14, 2, 16, 18, 20, 6, 12, 10]
In[5]:=	br = BR[Knot[10, 51]]
Out[5]=	BR[4, {1, 1, 2, -1, 2, 2, -3, 2, 2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 51]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 51]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 51]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, {2, 3}, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 51]][t]
Out[10]=	2 7 15 2 3 -19 + -- - -- + -- + 15 t - 7 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 51]][z]
Out[11]=	2 4 6 1 + 5 z + 5 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 51]}
In[13]:=	{KnotDet[Knot[10, 51]], KnotSignature[Knot[10, 51]]}
Out[13]=	{67, 2}
In[14]:=	Jones[Knot[10, 51]][q]
Out[14]=	-2 3 2 3 4 5 6 7 8 -6 - q + - + 9 q - 10 q + 12 q - 10 q + 8 q - 5 q + 2 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 51]}
In[16]:=	A2Invariant[Knot[10, 51]][q]
Out[16]=	-6 -4 -2 2 4 6 8 10 12 14 16 -1 - q + q - q + 2 q - 2 q + 3 q + q + 2 q + 3 q - q + 2 q - 18 20 24 > 2 q - 2 q - q
In[17]:=	HOMFLYPT[Knot[10, 51]][a, z]
Out[17]=	2 2 2 4 4 4 6 6 3 4 -2 2 3 z 7 z 3 z 4 z 4 z 3 z z z -1 - -- + -- + a - 2 z - ---- + ---- + ---- - z - -- + ---- + ---- + -- + -- 6 4 6 4 2 6 4 2 4 2 a a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 51]][a, z]
Out[18]=	2 2 2 3 4 -2 2 z 3 z 9 z 5 z 2 z 8 z 8 z -1 + -- + -- - a + --- - --- - --- - --- + a z + 3 z + -- - ---- - ---- + 6 4 9 7 5 3 8 6 4 a a a a a a a a a 2 3 3 3 3 4 4 4 4 z 3 z 5 z 21 z 15 z 3 4 4 z 9 z 13 z > ---- - ---- + ---- + ----- + ----- - 2 a z - 6 z - ---- + ---- + ----- - 2 9 7 5 3 8 6 4 a a a a a a a a 4 5 5 5 5 5 6 6 6 z z 6 z 16 z 16 z 6 z 5 6 2 z 6 z > ---- + -- - ---- - ----- - ----- - ---- + a z + 3 z + ---- - ---- - 2 9 7 5 3 a 8 6 a a a a a a a 6 6 7 7 7 7 8 8 8 9 9 12 z z 3 z 5 z 6 z 4 z 3 z 6 z 3 z z z > ----- - -- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + -- + -- 4 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 51]], Vassiliev[3][Knot[10, 51]]}
Out[19]=	{5, 8}
In[20]:=	Kh[Knot[10, 51]][q, t]
Out[20]=	3 1 2 1 4 2 q 3 5 5 2 5 q + 5 q + ----- + ----- + ---- + --- + --- + 6 q t + 4 q t + 6 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 6 q t + 4 q t + 6 q t + 4 q t + 4 q t + q t + 4 q t + 13 6 15 6 17 7 > q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 51], 2][q]
Out[21]=	-7 3 -5 9 16 -2 34 2 3 4 -37 + q - -- + q + -- - -- - q + -- - 16 q + 73 q - 52 q - 43 q + 6 4 3 q q q q 5 6 7 8 9 10 11 12 > 107 q - 52 q - 67 q + 117 q - 39 q - 73 q + 95 q - 17 q - 13 14 15 16 17 18 19 20 21 > 59 q + 55 q - q - 33 q + 21 q + 3 q - 11 q + 5 q + q - 22 23 > 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1051

1050 1052