© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table:

1074 1076

KnotPlot

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KnotPlot

PD Presentation:

X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X13,16,14,17 X7,15,8,14 X15,7,16,6 X17,20,18,1 X9,19,10,18 X19,9,20,8

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 14 18 2 16 6 20 8

Minimum Braid Representative: Length is 12, width is 5 Braid index is 5

A Morse Link Presentation:

3D Invariants:

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

Alexander Polynomial:

- t-3 + 7t-2 - 19t-1 + 27 - 19t + 7t2 - t3

Conway Polynomial:

1 + z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{1042, ...}

Determinant and Signature:

{81, 0}

Jones Polynomial:

q-4 - 4q-3 + 7q-2 - 10q-1 + 14 - 13q + 12q2 - 10q3 + 6q4 - 3q5 + q6

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

{...}

A2 (sl(3)) Invariant:

q-12 - 2q-10 + q-8 - 3q-4 + 4q-2 + 3q2 + q4 - q6 + 2q8 - 3q10 - 2q16 + q18 + q20

HOMFLY-PT Polynomial:

a-6 - 3a-4 - 3a-4z2 + 3a-2 + 6a-2z2 + 3a-2z4 - 4z2 - 3z4 - z6 + a2z2 + a2z4

Kauffman Polynomial:

- a-6 + 3a-6z2 - 3a-6z4 + a-6z6 - 3a-5z + 9a-5z3 - 9a-5z5 + 3a-5z7 - 3a-4 + 12a-4z2 - 9a-4z4 - 3a-4z6 + 3a-4z8 - 7a-3z + 24a-3z3 - 29a-3z5 + 9a-3z7 + a-3z9 - 3a-2 + 20a-2z2 - 21a-2z4 - 4a-2z6 + 7a-2z8 - 5a-1z + 17a-1z3 - 29a-1z5 + 13a-1z7 + a-1z9 + 15z2 - 24z4 + 7z6 + 4z8 - az - az3 - 5az5 + 7az7 + 4a2z2 - 8a2z4 + 7a2z6 - 3a3z3 + 4a3z5 + a4z4

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

{0, -1}

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=0 is the signature of 1075. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                     1 j = 11                   2   j = 9                 4 1   j = 7               6 2     j = 5             6 4       j = 3           7 6         j = 1         7 6           j = -1       4 8             j = -3     3 6               j = -5   1 4                 j = -7   3                   j = -9 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-12 - 4q-11 + 3q-10 + 10q-9 - 24q-8 + 12q-7 + 34q-6 - 68q-5 + 24q-4 + 77q-3 - 119q-2 + 24q-1 + 124 - 146q + 6q2 + 144q3 - 134q4 - 18q5 + 131q6 - 91q7 - 33q8 + 91q9 - 41q10 - 30q11 + 43q12 - 9q13 - 15q14 + 11q15 - 3q17 + q18
3	q-24 - 4q-23 + 3q-22 + 6q-21 - 4q-20 - 16q-19 + 9q-18 + 37q-17 - 28q-16 - 61q-15 + 43q-14 + 118q-13 - 77q-12 - 189q-11 + 106q-10 + 293q-9 - 132q-8 - 423q-7 + 146q-6 + 556q-5 - 123q-4 - 705q-3 + 99q-2 + 799q-1 - 9 - 899q - 48q2 + 912q3 + 155q4 - 917q5 - 229q6 + 854q7 + 314q8 - 773q9 - 377q10 + 662q11 + 414q12 - 525q13 - 434q14 + 391q15 + 414q16 - 249q17 - 377q18 + 140q19 + 303q20 - 46q21 - 227q22 - 10q23 + 152q24 + 32q25 - 86q26 - 36q27 + 42q28 + 26q29 - 16q30 - 15q31 + 6q32 + 5q33 - 3q35 + q36
4	q-40 - 4q-39 + 3q-38 + 6q-37 - 8q-36 + 4q-35 - 19q-34 + 22q-33 + 27q-32 - 50q-31 + 11q-30 - 52q-29 + 101q-28 + 92q-27 - 194q-26 - 36q-25 - 106q-24 + 359q-23 + 293q-22 - 526q-21 - 303q-20 - 246q-19 + 951q-18 + 848q-17 - 1026q-16 - 1020q-15 - 696q-14 + 1880q-13 + 1997q-12 - 1401q-11 - 2188q-10 - 1747q-9 + 2786q-8 + 3676q-7 - 1230q-6 - 3346q-5 - 3334q-4 + 3115q-3 + 5293q-2 - 403q-1 - 3863 - 4918q + 2650q2 + 6193q3 + 725q4 - 3539q5 - 5919q6 + 1648q7 + 6144q8 + 1749q9 - 2582q10 - 6159q11 + 441q12 + 5316q13 + 2499q14 - 1286q15 - 5712q16 - 743q17 + 3932q18 + 2859q19 + 100q20 - 4639q21 - 1640q22 + 2234q23 + 2642q24 + 1231q25 - 3065q26 - 1923q27 + 632q28 + 1830q29 + 1707q30 - 1430q31 - 1495q32 - 356q33 + 789q34 + 1415q35 - 297q36 - 721q37 - 559q38 + 63q39 + 755q40 + 112q41 - 150q42 - 316q43 - 147q44 + 244q45 + 91q46 + 35q47 - 86q48 - 87q49 + 45q50 + 18q51 + 26q52 - 9q53 - 22q54 + 6q55 + 5q57 - 3q59 + q60

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, 75]]
Out[2]=	PD[X[1, 4, 2, 5], X[5, 12, 6, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], > X[13, 16, 14, 17], X[7, 15, 8, 14], X[15, 7, 16, 6], X[17, 20, 18, 1], > X[9, 19, 10, 18], X[19, 9, 20, 8]]
In[3]:=	GaussCode[Knot[10, 75]]
Out[3]=	GaussCode[-1, 4, -3, 1, -2, 7, -6, 10, -9, 3, -4, 2, -5, 6, -7, 5, -8, 9, -10, > 8]
In[4]:=	DTCode[Knot[10, 75]]
Out[4]=	DTCode[4, 10, 12, 14, 18, 2, 16, 6, 20, 8]
In[5]:=	br = BR[Knot[10, 75]]
Out[5]=	BR[5, {1, -2, 1, -2, 3, -2, -2, 4, -3, 2, 4, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 75]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 75]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 75]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 2}
In[10]:=	alex = Alexander[Knot[10, 75]][t]
Out[10]=	-3 7 19 2 3 27 - t + -- - -- - 19 t + 7 t - t 2 t t
In[11]:=	Conway[Knot[10, 75]][z]
Out[11]=	4 6 1 + z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 42], Knot[10, 75]}
In[13]:=	{KnotDet[Knot[10, 75]], KnotSignature[Knot[10, 75]]}
Out[13]=	{81, 0}
In[14]:=	Jones[Knot[10, 75]][q]
Out[14]=	-4 4 7 10 2 3 4 5 6 14 + q - -- + -- - -- - 13 q + 12 q - 10 q + 6 q - 3 q + q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 75]}
In[16]:=	A2Invariant[Knot[10, 75]][q]
Out[16]=	-12 2 -8 3 4 2 4 6 8 10 16 18 20 q - --- + q - -- + -- + 3 q + q - q + 2 q - 3 q - 2 q + q + q 10 4 2 q q q
In[17]:=	HOMFLYPT[Knot[10, 75]][a, z]
Out[17]=	2 2 4 -6 3 3 2 3 z 6 z 2 2 4 3 z 2 4 6 a - -- + -- - 4 z - ---- + ---- + a z - 3 z + ---- + a z - z 4 2 4 2 2 a a a a a
In[18]:=	Kauffman[Knot[10, 75]][a, z]
Out[18]=	2 2 2 -6 3 3 3 z 7 z 5 z 2 3 z 12 z 20 z -a - -- - -- - --- - --- - --- - a z + 15 z + ---- + ----- + ----- + 4 2 5 3 a 6 4 2 a a a a a a a 3 3 3 4 4 2 2 9 z 24 z 17 z 3 3 3 4 3 z 9 z > 4 a z + ---- + ----- + ----- - a z - 3 a z - 24 z - ---- - ---- - 5 3 a 6 4 a a a a 4 5 5 5 21 z 2 4 4 4 9 z 29 z 29 z 5 3 5 6 > ----- - 8 a z + a z - ---- - ----- - ----- - 5 a z + 4 a z + 7 z + 2 5 3 a a a a 6 6 6 7 7 7 8 z 3 z 4 z 2 6 3 z 9 z 13 z 7 8 3 z > -- - ---- - ---- + 7 a z + ---- + ---- + ----- + 7 a z + 4 z + ---- + 6 4 2 5 3 a 4 a a a a a a 8 9 9 7 z z z > ---- + -- + -- 2 3 a a a
In[19]:=	{Vassiliev[2][Knot[10, 75]], Vassiliev[3][Knot[10, 75]]}
Out[19]=	{0, -1}
In[20]:=	Kh[Knot[10, 75]][q, t]
Out[20]=	8 1 3 1 4 3 6 4 3 - + 7 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 6 q t + 7 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 6 q t + 6 q t + 4 q t + 6 q t + 2 q t + 4 q t + q t + 11 5 13 6 > 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 75], 2][q]
Out[21]=	-12 4 3 10 24 12 34 68 24 77 119 24 124 + q - --- + --- + -- - -- + -- + -- - -- + -- + -- - --- + -- - 146 q + 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 2 3 4 5 6 7 8 9 10 > 6 q + 144 q - 134 q - 18 q + 131 q - 91 q - 33 q + 91 q - 41 q - 11 12 13 14 15 17 18 > 30 q + 43 q - 9 q - 15 q + 11 q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1075

1074 1076