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

10116 10118

KnotPlot

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KnotPlot

PD Presentation:

X1627 X5,16,6,17 X13,1,14,20 X7,15,8,14 X19,9,20,8 X3,11,4,10 X11,5,12,4 X9,19,10,18 X17,13,18,12 X15,2,16,3

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 16 14 18 4 20 2 12 8

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 Chiral 2 3 3 / NotAvailable 1

Alexander Polynomial:

2t-3 - 10t-2 + 24t-1 - 31 + 24t - 10t2 + 2t3

Conway Polynomial:

1 + 2z2 + 2z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a23, K11a111, ...}

Determinant and Signature:

{103, 2}

Jones Polynomial:

- q-2 + 4q-1 - 8 + 13q - 16q2 + 18q3 - 16q4 + 13q5 - 9q6 + 4q7 - q8

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

{...}

A2 (sl(3)) Invariant:

- q-6 + 2q-4 - q-2 - 1 + 4q2 - 3q4 + 3q6 + 3q12 - 3q14 + 3q16 - 2q18 - 2q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6 - a-6z2 - a-6z4 + a-4 + 2a-4z2 + 2a-4z4 + a-4z6 + a-2 + 2a-2z2 + 2a-2z4 + a-2z6 - z2 - z4

Kauffman Polynomial:

- a-9z3 + a-9z5 + a-8z2 - 5a-8z4 + 4a-8z6 - 3a-7z + 8a-7z3 - 14a-7z5 + 8a-7z7 + a-6 - 3a-6z2 + 6a-6z4 - 12a-6z6 + 8a-6z8 - 5a-5z + 21a-5z3 - 29a-5z5 + 10a-5z7 + 3a-5z9 + a-4 - 4a-4z2 + 17a-4z4 - 28a-4z6 + 15a-4z8 - 3a-3z + 18a-3z3 - 26a-3z5 + 9a-3z7 + 3a-3z9 - a-2 + 2a-2z2 - 8a-2z6 + 7a-2z8 - a-1z + 5a-1z3 - 11a-1z5 + 7a-1z7 + 2z2 - 6z4 + 4z6 - az3 + az5

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

{2, 3}

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 10117. 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                   3   j = 13                 6 1   j = 11               7 3     j = 9             9 6       j = 7           9 7         j = 5         7 9           j = 3       6 9             j = 1     3 8               j = -1   1 5                 j = -3   3                   j = -5 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-7 - 4q-6 + 3q-5 + 11q-4 - 28q-3 + 10q-2 + 54q-1 - 85 - 3q + 144q2 - 144q3 - 54q4 + 238q5 - 162q6 - 116q7 + 279q8 - 132q9 - 151q10 + 245q11 - 70q12 - 142q13 + 156q14 - 11q15 - 91q16 + 62q17 + 11q18 - 32q19 + 11q20 + 4q21 - 4q22 + q23
3	- q-15 + 4q-14 - 3q-13 - 6q-12 + 4q-11 + 19q-10 - 11q-9 - 51q-8 + 27q-7 + 109q-6 - 26q-5 - 225q-4 - 5q-3 + 389q-2 + 113q-1 - 590 - 311q + 768q2 + 626q3 - 910q4 - 984q5 + 936q6 + 1386q7 - 887q8 - 1733q9 + 739q10 + 2025q11 - 556q12 - 2195q13 + 312q14 + 2293q15 - 87q16 - 2255q17 - 170q18 + 2138q19 + 394q20 - 1904q21 - 600q22 + 1592q23 + 742q24 - 1222q25 - 793q26 + 832q27 + 752q28 - 485q29 - 621q30 + 213q31 + 456q32 - 59q33 - 275q34 - 21q35 + 145q36 + 34q37 - 65q38 - 19q39 + 22q40 + 8q41 - 6q42 - 4q43 + 4q44 - q45
4	q-26 - 4q-25 + 3q-24 + 6q-23 - 9q-22 + 5q-21 - 18q-20 + 24q-19 + 34q-18 - 65q-17 - 11q-16 - 65q-15 + 156q-14 + 223q-13 - 228q-12 - 272q-11 - 418q-10 + 521q-9 + 1128q-8 - 61q-7 - 1000q-6 - 2079q-5 + 355q-4 + 3200q-3 + 1945q-2 - 1045q-1 - 5685 - 2399q + 4878q2 + 6525q3 + 2126q4 - 9269q5 - 8428q6 + 3259q7 + 11381q8 + 8999q9 - 9723q10 - 15051q11 - 2078q12 + 13394q13 + 16639q14 - 6616q15 - 19075q16 - 8455q17 + 12073q18 + 21961q19 - 2028q20 - 19780q21 - 13455q22 + 8872q23 + 24171q24 + 2447q25 - 17947q26 - 16540q27 + 4657q28 + 23528q29 + 6586q30 - 13824q31 - 17645q32 - 491q33 + 19812q34 + 9848q35 - 7514q36 - 15867q37 - 5450q38 + 13037q39 + 10486q40 - 763q41 - 10783q42 - 7709q43 + 5377q44 + 7568q45 + 3246q46 - 4545q47 - 6096q48 + 346q49 + 3182q50 + 3266q51 - 534q52 - 2805q53 - 901q54 + 419q55 + 1495q56 + 456q57 - 686q58 - 415q59 - 203q60 + 349q61 + 214q62 - 87q63 - 53q64 - 88q65 + 46q66 + 38q67 - 14q68 + 2q69 - 13q70 + 6q71 + 4q72 - 4q73 + q74

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, 117]]
Out[2]=	PD[X[1, 6, 2, 7], X[5, 16, 6, 17], X[13, 1, 14, 20], X[7, 15, 8, 14], > X[19, 9, 20, 8], X[3, 11, 4, 10], X[11, 5, 12, 4], X[9, 19, 10, 18], > X[17, 13, 18, 12], X[15, 2, 16, 3]]
In[3]:=	GaussCode[Knot[10, 117]]
Out[3]=	GaussCode[-1, 10, -6, 7, -2, 1, -4, 5, -8, 6, -7, 9, -3, 4, -10, 2, -9, 8, -5, > 3]
In[4]:=	DTCode[Knot[10, 117]]
Out[4]=	DTCode[6, 10, 16, 14, 18, 4, 20, 2, 12, 8]
In[5]:=	br = BR[Knot[10, 117]]
Out[5]=	BR[4, {1, 1, 2, 2, -3, 2, -1, 2, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 117]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 117]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 117]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 117]][t]
Out[10]=	2 10 24 2 3 -31 + -- - -- + -- + 24 t - 10 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 117]][z]
Out[11]=	2 4 6 1 + 2 z + 2 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 117], Knot[11, Alternating, 23], Knot[11, Alternating, 111]}
In[13]:=	{KnotDet[Knot[10, 117]], KnotSignature[Knot[10, 117]]}
Out[13]=	{103, 2}
In[14]:=	Jones[Knot[10, 117]][q]
Out[14]=	-2 4 2 3 4 5 6 7 8 -8 - q + - + 13 q - 16 q + 18 q - 16 q + 13 q - 9 q + 4 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 117]}
In[16]:=	A2Invariant[Knot[10, 117]][q]
Out[16]=	-6 2 -2 2 4 6 12 14 16 18 -1 - q + -- - q + 4 q - 3 q + 3 q + 3 q - 3 q + 3 q - 2 q - 4 q 20 22 24 > 2 q + 2 q - q
In[17]:=	HOMFLYPT[Knot[10, 117]][a, z]
Out[17]=	2 2 2 4 4 4 6 6 -6 -4 -2 2 z 2 z 2 z 4 z 2 z 2 z z z -a + a + a - z - -- + ---- + ---- - z - -- + ---- + ---- + -- + -- 6 4 2 6 4 2 4 2 a a a a a a a a
In[18]:=	Kauffman[Knot[10, 117]][a, z]
Out[18]=	2 2 2 2 3 -6 -4 -2 3 z 5 z 3 z z 2 z 3 z 4 z 2 z z a + a - a - --- - --- - --- - - + 2 z + -- - ---- - ---- + ---- - -- + 7 5 3 a 8 6 4 2 9 a a a a a a a a 3 3 3 3 4 4 4 5 8 z 21 z 18 z 5 z 3 4 5 z 6 z 17 z z > ---- + ----- + ----- + ---- - a z - 6 z - ---- + ---- + ----- + -- - 7 5 3 a 8 6 4 9 a a a a a a a 5 5 5 5 6 6 6 6 14 z 29 z 26 z 11 z 5 6 4 z 12 z 28 z 8 z > ----- - ----- - ----- - ----- + a z + 4 z + ---- - ----- - ----- - ---- + 7 5 3 a 8 6 4 2 a a a a a a a 7 7 7 7 8 8 8 9 9 8 z 10 z 9 z 7 z 8 z 15 z 7 z 3 z 3 z > ---- + ----- + ---- + ---- + ---- + ----- + ---- + ---- + ---- 7 5 3 a 6 4 2 5 3 a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 117]], Vassiliev[3][Knot[10, 117]]}
Out[19]=	{2, 3}
In[20]:=	Kh[Knot[10, 117]][q, t]
Out[20]=	3 1 3 1 5 3 q 3 5 5 2 8 q + 6 q + ----- + ----- + ---- + --- + --- + 9 q t + 7 q t + 9 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 > 9 q t + 7 q t + 9 q t + 6 q t + 7 q t + 3 q t + 6 q t + 13 6 15 6 17 7 > q t + 3 q t + q t
In[21]:=	ColouredJones[Knot[10, 117], 2][q]
Out[21]=	-7 4 3 11 28 10 54 2 3 4 -85 + q - -- + -- + -- - -- + -- + -- - 3 q + 144 q - 144 q - 54 q + 6 5 4 3 2 q q q q q q 5 6 7 8 9 10 11 12 > 238 q - 162 q - 116 q + 279 q - 132 q - 151 q + 245 q - 70 q - 13 14 15 16 17 18 19 20 > 142 q + 156 q - 11 q - 91 q + 62 q + 11 q - 32 q + 11 q + 21 22 23 > 4 q - 4 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10117

10116 10118