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

821 92

KnotPlot

This page is passe. Go here instead!    The Alternating Knot 91    Also known as "The Nonoil Knot", following the trefoil knot, the cinquefoil knot and the septoil knot. Visit 91's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 91's page at Knotilus! Acknowledgement

KnotPlot

PD Presentation:

X1,10,2,11 X3,12,4,13 X5,14,6,15 X7,16,8,17 X9,18,10,1 X11,2,12,3 X13,4,14,5 X15,6,16,7 X17,8,18,9

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

10 12 14 16 18 2 4 6 8

Minimum Braid Representative: Length is 9, width is 2 Braid index is 2

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index Reversible 4 4 2 / 4 1

Alexander Polynomial:

t-4 - t-3 + t-2 - t-1 + 1 - t + t2 - t3 + t4

Conway Polynomial:

1 + 10z2 + 15z4 + 7z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{9, -8}

Jones Polynomial:

- q-13 + q-12 - q-11 + q-10 - q-9 + q-8 - q-7 + q-6 + q-4

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

{...}

A2 (sl(3)) Invariant:

- q-38 - q-36 - q-34 + q-22 + q-20 + 2q-18 + q-16 + q-14

HOMFLY-PT Polynomial:

5a8 + 20a8z2 + 21a8z4 + 8a8z6 + a8z8 - 4a10 - 10a10z2 - 6a10z4 - a10z6

Kauffman Polynomial:

5a8 - 20a8z2 + 21a8z4 - 8a8z6 + a8z8 - 4a9z + 10a9z3 - 6a9z5 + a9z7 + 4a10 - 14a10z2 + 16a10z4 - 7a10z6 + a10z8 - a11z + 6a11z3 - 5a11z5 + a11z7 + 3a12z2 - 4a12z4 + a12z6 + a13z - 3a13z3 + a13z5 - 2a14z2 + a14z4 - a15z + a15z3 + a16z2 + a17z

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

{10, -30}

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=-8 is the signature of 91. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -7                   1 j = -9                   1 j = -11               1     j = -13                     j = -15           1 1       j = -17                     j = -19       1 1           j = -21                     j = -23   1 1               j = -25                     j = -27 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-35 - q-34 + q-32 - q-31 + q-29 - q-28 - q-27 + q-26 - q-25 + q-23 - q-22 + q-20 - q-19 + q-17 - q-16 + q-14 - q-13 + q-11 + q-8
3	- q-66 + q-65 - q-62 + q-61 + q-57 - q-55 + q-53 - q-51 + q-49 - q-47 + q-45 - q-43 - q-39 + q-36 - q-35 + q-32 - q-31 + q-28 - q-27 + q-24 - q-23 + q-20 - q-19 + q-16 + q-12
4	q-106 - q-105 + q-101 - q-100 - q-98 + q-96 - q-93 + q-91 - q-88 + q-86 - q-83 + 2q-81 - q-78 + q-76 - q-73 + q-71 - q-68 + q-66 - q-63 + q-61 - q-58 + q-56 - q-55 - q-53 + q-51 - q-50 + q-46 - q-45 + q-41 - q-40 + q-36 - q-35 + q-31 - q-30 + q-26 - q-25 + q-21 + q-16
5	- q-155 + q-154 - q-149 + q-148 + q-147 - q-144 - q-143 + q-142 + q-141 - q-138 - q-137 + q-136 + q-135 - q-132 - q-131 + q-129 - q-126 + q-123 - q-120 + q-117 - q-114 + q-111 - q-108 + q-105 + q-104 - q-102 + q-99 + q-98 - q-97 - q-96 + q-93 + q-92 - q-91 - q-90 + q-87 + q-86 - q-85 - q-84 + q-81 + q-80 - q-79 - q-78 + q-75 + q-74 - q-73 - q-72 + q-68 - q-67 - q-66 + q-62 - q-61 + q-56 - q-55 + q-50 - q-49 + q-44 - q-43 + q-38 - q-37 + q-32 - q-31 + q-26 + q-20
6	q-213 - q-212 + q-206 - 2q-205 + q-202 + q-199 - 2q-198 + q-195 + q-192 - 2q-191 + 2q-188 + q-185 - 2q-184 - q-183 + 2q-181 + q-178 - 2q-177 - q-176 + 2q-174 + q-171 - 2q-170 - q-169 + 2q-167 + q-164 - 2q-163 - 2q-162 + 2q-160 + q-157 - 2q-156 - q-155 + 2q-153 + q-150 - 2q-149 - q-148 + 2q-146 + q-143 - 2q-142 - q-141 + 2q-139 + q-136 - 2q-135 - q-134 + 2q-132 + q-129 - 2q-128 + 2q-125 + q-122 - 2q-121 + q-118 + q-115 - 2q-114 + q-111 + q-108 - 2q-107 + q-104 + q-101 - 2q-100 + q-97 + q-94 - 2q-93 + q-90 + q-87 - 2q-86 + q-80 - 2q-79 + q-73 - q-72 + q-66 - q-65 + q-59 - q-58 + q-52 - q-51 + q-45 - q-44 + q-38 - q-37 + q-31 + q-24
7	- q-280 + q-279 + q-271 - q-269 + q-263 - q-261 - q-253 + q-250 - q-245 + q-242 - q-237 + q-234 + q-226 - q-222 + q-218 - q-214 + q-210 - q-206 + q-202 - q-198 - q-190 + q-185 - q-182 + q-177 - q-174 + q-169 - q-166 + q-161 - q-158 + q-153 + q-145 - q-139 + q-137 - q-131 + q-129 - q-123 + q-121 - q-115 + q-113 - q-107 + q-105 - q-99 - q-91 + q-84 - q-83 + q-76 - q-75 + q-68 - q-67 + q-60 - q-59 + q-52 - q-51 + q-44 - q-43 + q-36 + q-28

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[9, 1]]
Out[2]=	PD[X[1, 10, 2, 11], X[3, 12, 4, 13], X[5, 14, 6, 15], X[7, 16, 8, 17], > X[9, 18, 10, 1], X[11, 2, 12, 3], X[13, 4, 14, 5], X[15, 6, 16, 7], > X[17, 8, 18, 9]]
In[3]:=	GaussCode[Knot[9, 1]]
Out[3]=	GaussCode[-1, 6, -2, 7, -3, 8, -4, 9, -5, 1, -6, 2, -7, 3, -8, 4, -9, 5]
In[4]:=	DTCode[Knot[9, 1]]
Out[4]=	DTCode[10, 12, 14, 16, 18, 2, 4, 6, 8]
In[5]:=	br = BR[Knot[9, 1]]
Out[5]=	BR[2, {-1, -1, -1, -1, -1, -1, -1, -1, -1}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{2, 9}
In[7]:=	BraidIndex[Knot[9, 1]]
Out[7]=	2
In[8]:=	Show[DrawMorseLink[Knot[9, 1]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 1]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 4, 4, 2, 4, 1}
In[10]:=	alex = Alexander[Knot[9, 1]][t]
Out[10]=	-4 -3 -2 1 2 3 4 1 + t - t + t - - - t + t - t + t t
In[11]:=	Conway[Knot[9, 1]][z]
Out[11]=	2 4 6 8 1 + 10 z + 15 z + 7 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 1]}
In[13]:=	{KnotDet[Knot[9, 1]], KnotSignature[Knot[9, 1]]}
Out[13]=	{9, -8}
In[14]:=	Jones[Knot[9, 1]][q]
Out[14]=	-13 -12 -11 -10 -9 -8 -7 -6 -4 -q + q - q + q - q + q - q + q + q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 1]}
In[16]:=	A2Invariant[Knot[9, 1]][q]
Out[16]=	-38 -36 -34 -22 -20 2 -16 -14 -q - q - q + q + q + --- + q + q 18 q
In[17]:=	HOMFLYPT[Knot[9, 1]][a, z]
Out[17]=	8 10 8 2 10 2 8 4 10 4 8 6 10 6 5 a - 4 a + 20 a z - 10 a z + 21 a z - 6 a z + 8 a z - a z + 8 8 > a z
In[18]:=	Kauffman[Knot[9, 1]][a, z]
Out[18]=	8 10 9 11 13 15 17 8 2 10 2 5 a + 4 a - 4 a z - a z + a z - a z + a z - 20 a z - 14 a z + 12 2 14 2 16 2 9 3 11 3 13 3 15 3 > 3 a z - 2 a z + a z + 10 a z + 6 a z - 3 a z + a z + 8 4 10 4 12 4 14 4 9 5 11 5 13 5 > 21 a z + 16 a z - 4 a z + a z - 6 a z - 5 a z + a z - 8 6 10 6 12 6 9 7 11 7 8 8 10 8 > 8 a z - 7 a z + a z + a z + a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[9, 1]], Vassiliev[3][Knot[9, 1]]}
Out[19]=	{10, -30}
In[20]:=	Kh[Knot[9, 1]][q, t]
Out[20]=	-9 -7 1 1 1 1 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 27 9 23 8 23 7 19 6 19 5 15 4 15 3 q t q t q t q t q t q t q t 1 > ------ 11 2 q t
In[21]:=	ColouredJones[Knot[9, 1], 2][q]
Out[21]=	-35 -34 -32 -31 -29 -28 -27 -26 -25 -23 -22 q - q + q - q + q - q - q + q - q + q - q + -20 -19 -17 -16 -14 -13 -11 -8 > q - q + q - q + q - q + q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 91

821 92